text:
\begin{document}
\maketitle

\begin{abstract}
This present analysis deals with the regularity of the solutions of bilinear control
systems of the type $x'=(A+u(t)B)x$
where the state $x$ belongs to some complex infinite dimensional
Hilbert space, the (possibly unbounded) linear operators $A$ and
$B$ are skew-adjoint and the control $u$ is a real valued function.
Such systems arise for instance in
quantum control with the bilinear Schr\"{o}dinger equation.

Under some hypotheses on the commutator of the operators $A$ and $B$, it is possible to
define the solution
of the system for controls in the set of Radon measures and to obtain precise a priori
energy estimates of the solutions. This leads to a natural extension of a celebrated
non-controllability result of Ball, Marsden and Slemrod.

Moreover,  upper bounds on the error made by
replacing the original systems by its finite dimensional Galerkin approximation are obtain
from  the energy estimates. This allows  to use
geometric (finite dimensional) techniques to obtain approximate controllability results for
the original infinite dimensional problem.

\end{abstract}

\newpage

\tableofcontents

\section{Introduction}
\subsection{Physical context}

The state of a quantum system evolving on a finite dimensional Riemannian
manifold $\Omega$, with associated measure $\mu$, is described by its \emph{wave
function},
modeled as
a point in the unit sphere of $L^2(\Omega, \mathbf{C})$. A physical
quantity or observable $\mathcal O$ (such as energy or position), is in turn
modeled by a Hermitian
operator $O:D(O)\subset L^2(\Omega, \mathbf{C}) \to L^2(\Omega, \mathbf{C})$
such that the expected value of quantity $\mathcal O$ for a system with wave
function $\psi$ in $D(O)$ is $\int_{\Omega}\bar{\psi}O\psi \mathrm{d}\mu$. For
instance, a system with wave
function $\psi$ will be in a subset $\omega$ of $\Omega$ with the probability
$\displaystyle{\int_{\omega} |\psi|^2 \mathrm{d}\mu}$.

In the absence of interaction with the environment and neglecting the
relativistic effects, the time evolution of the
wave function is given by the Schr\"odinger equation
$$
\rmi \frac{\partial \psi}{\partial t}=-\frac{1}{2}\Delta \psi +V(x) \psi(x,t),
$$
where $\Delta$ is the Laplace-Beltrami operator on $\Omega$
and
$V:\Omega\rightarrow \mathbf{R}$ is a real function (usually called potential)
accounting for the physical properties of the system.  When submitted to an
excitation by an external electric field ({\it e.g.} a laser), the Schr\"odinger
equation reads
\begin{equation}\label{eq:blse}
\rmi \frac{\partial \psi}{\partial t}=-\frac{1}{2}\Delta \psi +V(x) \psi(x,t)
+u(t)
W(x) \psi(x,t),
\end{equation}
where $W:\Omega\rightarrow \mathbf{R}$ is a real function accounting for the
physical properties of the laser and $u$ is a real function of the time
accounting for the intensity of the laser.

For skew-symmetry  reasons, the unit sphere $\mathbf{S}$ of $L^2(\Omega,\C)$ is invariant by
the dynamics of~(\ref{eq:blse}).
 A natural question, with many practical implications, is whether it is possible
or not to find a control $u$ that steers the quantum system  from a given
initial wave function in $\mathbf{S}$ to a given target wave function in $\mathbf{S}$.

\subsection{State of the art}

If $\Omega = (-1,1), V=0$, and $W: x \mapsto x$, with Dirichlet
boundary conditions, then the system~(\ref{eq:blse}) is exactly controllable (see
\cite{camillo}) in
$H^3_{(0)}(\Omega)\cap \mathbf{S}$. The results extends to  control potentials $W$
sufficiently irregular, i.e., such that the sequence $\left (\int_{-1}^{1}
W(x)\cos(kx) \cos(x) \mathrm{d}x \right )_{k\in \N}$ converges toward zero slowly enough
(see also~\cite{fratelli-nersesyan}).

Our point of view in the present analysis was somehow the opposite. Starting with smooth potentials
we tried to give upper bounds on the attainable set.
To the best of our knowledge, no description of the attainable set is currently
available for systems that do not satisfy the hypotheses of \cite{camillo}, for
instance when $\Omega$ is a compact manifold, $V=0$ and $W$ is smooth ($C^\infty$) function.
The regularity of $W$ is a no-go for the linear test (see
\cite{Coron-libro}) used in \cite{camillo}. Smooth potentials are the most relevant
from the physical point of view, see  \cite[Figure 7a, page 35]{Cohen} and physicists pay a
a particular attention to potentials producing smooth wave function, see
Cohen-Tannoudji et al., \cite[Section II-A-1, page 94]{Cohen}~:
\begin{quote}
 ``From a physical point of view, it is clear that the set
$L^2(\Omega,\mathbf{C})$ is too wide in scope:
given the meaning attributed to $|\psi(x, t)|^2$, the wave functions which are
actually
used possess certain properties of regularity. We can only retain the functions
$\psi(x, t)$ which are everywhere defined, continuous, and infinitely
differentiable''
\end{quote}
As a consequence it seems that relevant models have to include smooth potentials.

 Equation~(\ref{eq:blse}) belongs to a class called {\it bilinear   Schr\"odinger
 equation} in the control community,
 the term {\it bilinear} referring to the linear dependence with respect to
 $\psi$ and the affine dependence with respect to $u$. A (single input) bilinear control
 system in  a  vector space $\Hc$ is given by an evolution equation
\begin{equation}\label{EQ_bilinear_abstrait}
\frac{d}{dt}x=Ax+u(t)Bx
\end{equation}
 where  $A$ and $B$ are two (possibly unbounded)
  linear operators on $\Hc$ and $u$ is a real-valued function usually called
 control. Well-posedness of bilinear evolution equations of
 type~(\ref{EQ_bilinear_abstrait}) for a given
 control $u$  is usually a difficult question.   If $K$ is a subset of $\R$ such that for
 every $u$ in $K$,
  $A+uB$ generates a contraction semi-group $t\mapsto e^{t(A+uB)}$, then  for  every
  piecewise constant   function of the form
  $u=\sum_{j=1}^p u_j \mathbf{1}_{[\tau_j,\tau_{j+1})}$ where $u_1,\ldots,u_p \in K$ and
  $\tau_1<\tau_2<\ldots <\tau_{p+1}$ one defines the associated propagator of
  (\ref{EQ_bilinear_abstrait}) by
  $$\Upsilon^u_{t,\tau_1}=e^{(t-\tau_j)(A+u_jB)}\circ e^{(\tau_j-\tau_{j-1})
  (A+u_{j-1}B)}\circ \cdots \circ
   e^{(\tau_2-\tau_1)  (A+u_1B)}$$ for every $t$ in $(\tau_j,\tau_{j+1})$. The solution
   of~(\ref{EQ_bilinear_abstrait}) with
  initial value $x_0$ at time $\tau_1$ is $t\mapsto \Upsilon^u_{t,\tau_1}\psi_0$. When
  $\tau_1=0$, we will   denote $\Upsilon^u_{t,0}=\Upsilon^u_t$.

Since their precise description seems out of reach, one could try to
use the regularity of the solutions of~\eqref{eq:blse} to bound the attainable sets of
bilinear systems.

  Ball, Marsden, and Slemrod proved the general
 result
\begin{theorem}[Theorem 3.6 in \cite{BMS}]\label{THE_BMS}
Let $\X$ be  an infinite dimensional Banach space, $A$ generates a $C^0$ semi-group of
bounded linear
operators on $\X$ and $B$ be a bounded linear operator on $\X$. Then for any $T\geq 0$,  the
input-output mapping $u\mapsto
\Upsilon^u_{T,0}$ admits a unique continuous extension to the set of integrable functions on
$[0,T]$ endowed with
the  $L^1$ topology, and the attainable set
\begin{equation}\label{EQ_attainable_set}
\bigcup_{r > 1} \bigcup_{T\geq 0} \bigcup_{u \in L^r([0,T],\R)} \{ \Upsilon^u_{t,0}\psi_0,
t\in [0,T]\}
\end{equation}
is a meager set in $\X$ and hence has dense complement.
\end{theorem}
  Theorem \ref{THE_BMS} implies (see~\cite{turinici}) that equation \eqref{eq:blse} is not
 controllable in (the Hilbert unit sphere of) $L^{2}(\Omega)$ when $\psi\mapsto
 W \psi$ is bounded in $L^2(\Omega)$.
 Moreover,
 in the case in which  $\Omega$ is a domain of $\R^n$ and $W:\Omega\to \R$ is $C^2$, if
 the control $u$ is restrained to $L^p([0,+\infty),\R)$ with $p>1$, then
 equation~\eqref{eq:blse}
 is neither controllable in the Hilbert sphere $\mathbf{S}$ of $L^2(\Omega)$ nor
 in the natural functional space  where the problem is formulated, namely the
 intersection of  $\mathbf{S}$ with the Sobolev spaces $H^2(\Omega)$ and
 $H^1_0(\Omega)$.

 The main argument of~\cite{BMS} is as follows. If a sequence $(u_n)_{n\in \N}$ converges weakly in $L^p$ toward some
 function $u$ in
 $L^p$, then for every $\psi_0$ in $\X$, for every $t\geq 0$,
  $\displaystyle{\left (\Upsilon_{t,0}^{u_n} \psi_0 \right )_{n\in N}}$ converges toward
  $\Upsilon_{t,0}^{u}
  \psi_0$ in $\X$. By Alaoglu-Bourbaki theorem,  one can extract from any bounded sequence in
  $L^p$ a subsequence that  weakly converges in the (topological) dual of the (topological) dual of $L^p$,
  that is $L^p$ provided $p>1$.
 Hence the attainable set (\ref{EQ_attainable_set}) of \eqref{eq:blse} from $\psi_0$ with $L^p$, $p>1$,
 controls  can be written as a
 countable union of relatively compact subsets of $\X$. This union is a meager set, hence has
 empty interior  in $\X$ by Baire theorem.

 \begin{remark}\label{Remark:OpenQuestionBMS}
 The above argument does not hold anymore if one considers controls in $L^1$, since $L^1$ is not a reflexive
 space.    This is the content of \cite[Remark 3.8]{BMS}, where the question of the possible extension of
 the above result to $r=1$ is qualified as open except in the so called (\cite{Slemrod}) diagonal case, see \cite[Theorem 5.5]{BMS}.
  \end{remark}

\subsection{Main results}\label{SEC_main_results}

\subsubsection{Upper bound for attainable set of bilinear control systems}
Our aim is to give upper bounds for attainable set of bilinear control systems.
Our main result is the following
\begin{theorem}[Proved in Section \ref{SEC_attainable_set_interaction}]\label{THE_main_introduction}
Let $\Hc$ be an infinite dimensional separable Hilbert space, $A$ be a maximal dissipative operator on $\Hc$ with domain $D(A)$ and $B$ an
operator on $\Hc$ such that $B-c$ and $-B-c'$ generate contraction semi-groups leaving $D(A)$ invariant for some real constants $c$ and $c'$.

If $A+uB$ is maximal dissipative with domain $D(A)$ for every $u$ in $\R$ and if
the map $t \in \mathbf{R} \mapsto e^{tB} A e^{-tB} \in L(D(A),\Hc)$ is locally Lipschitz, then, (i) for every $T>0$, there
exists a unique continuous extension to $L^1([0,T],\R)$ of the input-output mapping of (\ref{EQ_bilinear_abstrait})
$u\mapsto \Upsilon^u_{T,0} \in  L(\Hc,\Hc)$, and (ii) for every
$\psi_0$ in $H$,  the set
$$\bigcup_{\alpha \geq 0} \bigcup_{T\geq 0} \bigcup_{u \in L^1([0,T],\R)} \{\alpha \Upsilon^u_{t,0}\psi_0,
t\in [0,T]\}$$
is a meager set in $\Hc$ and hence has dense complement.
\end{theorem}

In the special case where the control operator $B$ is bounded, we obtain a simplified
statement similar to the one of \cite{BMS} and dealing with $L^1$ controls:
\begin{proposition}[Proved in Section \ref{Sec:BoundedPotential}] \label{PRO_introduction_BMS_Bborne}
Let $\Hc$ be  an infinite dimensional {separable  Hilbert} space, $A$ generate a $C^0$ semi-group of bounded linear
operators on $\Hc$ and $B$ be a bounded linear operator on $\Hc$. Then for every $T>0$, there
exists a unique continuous extension to $L^1([0,T],\R)$ of the input-output mapping of
(\ref{EQ_bilinear_abstrait})
$u\mapsto \Upsilon^u_{T,0} \in  L(\Hc,\Hc)$ and, for every $\psi_0$ in $\Hc$,
$$ \bigcup_{\alpha \geq 0} \bigcup_{T\geq 0} \bigcup_{u \in L^1([0,T],\R)} \{\alpha  \Upsilon^u_{t,0}\psi_0,
t\in [0,T]\}$$
is a meager set in $\Hc$ and hence has dense complement.
\end{proposition}
{ These results settle the open question by Ball, Marsden and Slemrod mentioned in Remark~\ref{Remark:OpenQuestionBMS}.}

The Lipschitz hypothesis in Theorem \ref{THE_main_introduction} is crucial for our analysis when $B$ is
unbounded  but may be difficult to check in practice.
 For bilinear systems encountered in quantum physics, one can take advantage of the skew-adjointness of the operators to make the analysis simpler. For instance, it is possible to  replace the Lipschitz assumption
 of Theorem \ref{THE_main_introduction} by a hypothesis of boundedness of the commutator of operators $A$ and $B$:
\begin{theorem}[Proved in Section \ref{SEC_Higher_regularity}] \label{THE_intro_weak_coupling}
Let $\Hc$ be an infinite dimensional separable Hilbert space, $k$ a positive
number, $A$ and $B$ be two skew-adjoint operators such that (i) $A$ is invertible,
(ii) $B$ is $A$-bounded with $\|B\|_A=0$, see \eqref{def:Abound},
(iii) for every $u$ in $\mathbf{R}$, $|A+uB|^{k/2}$ is self-adjoint with domain
$D(|A|^{k/2})$ and
(iv) there
exists a constant $c_k(A,B)$ such that, for every $\psi$ in $D(|A|^k)$,
$|\Re \langle |A|^k \psi, B\psi \rangle |
\leq c_k(A,B) |\langle |A|^k \psi, \psi \rangle |$.
Then, for every $T>0$, there exists a unique continuous
extension to $BV([0,T],\R)$ of the input-output mapping $u\mapsto \Upsilon^u_T$.
Moreover, for every
$s<1+k/2$ and every $\psi_0$ in $D(|A|^s)$, the set
$$ \bigcup_{\alpha \geq 0} \bigcup_{T\geq 0}
\bigcup_{u \in BV([0,T],\R)} \{ \alpha \Upsilon^u_{t,0}\psi_0, t\in [0,T]\}$$
 has dense complement in $D(|A|^s)$.
\end{theorem}

The key of our analysis is a set of estimates of the energy of the solution
of~\eqref{eq:blse} in terms of
the $L^1$ norm of the control $u$. From these estimates, one can deduce an upper bound of
the distance
between the solutions of the infinite dimensional system~\eqref{eq:blse} and the solutions
of finite dimensional approximations of this PDE.
 To this end, we use the notion of
\emph{weak-coupling} (see Definition~\ref{Def:WeakCoupling}). This notion has
already been introduced in \cite{BoussaidCaponigroChambrion}, with a slightly narrower
sense and proved its effectiveness both for theoretical analysis (\cite{time},
\cite{quadratique}) and for numerical simulations (\cite{ACC_QG}).
Here, we apply results of geometric control theory  on the (finite-dimensional) Galerkin
approximations of the system to prove:
\begin{theorem}[Proved in Section \ref{SEC_equivalence}]\label{THE_main_result_equivalence}
Let $\Omega$ be a Riemannian compact manifold, $r$ in $\mathbf{N}$, $r\geq 2$,
 $V,W:\Omega \to \R$ be two functions of class $C^r$, $a\neq b$ two real numbers,
  and $\psi_0$ be an eigenvector of $-\Delta +V$. Then the closure  for the  $H^{r}(\Omega, \mathbf{C})$ topology of the attainable set of \eqref{eq:blse}
  $$
  \bigcup_{T\geq 0} \bigcup_{u \in \mathcal{U}} \{\Upsilon^u_{t,0}\psi_0| t \in [0,T] \}
  $$
  is the same if the set of admissible controls $\mathcal U$ is equal to $L^1$ controls or to the set of
  piecewise constant controls  taking value in $\{a,b\}$.
\end{theorem}

\subsubsection{Higher regularity}
Under some regularity assumptions on the control potential with respect to the scale
of the uncontrolled operator, we are able to extend the results of the previous section.
For instance we can replace the ambient Hilbert space by any iterated domain of
the uncontrolled part up to the maximal regularity allowed by the control potential. The
immediate outcome of this extension is that the exact controllability is linked to
the lack of regularity of the uncontrolled potential.

Let us recall the main result of \cite{camillo}. Below $H^s_{(0)}((0,1),\mathbb{C})$ will denote
the domain of $|A|^{s/2}$ where $A$ is the Laplace-Dirichlet operator on $(0,1)$, and $\phi_1$ its first eigenvector normalized.
\begin{unumberedtheorem}
Let $T>0$ and $\mu \in H^{3}((0,1),\mathbb{R})$ be such that
exists $c>0$ such that $
\frac{c}{k^3} \leqslant |\langle \mu \varphi_1 , \varphi_k \rangle |$,
for all $k \in \mathbb{N}^*$.
There exists $\delta>0$ and a $C^1$ map
$\Gamma:  \mathcal{V}_T  \rightarrow  L^2((0,T),\mathbb{R})$
where
$$\mathcal{V}_T := \{ \psi_f \in H^3_{(0)}((0,1),\|\psi_f\|=1,\mathbb{C}) ;
\| \psi_f - \psi_1(T) \|_{H^3} < \delta  \},$$
such that, $\Gamma( \psi_1(T) ) =0$ and for every $\psi_f \in  \mathcal{V}_T$,
the solution of
\begin{equation}\label{Eq:PotetialWell}
\left\lbrace\begin{array}{l}
i \frac{\partial \psi}{\partial t}(t,x) =
- \frac{\partial^2 \psi}{\partial x^2}(t,x)
- u(t) \mu(x) \psi (t,x) , x \in (0,1), t \in (0,T), \\
\psi(t,0)=\psi(t,1)=0,
\end{array}\right.
\end{equation}
with initial condition $\phi(0)=\phi_1$ and control $u=\Gamma(\psi_f)$ satisfies $\psi(T)=\psi_f$.
\end{unumberedtheorem}

The above result applies for instance when $\mu:x\mapsto x$ is the identity function.
The techniques introduced in this paper provide
estimates from above and from below for the attainable set when using different classes of admissible controls:
\begin{proposition}[Proved in Section~\ref{Sec:PotentialWell}] Let $\mu :x\mapsto x$ be the
identity function on $(0,1)$. Then, for every $T>0$, the input-output mapping $u\mapsto
\Upsilon^u_T$ associated with \eqref{Eq:PotetialWell} admits a unique continuous extension
to the set $\radon{(0,T]}$ of the Radon measures.

 The attainable set from $\phi_1$ with Radon controls,
 $\displaystyle{\mathcal{A}tt_{\mathcal R}(\phi_1)=\bigcup_{T\geq 0}
 \bigcup_{u \in \radon{(0,T]}}\{\Upsilon^u_{t,0}\phi_1|0\leq t \leq T\},}
 $
satisfies $\displaystyle{\mathcal{A}tt_{\mathcal R}(\phi_1) \subset \bigcap_{s<5/2}
H^s_{(0)}((0,1),\mathbb{C})}$ and
$\displaystyle{\mathcal{A}tt_{\mathcal R}(\phi_1) \not \subset  H^{5/2}_{(0)}((0,1),
\mathbb{C})}$.

The attainable set from $\phi_1$ with BV controls,
$\displaystyle{\mathcal{A}tt_{BV}(\phi_1)=\bigcup_{T\geq 0}
 \bigcup_{u \in BV{(0,T]}}\{\Upsilon^u_{t,0}\phi_1|0\leq t \leq T\}},
 $

 is a $H^s$-dense  subset of  the intersection
of the $L^2$ unit sphere and $H^{s}_{(0)}((0,1),\mathbb{C})$ for every $s<9/2$.
\end{proposition}

\subsection{Organization of our analysis}

The scheme of the proofs of Theorem \ref{THE_main_introduction} and Proposition
\ref{PRO_introduction_BMS_Bborne} follows the structure of the proof of Theorem 3.6 in \cite{BMS}.
The lack of reflectiveness of  $L^1$ leads us to consider controls
as Radon measures, the weak-compactness of bounded sequences is ensured by Helly selection
theorem. The  main difficulty
is  to define a continuous input-output mapping associated with
(\ref{EQ_bilinear_abstrait}) in such
a way to guarantee compactness properties of the attainable sets.

In Section~\ref{sec:abstract-framework} we consider  bilinear evolution
 equations (that are not necessarily conservative) from an abstract point of view and
 we define the solution for controls with bounded variations. We also prove the
 well-posedness
 within this framework and prove the continuity of the propagators with respect
 to the control parameters.

 In Section~\ref{Sec:Interaction}, we use a reparametrization, inspired by physics, which we call
 the interacting framework, to extend the results of Section~\ref{sec:abstract-framework} to the case where
 the control is a Radon measure. This provides a proof of Theorem \ref{THE_main_introduction}. A special
 paragraph (Section \ref{Sec:BoundedPotential}) is dedicated to
 the case where $B$ is bounded and to the proof of Proposition
\ref{PRO_introduction_BMS_Bborne}.

 When considering closed quantum systems, the operators $A$ and $B$ appearing in
  (\ref{EQ_bilinear_abstrait})  are skew-adjoint.
  Section~\ref{Sec:HigherOrderFEPS}
 is devoted to the regularity analysis of the solution obtained so far when further
 assumption are made on the control potential.
 The energy estimates we obtain are then used in Section~\ref{Sec:GGA} which is dedicated to
 finite dimensional approximation of our solutions. Some consequences for control are expanded in
  Section~\ref{SEC_equivalence} which contains  a proof of Theorem \ref{THE_main_result_equivalence}. These results are applied
 in Section~\ref{sec:example} to various examples.

\subsection{Notations}

All along this analysis, $T$ is a positive real.

\paragraph{Bounded operators space} Let $\X$ and $\Y$ be two Banach spaces,
$L(\X,\Y)$ is the space of linear bounded operator acting on $\X$ with values in
$\Y$. If $\X=\Y$ we write $L(\X)$.

\paragraph{Maximal dissipative operators} An operator $A$ on $\Hc$ is dissipative
if for any $\phi\in D(A)$, $\Re\langle \phi, A\phi\rangle \leq 0$. It is maximal dissipative if it has no proper
dissipative extension.

\paragraph{Weak and strong topology} Let $(A_n)_{n\in\N}$ a sequence in
$L(\X,\Y)$, let $A$ in $L(\X,\Y)$. Then $A_n$ converges to $A$ in the strong sense
if for any $\psi$ in $\X$, $(A_n\psi)_{n\in\N}$ converges to $A\psi$ in $\Y$ and $A_n$
converges to $A$ in the weak sense if for any $\psi$ in $\X$ and $\phi$ in ${\Y}^{\ast}$,
 the topological dual of $\Y$, $(\phi(A_n\psi))_{n\in\N}$ converges to $\phi(A\psi)$
 in $\C$.

\paragraph{Graph topology} Consider a self-adjoint operator $A$ acting on a
Hilbert space $\Hc$ with domain $D(A)$, the graph topology on $D(A)$ is the
topology associated to the norm $\psi\in D(A)\mapsto \|\psi\|_\Hc+\|A\psi\|_\Hc \in
\R^+$.

\paragraph{Bounded variation function}
\label{DEF_bounded_variation}
Let $I\subset \R$ be an interval. A family $t\in I \mapsto u(t) \in E $, $E$ a subset
of a Banach space $\X$, is in $BV(I,E)$
if there exists $N\geq 0$ such that
\[
 \sum_{j=1}^n \|u(t_j)-u(t_{j-1})\|_X \leq N
\]
for any partition $a=t_0<t_1<\ldots<t_n=b$ of the interval $(a,b)$.
The mapping
\[
u\in BV(I,E) \mapsto \sup_{a=t_0<t_1<\ldots<t_n=b}\sum_{j=1}^n
\|u(t_j)-u(t_{j-1})\|_{\X}
\]
is a semi-norm on $BV(I,E)$ that we
denote with $\|\cdot\|_{BV(I,E)}$.
The semi-norm in $BV(I,E)$ is also called total variation.

\noindent  We say that $(u_{n})_{n\in \N} \in BV(I,E)$ converges to $u \in BV(I,E)$ if
$(u_{n})_{n\in N} $ is a bounded sequence in $BV(I,E)$
pointwise convergent to $u \in BV(I,E)$.

\noindent{ The Jordan decomposition theorem provides that any bounded variation function is the difference of two nondecreasing bounded functions. This coupled to Helly's theorem provides the famous Helly's selection theorem :
\begin{unumberedtheorem}
Let $(f_n)_{n\in\N}$ be a sequence in $BV(I,\R)$, where $I$ is a compact interval. If
\begin{enumerate}
 \item there exists $M>0$ such that for all $n\in\N$, $\|f_n\|_{BV(I,\R)}<M$,
 \item there exists $x_0\in I$ such that $(f_n(x_0))_{n\in\N}$ is bounded.
\end{enumerate}
Then $(f_n)_{n\in\N}$ has a pointwise convergent subsequence.
\end{unumberedtheorem}
}
\paragraph{Radon measure}
For a real  interval $I$, we consider the space $\radon{I}$ of (signed) Radon measures on an real interval $I$.
Recall that a positive Radon measure is a measure on borelians which is locally finite and inner regular. { A
signed Radon measure $\mu$ is (by means of Hahn-Jordan decomposition) the difference $\mu=\mu^+-\mu^-$
of two positive Radon measures $\mu^+$ and $\mu^-$ with disjoint
support. We denote $|\mu|=\mu^++\mu^-$. The total variation of $\mu$ is $|\mu|(I)$. }

We say that
$(u_{n})_{n\in \N} \in \radon{I}$ converges to $u \in \radon{I}$ if
$\sup_{n}|u_{n}|(I) < +\infty$ (uniformly bounded total variations) and $u_{n}((0,t]) \to u((0,t])$ for  every
$t \in I$ as $n$ tends to $\infty$.
\noindent  Note that any cumulative function of a Radon measure is locally of bounded variation
and the associated total variation (which do not depend on the choice of the
cumulative function) coincides with the total variation of the Radon measure.

\noindent  Every function in $L^{1}_{\mathrm{loc}}(I)$ can be considered as Radon measure {as the density of an absolutely continuous measure}.
When it does not create ambiguity we will identify the function with the
associated Radon measure.
\begin{UnnumberedLemma}
 Let $(u_n)_{n\in\N}$ be a sequence of integrable function over $I$ converging in $L^1(I)$ to some $u$. Let $(\mu_n)_{n\in\N}$
 defined by
 \[
  \mu_n(J)=\int_Ju_n d\lambda, \forall J \subset I \mbox{ borelian } \forall n \in \N.
 \]
 Then  $(\mu_n)_{n\in\N}$ converges to $\mu$ in $\radon{I}$ where
 \[
  \mu(J)=\int_Ju d\lambda, \forall J \subset I \mbox{ borelian}.
 \]
\end{UnnumberedLemma}
We remark that for $u$ in $L^{1}(I)$ the total variation of the associated Radon measure is the $L^1$-norm of $u$.

\noindent The total variation is a norm on the subspace of $\radon{I}$ formed by measures with finite total variation.   Note that the notion of convergence we use is \emph{not} the same as the topology induced by the norm of total variation, see also Remark \ref{Rem:NonUniformDensity}.

\begin{remark}
For properties of bounded variation functions and signed Radon measures, we refer to Ambrosio, Fusco and Pallara~\cite{AmbrosioFuscoPallara}.
\end{remark}
\paragraph{Other notations :} For any real interval $I$, we define
\[
\Delta_{I}:=\{(s,t)\in I^2\mid \, s\leq t\,\}.
\]

\noindent In a metric space $E$, the notation $B_E(v_0,r)$ stands for the open ball of radius $r$ and center $v_0$ in $E$.

\noindent For a densely defined operator $B$ on a Hilbert space, $B^\ast$ stands for its adjoint.
Recall that $B^\ast$ is densely defined if and only if $B$ is closable, {in which case} $B^\ast$ is closed.

\section{Well-posedness and continuity for BV controls}\label{sec:abstract-framework}

In this section, we present some  global well-posedness results for a
class of non autonomous perturbations of skew-adjoint linear Cauchy problem and
as well as a
continuity criteria for converging problem.

\subsection{Abstract framework: definitions and notations}
In the sequel we will consider a general framework, the classical
definitions and tools we use can be found in \cite[Section
X.8]{reed-simon-2}, as well as the associated notes and problems. The only
difference is that we consider the Hilbert framework and we have an opposite
sign for the generators  and thus following~\cite{Phillips} we use the word
dissipative instead of accretive, see also~\cite[Notes of Section
X.8]{reed-simon-2}. We also restrict our analysis to the Hilbert space framework.
In this case generators of contraction semi-groups and maximal dissipative
operators coincide, see~\cite[Theorem 1.1.3]{Phillips}. {These framework is imposed for
convenience as the existence of an hermitian product simplifies many parts of the analysis.}

Let $\Hc$ be a separable Hilbert space (possibly infinite dimensional) with scalar
product $\langle \cdot,\cdot\rangle$ and $\|\cdot\|$ the corresponding norm,
$A, B$ be two (possibly unbounded) dissipative operators on $\Hc$.  We consider the
formal bilinear control system
\begin{equation}\label{eq:main}
\frac{d}{dt} \psi(t)=A \psi(t) + u(t) B \psi(t),
\end{equation}
where the scalar control $u$ is to be chosen in a set of real functions.

Given an initial data $\psi(0)=\psi_0 \in \Hc$, there is no reason for the
system~\eqref{eq:main} to be well-posed. Indeed even the definition of $A+B$ is
not obvious when $A$ and $B$ are unbounded. To this effect it is
usually assumed that the operators $A$ and $B$ satisfy the following
condition.
\begin{definition}\label{Def:KatoRellich}
$B$ is said \emph{relatively bounded} with respect to $A$, or $A$-bounded, if
$D(A) \subset D(B)$ and there exist $a, b>0$ such that for every $\psi$ in $D(A)$,
$\|B \psi \|\leq a \|A \psi\| +b \|\psi \|$.
\end{definition}
It is well-known that if $A$ is skew-adjoint and $B$ skew-symmetric, from Kato--Rellich theorem, (see for
example~\cite[Theorem X.12]{reed-simon-2}),
if $B$ is relatively bounded with respect to $A$, then for every real constant
$u$ such that $|u| < 1/a$, $A+uB$ is skew-adjoint with domain $D(A)$ and
generates a group of unitary operators. The system~\eqref{eq:main} is then
well-posed for every initial condition\footnote{In the skew-adjoint
case the norm of the solutions of \eqref{eq:main} is conserved.}. From~\cite[Corollary of
theorem X.50]{reed-simon-2}, these can be extended
to the dissipative case when $A$ is maximal dissipative, $B$ dissipative and $u$ non-negative.

In most of the examples considered in Section~\ref{sec:example}, we consider the skew-adjoint case
and $a$ is arbitrary small, so that we can define the solutions of~\eqref{eq:main} for
every piecewise constant control $u$ with real values.

In the general cases, we will refer to the following assumptions.
\begin{assumption}\label{ass:ass}
$(A,B,K)$ is a triple with $A$ is a maximal dissipative operator on $\Hc$, $B$ an operator on $\Hc$ with $D(A)\subset D(B)$,
and $K$ a real interval containing $0$ is such that for any $u\in K$, $A+uB$ is
a maximal dissipative operator on $\Hc$ with domain
$D(A)$.
\end{assumption}

Assumption~\ref{ass:ass} implies that the operator $B$ is a bounded from $D(A)$ to $\Hc$ and
allows to define
\begin{equation}\label{def:Abound}
 \|B\|_{A}:=\inf_{\lambda >0}
\|B(\lambda-A)^{-1}\|.
\end{equation}
The possibly zero number $\|B\|_{A}$ is the lower bound of all possible constant $a$ in Definition~\ref{Def:KatoRellich}. Notice that actually
\begin{equation}\label{def:Abound2}
 \|B\|_{A}:=\liminf_{\lambda\to+\infty }
\|B(\lambda-A)^{-1}\|.
\end{equation}
Indeed, it is clear that $\inf_{\lambda >0}
\|B(\lambda-A)^{-1}\| \leq \liminf_{\lambda\to+\infty }
\|B(\lambda-A)^{-1}\|$. Conversely, for any $\epsilon>0$ there exists $b_\epsilon>0$ such that,
for every $\psi$ in $D(A)$,
$$\|B \psi \|\leq (\|B\|_{A}+\epsilon) \|A \psi\| +b_\epsilon \|\psi \|.$$
Hence, for every $\lambda$ such that $A-\lambda$ is invertible, for every $\phi$ in $\Hc$,

\[
\|B(A-\lambda)^{-1} \phi \|\leq (\|B\|_{A}+\epsilon)\|A(A-\lambda)^{-1} \phi\| +b_\epsilon\|(A-\lambda)^{-1}\phi \|
\]
and thus for $\lambda>0$, since $A$ is dissipative,
\[
\|B(A-\lambda)^{-1} \psi \|\leq (\|B\|_{A}+\epsilon)\|\psi\| +\frac{b_\epsilon}{\lambda}\|
\psi \|
\]
so that
\[
\lim_{\lambda\to +\infty}\|B(A-\lambda)^{-1}\|\leq (\|B\|_{A}+\epsilon).
\]
The conclusion $\lim_{\lambda\to +\infty}\|B(A-\lambda)^{-1}\|\leq \|B\|_{A}=\inf_{\lambda
>0}\|B(\lambda-A)^{-1}\|$ follows as $\epsilon$ tends to zero.

In the next section, we will consider the following assumption.
\begin{assumption}\label{ass:Interacting}
$(A,B,K)$, with $A$ a maximal dissipative operator on $\Hc$ and
$K$ a real interval containing $0$, is such that
\begin{enumerate}[label={{(\arabic{assumption}.\arabic{*})}}, ref={{(\arabic{assumption}.\arabic{*})}}]
\label{ASS_A_skew_adjoint2}
 \item there exists $c\geq 0$ and $c'\geq 0$ such that $B-c$
and $-B-c'$ generate contraction semigroups on $\Hc$ leaving $D(A)$ invariant,
\label{ASS_B_skew_symmetric}
\item for every $u\in \radon{(0,T]}$, with $u((0,t])\in K$ for any $t\in[0,T]$,
\[
t\in [0,T]\mapsto \mathcal{A}(t):=e^{u((0,t])B}Ae^{-u((0,t])B}
\]
is a family of
maximal dissipative operators with common domain $D(A)$
such that :
\begin{itemize}
 \item $\sup_{t\in [0,T]}
\left\|(1-\mathcal{A}(t))^{-1}\right\|_{{L}(\Hc,D(A))}<+\infty$,
 \item ${\mathcal
A}$ has bounded variation from $[0,T]$ to $L(D(A),\Hc)$.
\end{itemize}
\label{ASS_calA_bounded_variation}
\end{enumerate}
\end{assumption}

Notice that if $u\in \radon{(0,T]}$, $u((0,0])=0$ and $t\in [0,T]\mapsto u((0,t])$ is left continuous
due to the inner regularity of Radon measures.

The triple $(A,B,K)$ satisfies Assumption~\ref{ass:Interacting} for any
interval $K$ containing $0$ if $(A,B)$ satisfies the following one.
\begin{assumption}\label{ass:StrongInteraction}
$(A,B)$ is such that
\begin{enumerate}[label={{(\arabic{assumption}.\arabic{*})}}, ref={{(\arabic{assumption}.\arabic{*})}}]
 \item $A$ is a maximal dissipative operator on $\Hc$ with domain
$D(A)$,
\label{ASS_A_skew_adjoint3}
 \item there exists $c\geq 0$ and $c'\geq 0$ such that $B-c$
and $-B-c'$ generate contraction semigroups on $\Hc$ leaving $D(A)$ invariant,
\label{ASS_B_skew_symmetric2}
\item the map $t\in \R \mapsto e^{t B}Ae^{-t B} \in
L(D(A),\Hc)$ is locally Lipschitz.\label{ass:commutator}
\end{enumerate}
\end{assumption}
Assumption~\ref{ass:commutator} is a strong assumption on the regularity of $B$
with respect to the scale of $A$. Indeed it implies that $B$ is the generator of
a strongly continuous semigroup on $D(A)$. In Section~\ref{Sec:HigherOrderFEPS},
we consider higher regularity assumption.

\subsection{Propagators}\label{Sec:Propagators}

\begin{definition}[Propagator on a Hilbert space]\label{Definition:Propagator}
 Let $I$ be a real interval. A family $(s,t)\in
\Delta_I\mapsto X(s,t)$ of linear contractions, that is
Lipschitz maps with Lipschitz constant less than one, on a Hilbert space $\Hc$,
strongly
continuous in $t$ and $s$  and such that
\begin{enumerate}[label=(\roman*)]
\item for any $s<r<t$, $X(t,s)=X(t,r)X(r,s)$,
\item $X(t,t)=I_{\Hc}$,
\end{enumerate}
is called a contraction propagator on $\Hc$.
\end{definition}

\begin{assumption}\label{ass:kato-generale}
Let $I$ be a real interval and $\Dc$ dense subset of
$\Hc$
\begin{enumerate}[label={{(\arabic{assumption}.\arabic{*})}}, ref={{(\arabic{assumption}.\arabic{*})}}]
\item $A(t)$ is a maximal dissipative operator on $\Hc$ with domain $\Dc$ ,
\item \label{Ass:BV} $t\mapsto A(t)$ has bounded variation from $I$ to $L(\Dc,\Hc )$,
where $\Dc$ is endowed with the graph topology associated with $A(a)$ for $a=\inf
I$,
 \item \label{Ass:ResolventBound}
 $
 M:=\sup_{t\in I} \left\|(1-A(t))^{-1}\right\|_{L(\Hc,\Dc)}< \infty,
 $
\end{enumerate}
\end{assumption}
\begin{remark}
In Assumption \ref{Ass:BV}, the bounded variation of $t\mapsto A(t)$ ensures that any choice of
$a\in I$ will be equivalent.
\end{remark}

\begin{remark}\label{Remark:AssumptionKato}
As  $A(t)$ is a maximal dissipative operator, that is the generator of a contraction semigroup, its
resolvent set contains the positive half line and from Hille-Yosida theorem, see
Proposition~\ref{Prop:ThmHilleYosida},
any generator of a contraction semi-group satisfies
$$
 \sup_{t\in I} \left\|(1-A(t))^{-1}\right\|_{L(\Hc)}< \infty.
$$
Note that $\|(1-A(t))^{-1}\|_{L(\Hc,\Dc)}<+\infty$,
the essence of Assumption \ref{Ass:ResolventBound}
is that
$\left\|(1-A(t))^{-1}\right\|_{L(\Hc,\Dc)}$ is bounded uniformly with respect to
$t\in I$.

We do not assume $t \mapsto A(t)$ to be continuous. However as a
consequence of Assumption~\ref{Ass:BV} (see~\cite[Theorem~3]{Edwards})
it admits right and left limit in $L(\Dc,\Hc )$, $A(t - 0)=\lim_{\varepsilon
\to 0^{+}}{A(t-\varepsilon)}$, $A(t + 0)=\lim_{\varepsilon
\to 0^{+}}A(t+\varepsilon)$, for all $t \in I$, and $ A(t-0)= A(t+0)$ for all $t
\in I$ except an at most countable set.
\end{remark}

The core of our analysis is the following result due to Kato
(see~\cite[Theorem 2 and Theorem 3]{Kato1953}). It gives sufficient conditions
for the well-posedness of the system~\eqref{eq:main}.
\begin{theorem}\label{thm:kato}
If $t \in I \mapsto A(t)$ satisfies
Assumption~\ref{ass:kato-generale}, then
there exists a unique contraction propagator
$X:\Delta_{I}\to L(\Hc)$ such that
if $\psi_0\in \Dc$ then $X(t,s)\psi_0\in \Dc$ and for $(t,s)\in \Delta_{I}$
\[
 \|A(t)X(t,s)\psi_0\|\leq M e^{M\|A\|_{BV(I,L(\Dc,\Hc))}}\|A(s)\psi_0\|.
\]
and in this case $X(t,s)\psi_0$ is strongly left
differentiable in $t$ and  right differentiable in $s$ with
derivative (when $t=s$) $A(t+0)\psi_0$ and $-A(t-0)\psi_0$ respectively.

In the case in which $t \mapsto A(t)$ is continuous and
skew-adjoint,
if $\psi_0\in \Dc$  then $t\in(s,+\infty)\mapsto X(t,s)\psi_0$  is
strongly continuously differentiable in $\Hc$ with derivative $A(t)X(t,s)\psi_0$.
\end{theorem}
\begin{proof}
The statement of this theorem is  obtained by gathering statements
of~\cite{Kato1953}. The point which may not be stated clearly is
the existence of $C>0$ such that for $(t,s)\in \Delta_{I}$
\[
 \|A(t)X(t,s)\psi_0\|\leq C\|A(s)\psi_0\|,
\]
for any $\psi_0\in \Dc$. This is in~\cite[\S 3.10]{Kato1953} and $C=M\exp(MN)$ with
\[
M =\sup_{t\in I} \left\|(1-A(t))^{-1}\right\|_{L(\Hc,\Dc)}
\quad\text{ and }\quad N = \|A\|_{BV(I,L(\Dc,\Hc))}.\qedhere
\]
\end{proof}

We will consider $t\mapsto X(t,s)\phi_0$ as a ``mild'' solution in $\Hc$ of
\begin{equation}\label{Eq:NonAutonomousCauchy}
 \begin{cases}
  \frac{d}{dt} \phi(t)=A(t)\phi(t)\\
\phi(s)=\phi_0
 \end{cases}
\end{equation}
even if it is not differentiable. However in the case in which $t \mapsto A(t)$ is continuous and
skew-adjoint,
if $\phi_0\in \Dc$  then $t\mapsto X(t,s)\phi_0$ is a strong solution as it is
continuously differentiable in $\Hc$ and satisfies \eqref{Eq:NonAutonomousCauchy}, see~\cite{Kato1953}.

We write that $X:\Delta_{I}\to L(\Hc)$ is the propagator with generator $t\in I \mapsto A(t)$.
The second argument $s$ of $X(t,s)$ will be called the initial time.

\begin{remark}
If $(A,B,K)$ satisfies Assumption~\ref{ass:Interacting}, the operator $t \in [0,T]
\mapsto \mathcal{A}(t)\!:=e^{u((0,t])B}\!Ae^{-u((0,t])B}$ defined in
Assumption~\ref{ASS_calA_bounded_variation} satisfies
Assumption~\ref{ass:kato-generale} for any Radon measure $u$ on $(0,T)$ with
 $u((0,t])\in K$ for any $t\in(0,T]$. If $(A,B)$ satisfies Assumption~\ref{ass:StrongInteraction} then
$(A,B,\R)$ satisfies Assumption~\ref{ass:Interacting})
\end{remark}

The fact that Assumption~\ref{ass:ass} is stronger than
Assumption~\ref{ass:kato-generale} is the content of the following
lemma.
\begin{lemma}\label{Rem:NonInteractingCase}
If $(A,B,K)$ satisfies Assumption~\ref{ass:ass} and $u:[0,T]\mapsto K$
has bounded variation then $A(t):=A+u(t)B$ satisfies
Assumption~\ref{ass:kato-generale} with $I=[0,T]$.
\end{lemma}
\begin{proof}
The non obvious point to verify is Assumption~\ref{Ass:ResolventBound}. The map
\[
u\in K\mapsto (1-A)(1-A-uB)^{-1}
\]
is continuous. Indeed
\begin{align*}
\lefteqn{ (1-A)(1-A-uB)^{-1}-(1-A)(1-A-v B)^{-1}}\\
&=(1-A)\left((1-A-uB)^{-1}-(1-A-v B)^{-1}\right)\\
&=(v-u)(1-A)\left((1-A-uB)^{-1}B(1-A-v B)^{-1}\right)\\
&=(v-u)(1-A)\left((1-A-uB)^{-1}B(1-A)^{-1}(1-A)(1-A-v B)^{-1}\right)
\end{align*}
so that
\begin{align*}
\lefteqn{ (1-A)(1-A-u B)^{-1}-(1-A)(1-A-v B)^{-1}}\\
&-(v-u)(1-A)(1-A-uB)^{-1}B(1-A)^{-1}\left((1-A)(1-A-v B)^{-1}-(1-A)(1-A-u B)^{-1}\right)\\
&=(v-u)(1-A)\left((1-A-uB)^{-1}B(1-A)^{-1}(1-A)(1-A-u B)^{-1}\right)
\end{align*}
Define
\[
 C(u)=\|(1-A)(1-A-uB)^{-1}\|_{L(\Hc)}\quad\text{and}\quad b=\|B(1-A)^{-1}\|
\]
so that
\[
 (1-|v-u|bC(u))\|(1-A)(1-A-uB)^{-1}-(1-A)(1-A-v B)^{-1}\|
\leq|v-u|C(u)^2b(1-A)
\]
which provides the desired continuity. Then as $|u(t)-u(0)|\leq \|u\|_{BV(I)}$ for any $t\in I$,
$u(t)$ is in a bounded subset of $K$ for all $t\in I$ thus the closure of its image is compact and by Heine theorem
\[
 t\in I\mapsto \|(1-A-u(t)B)^{-1}\|_{L(\Hc,\Dc)}
\]
is bounded.
\end{proof}

\subsection{Continuity}

Our main concern on this analysis is the continuity of the solution in the control $u$.
This is indeed a consequence of work~\cite{Kato1953} by Kato as shown below.
\begin{definition}
A family $(A_n)_n$ of generators of contraction semi-groups
tends to $A$, a generator of a contraction semi-group, in the strong resolvent
sense if $\left((\lambda-A_n)^{-1} \phi\right)_n$ tends to $(\lambda-A)^{-1}\phi$
as $n$ goes to infinity, for every $\phi$ in $\Hc$ and some (and hence all, see
\cite{reed-simon-2}) $\lambda$ in $\mathbf{R}^+$.
\end{definition}

\begin{proposition}\label{prop:continuity} Let $(A_{n})_{n\in \N}$ and $A$
satisfy Assumption~\ref{ass:kato-generale} on a { bounded} real interval  $I$. Let
$(\Dc_n)_{n\in \N}$ and
$\Dc$ be their respective domains (for any
$t\in I$).
Let $X_n$ (respectively $X$) be the contraction propagator associated with
$A_n$ (respectively $A$).

Assume
\begin{enumerate}[label=(\roman*)]
 \item $\sup_{n\in N} \sup_{t\in I}\|(1-A_{n}(t))^{-1}\|_{L(\Hc,\Dc_n)} <
+ \infty$,
 \item $A_{n}(\tau)$ converges to $A(\tau)$ in the strong resolvent
sense for
almost every $\tau \in I$ as $n$ tends to infinity,
 \item $\sup_{n\in \N}\|A_{n}\|_{BV(I,L(\Dc_n,\Hc))} < + \infty$,
 \item $\sup_{n\in \N}\|A_{n}(a)\|_{L(\Dc_n,\Hc))} < + \infty$ for for $a=\inf
I$.
\end{enumerate}
Then $X_{n}(t,s)$ tends strongly to $X(t,s)$ locally uniformly in $s,t\in
I$.
\end{proposition}
\begin{proof}
From~\cite[\S 3.8]{Kato1953}, we deduce that one can replace both $X_n$ and $X$
by any of their Riemann products that is the propagator associated with a
piece-wise constant approximation, namely :
\[
\|\left(X_{n}(t,s;\Delta)-X_{n}(t,s)\right)\phi\|\leq Me^{MN}N|\Delta|\|A(a)\phi\|, \forall \phi\in\Ec
\]
where $\Delta:=\{s=t_0<t_1<\ldots<t_n=t\}$ is a partition of the interval
$(t,s)$,
$X_{n}(\Delta)$ is the propagator associated with
$
\displaystyle \sum_{j =1}^n A_{n,j-1}
\chi_{[t_{j-1},t_{j})}
$
where $A_{n,j-1}=A_n(t_{j-1})$,
\begin{align*}
 M &=\sup_{n\in \N} \sup_{t\in I}\{ \left\|(1-A_n(t))^{-1}\right\|_{L(\Hc,\Dc_n)},
 \left\|(1-A(t))^{-1}\right\|_{L(\Hc,\Dc)}\},\\
 N &= \max\{ \sup_{n\in\N}\|A_{n}\|_{BV(I,L(\Dc_n,\Hc))},\|A\|_{BV(I,L(\Dc,\Hc))}\},
\end{align*}
and $|\Delta|=\sup_{1\leq j \leq n}|t_j-t_{j-1}|$. We define similarly
$X(\Delta)$ and the following holds
\[
\| X(t,s;\Delta)-X(t,s)\|_{L(\Dc,\Hc)}\leq Me^{MN}N|\Delta|.
\]

Following the
proof of \cite[Theorem X.47a (Hille-Yosida)]{reed-simon-2} (see also
Proposition~\ref{Prop:ThmHilleYosida}), we have that
\[
 \left\|e^{t A_n(\tau)}\phi - e^{t A^\lambda_n(\tau)}\phi\right\| \leq t
\left\|A_n(\tau)\phi -A^\lambda_n(\tau)\phi\right\|,\quad \forall \phi \in
\mathcal{D}_n
\]
with $A^\lambda_n(\tau):=\lambda(\lambda-A_n(\tau))^{-1}A_n(\tau)$, with
$\lambda >0$, and
\[
 \left\|e^{t A(\tau)}\phi - e^{t A^\lambda(\tau)}\phi\right\| \leq t
\left\|A(\tau)\phi -A^\lambda(\tau)\phi\right\|,\quad \forall \phi \in
\mathcal{D}
\]
with $A^\lambda(\tau):=\lambda(\lambda-A(\tau))^{-1}A(\tau)$.

As $A_n$ and $A$ are generator of contraction semi-groups, then
$\{\lambda(\lambda-A_n(\tau))^{-1}\}$ is uniformly bounded in $n$ and $\tau$.
Notice that  $\lambda(\lambda-A_n(\tau))^{-1}\phi$ tends to $\phi$ as $\lambda$
goes to infinity for any
$\phi \in \Dc_n$ uniformly in $\tau$ and $n$ since from the assumptions we deduce
$\sup_{n\in
\N}\sup_{t\in I}\|A_{n}(t)\|_{L(\Dc_n,\Hc))} < + \infty$.
We
deduce that  $\lambda(\lambda-A_n(\tau))^{-1}$ tends strongly to the identity
as $\lambda$ goes to infinity uniformly in $\tau$ and $n$. Similarly
$A^\lambda(\tau)$ tends stronlgy to $A(\tau)$
uniformly in $\tau$ and $n$ as $\lambda$ goes to infinity.

{For $\phi\in\Hc$ and $\delta>0$ there exists $\psi \in \Dc$ and $\psi^n\in\Dc_n$ such
that
\[
 \|\phi-\psi\|\leq \delta\quad\mbox{ and }\quad \|\phi-\psi^n\|\leq \delta
\]
so that we can write
\begin{eqnarray*}
\left  \| e^{tA(\tau)}\phi -e^{tA_n(\tau)}\phi \right \|& \leq & 4\delta+\left \|
e^{tA(\tau)}\psi -
e^{tA^\lambda(\tau)}\psi \right \|+ \left \| e^{tA^\lambda (\tau)}\phi
-e^{tA^\lambda_n(\tau)}\phi \right
\|+
\left \| e^{tA^\lambda_n (\tau)}\psi^n -e^{tA_n(\tau)}\psi^n \right \|\\
&\leq& 4\delta+ t \left \| A (\tau)\psi -A^\lambda(\tau)\psi \right \|
+ t \left \| A_n
(\tau)\psi^n -
A^\lambda_n(\tau)\psi^n \right \| + \left \| e^{tA^\lambda (\tau)}\phi
-e^{tA^\lambda_n(\tau)}\phi \right \|.
\end{eqnarray*}}
As $e^{t A_n^\lambda(\tau)}=e^{-\lambda t } e^{t \lambda^2
(\lambda-A_n(\tau))^{-1}}$ and $e^{t A^\lambda(\tau)}=e^{-\lambda t } e^{t
\lambda^2
(\lambda-A(\tau))^{-1}}$ (see \cite[Theorem X.47a
(Hille-Yosida)]{reed-simon-2}),
\begin{eqnarray*}
 \left \| e^{tA^\lambda (\tau)}\phi -e^{tA^\lambda_n(\tau)}\phi \right \| & \leq
&
\left \|e^{-\lambda t } e^{t \lambda^2
(\lambda-A_n(\tau))^{-1}} \phi - e^{-\lambda t } e^{t \lambda^2
(\lambda-A(\tau))^{-1}} \phi \right \|\\
& \leq & e^{-\lambda t} \left \|e^{t \lambda^2
(\lambda-A_n(\tau))^{-1}} \phi - e^{t \lambda^2
(\lambda-A(\tau))^{-1}} \phi \right \|
\end{eqnarray*}
Recall that $\|
(\lambda-A_n(\tau))^{-1} \| \leq \frac{1}{\lambda}$ (see Proposition
\ref{Prop:ThmHilleYosida} with $\omega=0$) and hence $\|e^{t\lambda^2
(\lambda-A_n(\tau))^{-1}} \| \leq e^{\lambda t} $.
Duhamel's identity writes, for $0\leq t\leq T$,
\begin{eqnarray}
\lefteqn{
 \left \|e^{t \lambda^2
(\lambda-A_n(\tau))^{-1}} \phi - e^{t \lambda^2
(\lambda-A(\tau))^{-1}} \phi \right \|} \nonumber \\
 & =& \left \| \int_0^t \lambda^2 e^{(t-s) \lambda^2
(\lambda-A_n(\tau))^{-1}} \left \{
(\lambda-A_n(\tau))^{-1} -
(\lambda-A(\tau))^{-1} \right \} e^{s \lambda^2
(\lambda-A(\tau))^{-1}} \phi \; \mathrm{d}s \right \|
\label{EQ_identite_resolvante_Lebesgues}\\
&\leq &  \lambda^2 e^{T \lambda}
 \int_0^T \left \| \left \{
(\lambda-A_n(\tau))^{-1} -
(\lambda-A(\tau))^{-1} \right \} e^{s \lambda^2
(\lambda-A(\tau))^{-1}} \phi \right \|\; \mathrm{d}s \nonumber
\end{eqnarray}
The result follows from Lebesgue dominated convergence theorem, using the
convergence of
$A_{n}(\tau)$  to $A(\tau)$ in the strong resolvent sense for
almost every $\tau \in I$ as $n$ tends to infinity.

\end{proof}
\begin{remark}\label{Rem:AssumptionsPropositionContinuity}
Assuming that $\Dc_n=\Dc$, for all $n\in\N$, following \cite[Theorem VIII.25]{reed_simon}
and using Banach-Steinhaus Theorem
the assumptions of Proposition~\ref{prop:continuity} are
verified if
\begin{enumerate}[label=(\roman*)]
 \item $\sup_{n\in \N} \sup_{t\in
I}\|(1-A_{n}(t))^{-1}\|_{L(\Hc,\Dc)} < + \infty$,
 \item $A_{n}(\tau)$ converges to
$A(\tau)$ in the strong sense in $\Dc$ for almost every $\tau \in I$ as $n$
tends to infinity,
 \item $\sup_{n\in
\N}\|A_{n}\|_{BV(I,L(\Dc,\Hc))} < + \infty$.
\end{enumerate}
\end{remark}

Recall that $I$ is bounded, we have the following corollary.
\begin{corollary}\label{cor:continuity}
{Let $(A,B,K)$ satisfy Assumption~\ref{ass:Interacting}.
Let $(v_{n})_{n\in N} $ be a uniformly bounded sequence in $\radon{I}$ which is almost everywhere
convergent to $v \in \radon{I}$  with $v_n((0,t])\in K$  and $v((0,t])\in K$ for any $t\in(0,T]$ and  $v_n(\{0\})=v(\{0\})=0$.
Set $\mathcal{A}_{n}(t) = e^{-v_{n}((0,t])B}Ae^{v_{n}(0,t])B}$ and $\mathcal{A}(t) =
e^{-v((0,t])B}Ae^{v((0,t])B}$ and}
let $X_n$ (respectively $X$) be the contraction propagators associated with
$\mathcal{A}_n$ (respectively $\mathcal{A}$).

If $\sup_{n\in \N}\|\mathcal{A}_{n}\|_{BV(I,L(D(A),\Hc))} < + \infty$,
then $X_{n}(t,s)$ tends to $X(t,s)$ locally uniformly $s,t\in I$ (and hence, uniformly
in $I^2$).
\end{corollary}
\begin{proof}
Referring to the proof of Proposition~\ref{prop:continuity}, as
$X_{n}(t,s;\Delta)$ is a product of contractions so is $X_{n}(t,s)$.

The proof consist of checking that the assumptions of
Proposition~\ref{prop:continuity} are fulfilled:
\begin{enumerate}[label=(\roman*)]
\item We have $\sup_{n\in N} \sup_{t\in
I}\|(1-\mathcal{A}_{n}(t))^{-1}\|_{L(\Hc,\Dc)} <\infty$. Indeed as
\begin{align*}
\lefteqn{\| (1-A) (1-\mathcal{A}_{n}(t))^{-1}\|_{L(\Hc)}}\\& =\| (1-A)
e^{
v_{n}((0,t])B}(1-A)^{-1}e^{ -
v_{n}((0,t])B}\|_{L(\Hc)}\\
&\leq \|e^{
v_{n}((0,t])B}\|_{L(\Hc)}\| e^{-v_{n}((0,t])B} (1-A)  e^{
v_{n}((0,t])B}(1-A)^{-1}\|_{L(\Hc)}\|e^{ -
v_{n}((0,t])B}\|_{L(\Hc)}\\
& \leq \|e^{
v_{n}((0,t])B}\|_{L(\Hc)}\| (1-\mathcal{A}_{n}(t))
(1-A)^{-1}\|_{L(\Hc)}\|e^{ -
v_{n}((0,t])B}\|_{L(\Hc)} \\
& \leq\|e^{
v_{n}((0,t])B}\|_{L(\Hc)}\left(\|\mathcal{A}_{n}(t) -
\mathcal{A}_{n}(0)\|_{L(D(A),\Hc)} +
\|1-A\|_{L(D(A),\Hc)}\right) \|e^{ -
v_{n}((0,t])B}\|_{L(\Hc)}\\
& \leq \|e^{
v_{n}((0,t])B}\|_{L(\Hc)}\left(\|\mathcal{A}_{n}\|_{BV(I,L(D(A),\Hc))} +
1\right)\|e^{ -
v_{n}((0,t])B}\|_{L(\Hc)}.
\end{align*}
We used $v_n(\{0\})=0$ to set $\mathcal{A}_{n}(0)=A$. Then from Assumption~\ref{ASS_B_skew_symmetric}, there exists
$\omega \in \R$ such that
\begin{equation}\label{eq:EwpGrowthB}
\|e^{ vB}\|_{L(\Hc)} \leq e^{\omega |v|}, \, \forall v\in \R
\end{equation}
which provides the boundedness.
 \item The sequence $\mathcal{A}_{n}(t)$ tends to $\mathcal{A}(t)$ in the
strong resolvent sense.
Indeed from
\[
(1-\mathcal{A}_{n}(t))^{-1} - (1-\mathcal{A}(t))^{-1}=
e^{-v_n((0,t])B}(1-A)^{-1}e^{v_n((0,t])B} - e^{-v((0,t])B}(1-A)^{-1}e^{-v((0,t])B}
\]
we have
\begin{align*}
 (1-\mathcal{A}_{n}(t))^{-1} - (1-\mathcal{A}(t))^{-1}&=
(e^{-v_n((0,t])B}-e^{-v((0,t])B})(1-A)^{-1}(e^{v_n((0,t])B}-e^{v((0,t])B})\\
&\quad +(e^{-v_n((0,t])B}-e^{-v((0,t])B})(1-A)^{-1}e^{v((0,t])B}\\
&\quad +e^{-v((0,t])B}(1-A)^{-1}(e^{v_n((0,t])B}-e^{v((0,t])B})
\end{align*}
then using \eqref{eq:EwpGrowthB}
the boundedness of the sequence $(v_n)$ and the strong continuity of
$t\in \R \mapsto e^{tB}$, we conclude on the strong resolvent convergence.

\item From the assumptions, we have $\sup_{n\in
\N}\|\mathcal{A}_{n}\|_{BV(I,L(D(A),\Hc))} < + \infty$.
 \item As ${\mathcal A}_{n}(0)=A$, we have $\sup_{n\in
\N}\|{\mathcal A}_{n}(0)\|_{L(D(A),\Hc))} < + \infty$.\qedhere
\end{enumerate}
\end{proof}
\begin{remark}\label{Rem:CommAssumption}
To prove that $\sup_{n\in \N}\|\mathcal{A}_{n}\|_{BV(I,L(\Dc,\Hc))} < + \infty$
for
$\mathcal{A}_{n}(t) = e^{-u_{n}((0,t])B}Ae^{u_{n}((0,t])B}$, we use
Assumption~\ref{ass:commutator} as this provides $L_I(A,B)$ such that
for every $s,t \in I$,
\begin{equation}\label{eq:comm}
\|e^{-tB}Ae^{tB} -  e^{-sB}A e^{sB}\|_{L_I(\Dc,\Hc)} \leq   |t-s|
L_I(A,B).
\end{equation}
\end{remark}

\section{Interaction framework}\label{Sec:Interaction}
In this section we consider the framework of Assumption~\ref{ass:Interacting} or
\ref{ass:StrongInteraction}. We show that they indeed lead to weak solutions of \eqref{eq:main}
and consider an extension of a negative result of Ball, Marsden and Slemrod~\cite{BMS}
on the attainable set.

In a subsequent paragraph we analyze the source problem in order to obtain a Duhamel formula
and use it in the last paragraph to analyze perturbation by bounded control potentials.

\subsection{Generalized propagators}
In the following, we aim at explaining the link between Assumptions~\ref{ass:ass} and
\ref{ass:Interacting} and thus emphasize the fact that \eqref{eq:main} has also
solution for $u$ being a Radon measure.

We will use approximation of Radon measures by piecewise constant functions.
\begin{lemma}\label{LEM_approximation_measure}
For every $u \in \radon{(0,T]}$  there exists a sequence $(u_n)_n$
of piece-wise constant functions such that $\int_0^t u_n$ {tends to
$u((0,t))$ and $\int_0^t |u_n|$ tends to
$|u|((0,t])$ point-wise with respect
to $t$ in $[0,T]$ as $n$ tends to infinity with $\int_0^T |u_n|\leq
|u|((0,T))$.

Moreover, if the cumulative function $t\mapsto u((0,t))$ is $M$-Lipschitz continuous absolutely continuous and bounded on $[0,T]$ then $(u_n)_n$ can be chosen such that $|u_n|\leq M$. }
\end{lemma}
\begin{proof}
 Using Hahn decomposition~\cite{Doss1980}  (see also~\cite{Fischer} for the Hahn-Jordan decomposition), any
Radon measure $u$ is the difference of two non negative Radon measures $u^{+}$
and $u^{-}$ with disjoint supports. Hence its is equivalent to prove the
statement for positive Radon measures.

Let us assume $u$ is positive. {Then $U:t\in (0,T] \mapsto u((0,t))$ is a
decreasing function (with bounded variation). Except on a at most countable set, $U$ is
continuous. So $U$ is the sum of an increasing step function, possibly with infinite steps, and an increasing continuous function.
Both can be approximated by an increasing sequences
of increasing continuous piecewise affine functions.}

The last statement follows by considering approximation of Lipschitz continuous functions by continuous piecewise affine ones.
\end{proof}
\begin{remark}\label{Rem:NonUniformDensity}
This lemma explains our choice of topology over the Radon measures  while the total variation topology may seem more natural at first sight.
Note that in this lemma, we consider the left-continuous cumulative function of $u$. Indeed below, we need convergence of the measures for intervals of the form $[a,b)\cap (0,T]$ thus which corresponds to characteristic functions right-continuous on $[0,T]$.

For instance, for a positive $u \in \radon{(0,T]}$  the sequence $(u_n)_n$
of piece-wise constant functions such that $\int_t^T u_n$ tends to
$u((0,t])$ point-wise is, in our construction, an increasing sequence.

As each $t\mapsto \int_0^t u_n$ is continuous, if $t\mapsto u((0,t))$ is not, we see that
the same result for the total variation topology is excluded. In this case,
we notice that $t\mapsto u((0,t))- \int_0^t u_n$ cannot be, by any other construction,
both positive and increasing in $t$ as otherwise for any $t$ $|u((0,t))- \int_0^t u_n|\leq |u((0,T])- \int_0^T u_n|$
which will implies the convergence for the total variation topology.
\end{remark}

\begin{definition}
Let $(A,B,K)$ satisfy Assumptions~\ref{ass:ass} and~\ref{ass:Interacting}. {Let
$u\in \radon{[0,T]}$. For any $v\in  BV([0,T],K)$ with distributional derivative $u$ let
$t\mapsto
Y^{u}_{t}$ be the contraction propagator
with initial time $s=0$ associated with ${\mathcal A}_v(t):=e^{-v(t)B} A
e^{v(t)B}$. We define the \emph{generalized propagator} associated
with $A+u(t)B$ with initial time zero, to be
$\Upsilon^{\mathfrak{d}v}_{t,0}=e^{v(t)B} Y^{u}_{t}$ and $t$ in $[0,T]$ and for every $v$ in
$BV([0,T],K)$ such that $v'=u$ in the distributional sense.
}
\end{definition}

\begin{remark}
{Let $u\in \radon{[0,T]}$ and define $v_0(t)=u((0,t])$ the associated right-continuous
cumulative function and $v\in BV([0,T],\R)$ such that $v'=u$. Then $v-v_0$ is in
$BV([0,T],\R)$ and is almost everywhere $0$ as it is supported on the at most countable set
where $v$ is not right-continuous. A somehow pathological example could be $u=0$ and $v$ any
characteristic function
of a finite set.

The propagator $Y^{u}_{t}$ will not depend on the choice of $v$ being right-continuous,
or not, at its discontinuities. Indeed the later is a negligible set and a Duhamel formula
 provides the equality of the propagators. In the other hand, the factor $e^{vB}$ depends
 crucially on this choice.
This explains the the notation $\Upsilon^{\mathfrak{d}v}$ instead of $\Upsilon^{u}$.

We introduced this generalized propagator as imposing any extra requirement on the choice of $v$ will
lead to a loss in the compactness provided by Helly's selection theorem when we consider the
bounded case, Section~\ref{Sec:BoundedPotential}.
Nonetheless, except on this particular case we will only consider right-continuous $v_0$
keeping in mind that the generalized propagator $\Upsilon^{\mathfrak{d}v}$ will be deduce
multiplying with $e^{(v-v_0)B}$.

Compared to Definition~\ref{Definition:Propagator}, $X(t,s)=e^{v(t)B} Y^{u}_{t,s}e^{-v(s)B}$
defines a propagator when $v$ is continuous, that is when $u$ has no atoms. Otherwise, we no
longer require any continuity keeping in mind that $v_0$ the right-continuous cumulative
function of $u$ will lead to a right-continuous propagator which seems more compatible with
the initial condition requirement.}
\end{remark}

\begin{proposition}\label{prop:propagator-radon}
Let $(A,B,{K})$ satisfies Assumptions~\ref{ass:ass} and~\ref{ass:Interacting}. Let $t\mapsto Y^{u}_{t}$ be the contraction propagator
with initial time $s=0$ associated with ${\mathcal A}(t):=e^{-u((0,t])B} A
e^{u((0,t])B}$ for $u\in  \radon{(0,T]}$ and $\pro^u_{t}$ the one associated with $A+u(t)B$
with initial time $s=0$  for $u\in  BV([0,T],\R)$.
Then for every $\psi_{0} \in \Hc, t\in [0,T]$ the map $\pro_{t}(\psi_{0}):u\mapsto \pro^u_{t}(\psi_{0})
\in \Hc$
admits a unique continuous extension on $\radon{(0,T]}$ denoted $\pro_{t}(\psi_{0})$ which satisfies, { for every $u$ in $\radon{I}$,}
\begin{equation}\label{Eq:InteractionFramework}
\pro^{u}_{t} (\psi_{0}) = e^{u((0,t]) B} Y^{u}_{t}(\psi_{0}), \forall u\in  \radon{(0,T]}, \forall t\in[0,T].
\end{equation}
\end{proposition}
\begin{proof}
Let $u\in  L^{\infty}([0,T],\R)$
and consider the propagator $Y^{u}_{t}$ with generator
\[
t\in[0,T] \mapsto e^{- \int_{0}^{t}u(\tau)d\tau B} A
e^{\int_{0}^{t}u(\tau)d\tau B}
\]
and initial time $s=0$
given by Theorem~\ref{thm:kato}.
If, in particular,
 $u$ is left-continuous piecewise constant and $\psi_{0} \in D(A)$ then
the continuous function
$\Psi :t \mapsto e^{-\int_{0}^{t}u(\tau)d\tau B} \pro^{u}_{t}(\psi_{0})$  is
piecewise $C^{1}$ propagator with derivative $e^{- \int_{0}^{t}u(\tau)d\tau B} A
e^{\int_{0}^{t}u(\tau)d\tau B} \Psi(t)$ and
hence by uniqueness in Theorem~\ref{thm:kato},
$Y^{u}_{t}(\psi_{0}) = e^{-\int_{0}^{t}u(\tau)d\tau B} \pro^{u}_{t}(\psi_{0})$.

For every $u \in \radon{(0,T]}$ let $(u_{n})_{n\in \N}$ be a sequence of (left-continuous)
piecewise
constant functions on $[0,T]$ that converges to $u$ for the topology of
$\radon{(0,T]}$.

From Remark~\ref{Rem:CommAssumption} and Corollary~\ref{cor:continuity}, for
every $\psi_{0} \in \Hc$,
$Y^{u_{n}}_{t}(\psi_{0}) $ tends to $Y^{u}_{t}(\psi_{0}) $ as $n$  tends to
$\infty$.
We set $\pro^{u}_{t} (\psi_{0}) = e^{u((0,t]) B} Y^{u}_{t}(\psi_{0})$. Then
$\pro^{u_{n}}_{t} (\psi_{0})$ tends to $\pro^{u}_{t} (\psi_{0})$ as $n$ tends to
$\infty$.
\end{proof}

\emph{In the rest of this analysis, the definition of $\pro^{u}_{t}$ will be given
by \eqref{Eq:InteractionFramework}.} This means that
in the case where $v=u((0,t])$, we write $\Upsilon^{u}$ instead of $\Upsilon^{\mathfrak{d}v}$.

\begin{proposition}\label{prop:propagator-radon2}
Let $T\geq 0$. If $(A,B)$ satisfies
Assumption~\ref{ass:StrongInteraction} and $D(A)\subset D(B)$.  Then for every
$\psi_0$ in $D(A)$, for every $u \in \radon{(0,T]}$,
the map $t \mapsto
\pro^{u}_{t}(\psi_{0})$ is a weak solution of \eqref{eq:main} in the sense that it satisfies
\[
\int_{[0,T]} \langle f'(t),  \pro^{u}_{t}(\psi_{0}) \rangle dt =
\int_{[0,T]}\langle
f(t),  A \pro^{u}_{t}(\psi_{0}) \rangle dt+  \int_{(0,T]}\langle  f(t), B
\pro^{u}_{t}(\psi_{0})   \rangle du(t)\,,
\]
for every
$f \in C_0^{1}([0,T],\Hc)$.
\end{proposition}
\begin{proof}
For every $u \in \radon{(0,T]}$ let $(u_{n})_{n\in \N}$ be a sequence of
piecewise
constant functions on $[0,T]$ that converges to $u$ for the topology of
$\radon{(0,T]}$.

For every $f \in C_{0}^{1}([0,T], \Hc)$,
$$
- \int_{[0,T]} \langle f'(t),  \pro^{u_{n}}_{t}(\psi_{0}) \rangle dt =
\int_{[0,T]}\langle  f(t), A  \pro^{u_{n}}_{t}(\psi_{0}) \rangle dt+
\int_{[0,T]}\langle  f(t), B \pro^{u_{n}}_{t}(\psi_{0})   \rangle du_{n}(t)\,
$$
since from Theorem~\ref{thm:kato}, $Y^{u_{n}}_{t}(\psi_{0})\in D(A)$ for any
$t\in[0,T]$.

The left hand side converges to
$$-\int_{[0,T]} \langle f'(t),
\pro^{u}_{t}(\psi_{0}) \rangle dt,$$
$$\int_{[0,T]}\langle  f(t),  A \pro^{u_{n}}_{t}(\psi_{0}) \rangle dt
\to \int_{[0,T]}\langle  f(t), A  \pro^{u}_{t}(\psi_{0}) \rangle dt,$$
and
$$\int_{[0,T]}\langle  f(t),  B \pro^{u_{n}}_{t}(\psi_{0})   \rangle du_n(t)\to
\int_{(0,T]}\langle f(t),  B \pro^{u}_{t}(\psi_{0})   \rangle du(t)$$
as $n$ tends to $\infty$. { Indeed, the integrand is a bounded pointwise converging sequence of right-continuous regulated functions
which makes each of them the uniform limit of a sequence of piecewise constant functions which can be chosen to be right-continuous. Since
$$\int_{[0,T]}v(t) du_n(t)\to
\int_{(0,T]}v(t)  du(t)$$
for $v$ a piecewise constant right-continuous function, see Lemma~\ref{LEM_approximation_measure}
and $(|\mu_n|((0,T]))_{n\in\N}$ is a bounded sequence, the desired convergence holds.}

Eventually
\[
\left |\int_{(0,T]}\langle B f(t),
\pro^{u}_{t}(\psi_{0})    \rangle d(u-u_{n})(t) \right | \leq
\sup_{t\in [0,T]} \|Bf\|\|\psi_{0}\|
|u-u_{n}|(\mathrm{supp}f)\to_{n \to \infty} 0.\qedhere
\]
\end{proof}
\begin{remark}
The assumption $D(A)\subset D(B)$ is made for convenience to avoid the introduction
of auxiliary spaces such as $D(B^\ast)^\ast$, the topological dual of the domain
of the adjoint of $B$.

An interesting question would be to understand the relation between the
 assumptions associated to the two constructions
of propagators we have considered in this section. For example, on what extent
does Assumption~\ref{ass:StrongInteraction} ensure that $A+uB$ has a maximal
dissipative closure for $u\in \R$ ?

This seems to be a difficult question. However in the skew-adjoint case, the
following considerations are in place. Let $A$ and $B$ be skew-adjoint (symmetric
maximal dissipative) with $D(A)\cap D(B)$ dense. For any $v\in \Hc$, any $u\in D(A)$ the map
\[
t\in K \mapsto \langle (1-\varepsilon (B^2+A^2))^{-1}v, e^{tB}Ae^{-tB}(1-\varepsilon (B^2+A^2))^{-1}u\rangle
\]
is Lipschitz, its distributional derivative is bounded uniformly in $\varepsilon$ by
the Banach-Steinhaus theorem. So that $[A,B]\in L(D(A)\cap D(B),(D(A)\cap D(B))^\ast)$ extends to an operator such that
\[
[A,B]\in L(D(A),\Hc).
\]
with a slight abuse of notation and hence for any $u\in \R$
\[
[A,A+uB]\in L(D(A),\Hc).
\]
The Nelson commutator theorem, see Appendix~\ref{Sec:Nelson}, gives that $A+uB$ is
essentially skew-adjoint for any $u\in \R$.

Proposition~\ref{prop:propagator-radon} extends the notion of
solution to a very low regular case, namely Radon measure controls. To do so
we made some regularity assumptions on the control potential $B$ by using Assumption~\ref{ass:StrongInteraction}.
\end{remark}

\begin{corollary}\label{Cor:propagator-radon}
Let $T\geq 0$. {Assume} that $(A,B)$ satisfies
Assumption~\ref{ass:StrongInteraction} and $D(A)\subset D(B)$.  Then
for every $\psi_{0} \in  D(A)$,  for every $u \in \radon{(0,T]}$,  for every $ t\in [0,T]$,
$\pro^{u}_{t}(\psi_{0})\in D(A)$ and satisfies
\begin{equation}\label{Eq:StrongSolution}
 \pro^{u}_{t}(\psi_{0})  = \psi_{0}+
\int_{[0,t]} A\pro^{u}_{s}(\psi_{0})\,ds+  \int_{(0,t]}B\pro^{u}_{s}(\psi_{0})\,du(s)\,.
\end{equation}
\end{corollary}
\begin{proof}
From Theorem~\ref{thm:kato}, $\pro^{u}_{t}(\psi_{0})\in D(A)$.
From Proposition~\ref{prop:propagator-radon2}, the following holds
\[
\int_{[0,T]} \langle f'(t),  \pro^{u}_{t}(\psi_{0}) \rangle dt =
\int_{[0,T]}\langle A
f(t),   \pro^{u}_{t}(\psi_{0}) \rangle dt+  \int_{(0,T]}\langle B f(t),
\pro^{u}_{t}(\psi_{0})   \rangle du(t)\,,
\]
for $f \in C_0^{1}([0,T], \Hc)$. Then plugging
\[
 f(t)=\int_{[0,t]}f'(s)\,ds
\]
and using Fubini's theorem for Bochner integral~\cite[Theorem III.11.9]{DunfordSchwartz}, we have
\[
\int_{[0,T]} \langle f'(t),  \pro^{u}_{t}(\psi_{0}) \rangle dt =
-\int_{[0,T]}\langle
f'(s),  \int_{[s,T]} A\pro^{u}_{t}(\psi_{0})\,dt \rangle\, ds-  \int_{(0,T]}\langle f'(s),
\int_{[s,T)}  B\pro^{u}_{t}(\psi_{0}) \,  du(t)\rangle\, ds\,,
\]
for $f' \in C_0([0,T],  \Hc)$ orthogonal to $1$.  Hence
\[
t\in[0,T]\mapsto \pro^{u}_{t}(\psi_{0}) +
\int_{[s,T]} A\pro^{u}_{t}(\psi_{0})\,dt
+\int_{[s,T)}  B\pro^{u}_{t}(\psi_{0}) \,  du(t)
\]
is almost everywhere equal to a fixed constant, so that \eqref{Eq:StrongSolution} follows
for almost every $t\in[0,T]$. Recalling that both sides of \eqref{Eq:StrongSolution} are right-continuous
we obtain the corollary.
\end{proof}

\subsection{The attainable set}
\label{SEC_attainable_set_interaction}

\begin{proposition}\label{rem:compactness}
Let $T>0$. Let
$\psi_{0} \in \Hc$. Let
$(A,B)$ satisfy
Assumption~\ref{ass:StrongInteraction}.
Then,
for every $L>0$,   the set $\{\alpha \pro^{u}_{t}(\psi_0)\,:\, u\in
\radon{(0,T]}, |u|({ (0,T]}) \leq L, {t\in
{ [0,T]}}, |\alpha| \leq 1\}$ is relatively compact in
{$\Hc$}.
\end{proposition}
\begin{proof}
For every $u \in \radon{(0,T]}$ consider $v(t) = u((0,t])$. Then $v
\in BV([0,T])$ and
$\|v\|_{BV([0,T])} \leq L$. Note that
$\|v\|_{L^{\infty}(J)} = \sup_{t\in [0,T]}|v(t) - v(0)|
\leq \|v\|_{BV([0,T])} \leq L$
since {$v(0)=u(\varnothing )=0$}.
Consider a sequence $(u_{n})_{n\in \N} \subset \radon{(0,T]}$ such
that $ |u_{n}|((0,T])
\leq  L$ for every $n$.
The sequence $v_{n}: t \mapsto u_{n}((0,t])$,
from Helly's selection theorem (see for example~\cite{Helly, Natanson})
has a subsequence pointwise converging to some $v \in BV([0,T])$,
$\|v\|\leq L$. Let
$u$ the Radon measure associated to $v$. Thus the sequence $(u_{n})_{n\in \N}$
has a subsequence converging to $u \in \radon{(0,T]}$.
We relabel this convergent subsequence by $(u_{n})_{n\in \N}$.

From \eqref{eq:comm} we have that
$\mathcal{A}_{n}(t) = e^{-u_{n}((0,t])B}Ae^{u_{n}((0,t])B}$ is uniformly bounded
in $BV([0,T], L(\Dc,\Hc))$ by
$L\|[A,B]\|_{L(\Dc,\Hc)}$. By Corollary~\ref{cor:continuity},
$t \mapsto Y_{t}^{u_{n}}(\psi_0)$ converges uniformly for $t\in {
[0,T]}$ to $t
\mapsto Y_{t}^{u}(\psi_0)$.
{A special attention has to be paid to the sequence of factors $(t\mapsto e^{v_n((0,t])B})$,
no uniform convergence can be ensured, and
$t \mapsto \Upsilon_{t}^{u_{n}}(\psi_0)$ converges pointwise in $t\in {
[0,T]}$ to $t \mapsto \pro_{t}^{\mathfrak{d}v}(\psi_0)$.

To conclude the lemma, we notice that
for any sequence $(t_n)_n$,
$(v_n((0,t_n])_n$ is a bounded sequence and thus $(e^{v_n((0,t_n])B})$ is bounded
and has a strongly convergent subsequence.}
\end{proof}
\begin{remark}\label{Rem:Compactness}
 Note that the set $\{\pro^{u}(\psi_0)\,:\, u\in
L^1( [0,T], \R) , \|u\|_{L^1} \leq L \}$ is relatively compact
in
{$L^\infty({ [0,T]},\Hc)$}.
 However, despite the compactness of $[0,T]$, the set $\{\pro^{u}(\psi_0)\,:\,
u\in
\radon{J}, |u|((0,T) \leq L \}$ may be not relatively compact
in {$L^\infty([0,T],\Hc)$}. Indeed, if this set were relatively compact, the
trajectories of \eqref{eq:main} would be continuous, due to the density of
continuous trajectories, which is not the case in general. The obstructions
comes from the factor $e^{u((0,t])B}$ in \eqref{Eq:InteractionFramework}.
\end{remark}

From the above corollary the attainable set is contained in a countable union of
precompact sets.
\begin{corollary}\label{Cor:NoExactControllability}
If $\Hc$ is infinite dimensional and $\psi_{0} \in \Hc$.  If $(A,B)$ satisfies
Assumption~\ref{ass:StrongInteraction} then
$$
\bigcup_{L,T,a >0}  \left\{ \alpha \pro^{u}_{t}(\psi_{0}), u\in
\radon{(0,T]},
|u|((0,T]) \leq L, t\in [0,T], |\alpha|\leq a\right\}
$$
is  a meagre set (in the
sense of Baire).
\end{corollary}
\begin{proof}
 The union can be actually restricted to $L,T,a$ in a countable set, for instance $\N^3$.
Then Proposition~\ref{rem:compactness} tells that  each set of the union is
relatively compact in $\Hc$ and thus with empty
interior.
\end{proof}

\begin{proof}[Proof of Theorem \ref{THE_main_introduction}]
The well-posedness result for $L^1$
controls is a consequence of Proposition \ref{prop:propagator-radon2} proved for Radon
controls. Emptyness of the attainable set for $L^1$ controls is a consequence of Corollary
\ref{Cor:NoExactControllability} proved for Radon controls.
\end{proof}

\begin{remark}
This result represents an extension of \cite[Theorem~3.6]{BMS}, see
also~\cite{turinici}, in the Hilbert
case.

Similar results holds for bounded variation function in the
framework of Theorem~\ref{thm:kato} with $A(t)=A+u(t)B$ where $(A,B)$
satisfies Assumptions~\ref{ass:ass},
 see Section~\ref{Sec:HigherOrderFEPS}.
\end{remark}

\subsection{The source problem}
In this section we want to consider the {formal ``source problem''}
\begin{equation}\label{eq:source}
\frac{d}{dt} \psi(t)=A \psi(t) + u(t) B \psi(t)+v(t) F(t),
\end{equation}
where $F :[0,T]\mapsto \Hc$ is integrable and $v \in \radon{(0,T]}$.

Our aim is to establish a Duhamel formula in the Radon measure framework.
\begin{corollary}\label{Cor:Duhamel-Radon}
{Let $T\geq 0$. Let $u\in \radon{(0,T]}$. If $(A,B)$  satisfy
Assumption~\ref{ass:StrongInteraction} and $D(A)\subset D(B)$.
Let $X^{u}(t,s)=e^{u((0,t])B}Y^{u}(t,s)e^{u((0,s])B}$ where $Y^{u}(t,s)$ is the contraction propagator
associated with ${\mathcal A}(t)\!=\!e^{-u((0,t])B}\!Ae^{u((0,t])B}$\!\!\!\!.

For every $\psi_{0} \in  D(A)$, for every $ t\in [0,T]$ for every $u \in \radon{(0,T]}$,
$F\in L_{dv}^1([0,T],D(A))$ and $v \in \radon{(0,T]}$
such that
\[
\forall t\in(0,T],\,\int_{(0,t]}BF(r)v(\{r\})du(r)=0
\]
define
\[
 \psi(t)=X^{u}(t,0)\psi_0+\int_{(0,t]}X^{u}(t,s)F(s)dv(s).
\]
Then
\[
\psi(t)  = \psi_{0}+
\int_{[0,t]} A\psi(s)\,ds+  \int_{(0,t]}B\psi(s)\,du(s)
+\int_{(0,t]}F(s)dv(s).
\]

}
\end{corollary}
\begin{proof}
From Corollary~\ref{Cor:propagator-radon}, for $t\in[0,T]$ and $t_0\in[0,t]$ after a translation by $t_0$
we deduce
\[
X^{u}(t,t_0)\psi_{0}  = \psi_{0}  +
\int_{[t_0,t]} AX^{u}(s,t_0)\psi_{0}\,ds+  \int_{(t_0,t]}BX^{u}(s,t_0)\psi_{0}\,du(s).
\]

Then
\begin{align*}
\psi(t)&=\psi_{0}  +
\int_{[0,t]} AX^{u}(s,0)\psi_{0}\,ds+  \int_{(0,t]}BX^{u}(s,0)\psi_{0}\,du(s)\\
&+\int_{(0,t]} F(s)dv(s)  +
\int_{(0,t]}\int_{[s,t]} AX^{u}(r,s)F(s)drdv(s)+  \int_{(0,t]}\int_{(s,t]}BX^{u}(r,s)F(r)du(r)dv(s)\\
&=\psi_{0}  +\int_{(0,t]} F(s)dv(s) +\int_{[0,t]} AX^{u}(s,0)\psi_{0}\,ds
+\int_{(0,t]}\int_{[s,t]} AX^{u}(r,s)F(s)drdv(s)\\
&+\int_{(0,t]}BX^{u}(s,0)\psi_{0}\,du(s)
+\int_{(0,t]}\int_{(s,t]}BX^{u}(r,s)F(s)du(r)dv(s).
\end{align*}
So that we obtain
\begin{align*}
\psi(t)
&=\psi_{0}  +\int_{(0,t]} F(s)dv(s) +\int_{[0,t]} AX^{u}(r,0)\psi_{0}\,dr
+\int_{(0,t]}\int_{(0,r]} AX^{u}(r,s)F(s)dv(s)dr\\
&+\int_{(0,t]}BX^{u}(r,0)\psi_{0}\,du(r)
+\int_{(0,t]}\int_{(0,r)}BX^{u}(r,s)F(s)dv(s)du(r)\\
&=\psi_{0}  +\int_{(0,t]} F(s)dv(s) +\int_{[0,t]} A\left(X^{u}(r,0)\psi_{0}+\int_{(0,r]} X^{u}(r,s)F(s)dv(s)\right)dr\\
&+\int_{(0,t]}B\left(X^{u}(r,0)\psi_{0}
+\int_{(0,r)}X^{u}(r,s)F(s)dv(s)\right)du(r)
\end{align*}
which provides the conclusion as
\[
\int_{(0,t]}BF(r)v(\{r\})du(r)=0. \qedhere
\]

\end{proof}
\begin{remark}
{We can now consider that for $\psi_{0} \in  \Hc$,  for $u \in \radon{(0,T]}$,
$v \in \radon{(0,T]}$ with no atoms and $F\in L_{dv}^1([0,T],\Hc)$

\[
t\in [0,T]\mapsto \psi(t)=X^{u}(t,0)\psi_0+\int_{(0,t]}X^{u}(t,s)F(s)dv(s)
\]
is a mild solution of \eqref{eq:source}.}
\end{remark}

\subsection{Bounded control potentials}\label{Sec:BoundedPotential}

In this section only, we assume that $A$ is the generator of a
strongly continuous semigroup with domain $D(A)$ and $B$ is bounded. For every
$u$ in $\mathbf{R}$, $A+uB$ is also a generator of a strongly continuous
semigroup with domain $D(A)$.

Since $A$ generates a strongly continuous semi-group there exist $C_A>0$ and $\omega\in \R$
such that
\begin{equation}\label{eq:ExpGrowthA}
 \|e^{tA}\|\leq C_A e^{\omega t},\quad \forall t>0.
\end{equation}
For the equivalent norm
\[
N(\psi)=\sup_{t>0} \|e^{t(A-\omega)}\psi\|
\]
$A-\omega$ is the generator of a contraction semigroup. Let $B$ in $\Bc(\Hc)$
then $B$ is also bounded for the norm  $N$ and if $\|B\|_N$ is its norm, $A-\omega+u(t)B-R\|B\|_N$
for any $u$ with bounded variation and value in $B_\R(0,R)$
satisfies the assumptions of \cite{Kato1953} in the Banach space structure associated with $N$.
So that in this case the results of \S~\ref{Sec:Propagators} are still valid.
Below we show that it is actually as simple to consider Radon measures controls.

It is classical (see \cite{BMS}) that the input-output mapping $\Upsilon$ admits a unique continuous
continuation to $L^1(\mathbf{R},\mathbf{R})$. We consider below the extension to Radon measures.
Recall that $C_A$ is defined in \eqref{eq:ExpGrowthA}.

\begin{theorem}\label{Thm:ExistenceBoundedPotential}
Let $A$ be with domain $D(A)$ be the generator of a strongly continuous semigroup
and let $B$ be bounded. Then, the input-output mapping $\Upsilon$ admits a unique continuous
extension to the set of Radon measures.
Precisely, for any $u\in \radon{(0,T]}$ there exists a unique family of operators
$\Upsilon^u:\Delta_{[0,T]}\to L(\Hc)$ strongly
continuous in $t$ and $s$  such that
\begin{enumerate}[label=(\roman*)]
\item for any $s<r<t$, $\Upsilon^u_{t,s}=\Upsilon^u(t,r)\Upsilon^u(r,s)$,
\item $\Upsilon^u(t,t)=I_{\Hc}$,
\item if $u$ has bounded variation on $[0,T]$, for any $\psi_0\in \Hc$, $(s,t)\in\Delta_{[0,T]}\mapsto \Upsilon^u_{t,s}\psi_0$
is
strongly left
differentiable in $t$ and  right differentiable in $s$ with
derivative (when $t=s$) $(A+u(t+0)B)\psi_0$ and $-(A+u(t-0)B)\psi_0$ respectively,
\item the map $u\mapsto \Upsilon^u$ is continuous in $\radon{(0,T]}$ in the sense that
if $(u_l)_{l\in \mathbf{N}}$ is a
sequence of Radon measures $(0,T]$ and $u$ a Radon measure on $(0,T]$ such that
$u_l((0,T])$ tends to $u((0,T])$ and for every $t$ in $[0,T]$, $u_l((0,t))$
tends to $u((0,t))$ as $l$ tends to infinity with
$\sup_{l \in \mathbf{N}}|u_l|((0,T])<+ \infty$ then
\[
 \Upsilon^{u_l}_{t,s}\psi_0 \to \Upsilon^{u}_{t,s}\psi_0
\]
for any $\psi_0\in \Hc$.
\end{enumerate}
The family $\Upsilon^u$ is called propagator associated with $A+uB$.
Moreover $\Upsilon^u$ satisfies the following
estimate:
\[
 \|\Upsilon^u_{t,s}\|_{B(\Hc)}\leq C_Ae^{\omega|t-s|+|u|([s,t])C_A\|B\|}
\]
for any Radon measure $u$.
\end{theorem}
\begin{proof}
 Let $u:[0,T]\rightarrow \mathbf{R}$ be a piecewise constant function.
For every $\psi_0$ in $D(A)$, the map $t\mapsto \Upsilon^u_{t,s}\psi_0$ is
absolutely continuous and satisfies, for almost every $t$ in
$[s,T]$,
$$
\frac{\mathrm{d}}{\mathrm{d}t} \Upsilon^u_{t,s}\psi_0=(A+u(t)B)
\Upsilon^u_{t,s}\psi_0.
$$
Following \cite[Theorem X.69]{reed-simon-2}, we write, thanks to the
Duhamel's formula,
\[
\Upsilon^u_{t,s}\psi_0=e^{(t-s)A}\psi_0 + \int_{(s,t]}
e^{(t-s_1)A}u(s_1)B\Upsilon^u_{s_1,s}\psi_0 \mathrm{d}s_1
\]
and replacing iteratively $\Upsilon^u_{t,s}\psi_0$ by its expression $p$ times,
we
get
\begin{align*}
\Upsilon^u_{t,s}\psi_0=&e^{(t-s)A}\psi_0+\sum_{n=1}^\infty \int_{s<s_1\leq s_2 \leq \ldots \leq s_n \leq t}
e^{(t-s_n)A}Be^{(s_{n}-s_{n-1})A}\circ \cdots\\
& \quad \quad \cdots \circ  Be^{(s_2-s_1)A}B e^{(s_1-s)A}\psi_0 u(s_1)
u(s_2)\ldots u(s_n)\mathrm{d}s_1\ldots\mathrm{d}s_n.
\end{align*}

This allows us to extend the propagator to Radon measures. Namely let $u\in\radon{(0,T]}$,
we define the linear operator
$$
Y^u_{(n)}(t,s)\psi_0:=\int_{s<s_1\leq s_2 \leq \ldots \leq s_n\leq  t}\!\!\!\!\!\!\!
e^{(t-s_n)A}Be^{(s_{n}-s_{n-1})A}\ldots Be^{(s_2-s_1)A}B e^{(s_1-s)A} \psi_0\mathrm{d}u(s_1) \mathrm{d}u(s_2)\ldots
\mathrm{d}u(s_n).
$$
As $B$ is bounded,
$$
\|Y^u_{(n)}(t,s)\psi_0\|\leq e^{\omega t}
C_A^{n+1}\|B\|^n\|\psi_0\|\int_{s<s_1\leq s_2 \leq \ldots \leq s_n\leq t}
\mathrm{d}|u|(s_1) \mathrm{d}|u|(s_2)\ldots
\mathrm{d}|u|(s_n)
$$
and since $[s,t)^n$ contains the disjoint union of $\{s <s_{\sigma(1)}< s_{\sigma(2)} < \ldots < s_{\sigma(n)} \leq t\}$ over all permutations $\sigma$ of $\{1,2,\ldots,n\}$
\[
\|Y^u_{(n)}(t,s)\psi_0\|\leq e^{\omega t}
C_A^{n+1}\|B\|^n\|\psi_0\|\frac{|u|((s,t])^n}{n!}
\]
and the infinite expansion
\begin{equation}\label{Eq:UpsilonBoundedExpansion}
\Upsilon^u_{t,s} = \sum_{n=0}^\infty Y^u_{(n)}(t,s)
\end{equation}
converges in norm in the set $\Bc(H)$ of the bounded operators of $H$.

Let $(u_l)_{l\in \mathbf{N}}$ be a
sequence of piecewise constant functions on $(0,T]$ that converges to $u$ a Radon measure on $(0,T]$
in the sense that: $(u_l)_{l\in \mathbf{N}}$ is a
sequence of Radon measures $(0,T]$ and $u$ a Radon measure on $(0,T]$ such that
$u_l((0,T])$ tends to $u((0,T])$ and for every $t$ in $[0,T]$, $u_l((0,t))$
tends to $u((0,t))$ as $l$ tends to infinity with
$\sup_{l \in \mathbf{N}}|u_l|((0,T])<+ \infty$.

Before proving the convergence of $(Y^{u_l}_{(n)}(t,s)\psi_0)_l$ in $l$, we remark
that as in the proof of Proposition~\ref{prop:propagator-radon2}
for any right-continuous regulated function $F:[0,T]\to \Hc$
\[
 \int_{(s,t]}e^{(t-s_1)A}F(s_1)\mathrm{d}u_l(s_1)
\to \int_{(s,t]}e^{(t-s_1)A}F(s_1)\mathrm{d}u(s_1).
\]

For
every $(s,t)$ in $\Delta_{[0,T]}$, we have
\begin{align*}
 Y^{u_l}_{(n)}(t,s)\psi_0-Y^{u}_{(n)}(t,s)\psi_0&=
 \sum_{k=1}^n \int_{s< s_1\leq s_2 \leq \ldots \leq s_n \leq t}
 e^{(t-s_n)A}Be^{(s_{n}-s_{n-1})A}\ldots B e^{(s_2-s_1)A}\psi_0\times \\
 & \quad \times \mathrm{d}u_l(s_1)\ldots \mathrm{d}u_l(s_k)\mathrm{d}u(s_{k+1})\ldots \mathrm{d}u(s_n)\\
 &-\int_{s< s_1\leq s_2 \leq \ldots \leq s_n \leq t}
 e^{(t-s_n)A}Be^{(s_{n}-s_{n-1})A}\ldots B e^{(s_2-s_1)A}\psi_0\times\\
 & \quad \times  \mathrm{d}u_l(s_1)\ldots \mathrm{d}u_l(s_{k-1})\mathrm{d}u(s_k)\ldots \mathrm{d}u(s_n)
\end{align*}
so that for
\[
F(t)=B\int_{s<s_1\leq s_2\leq \ldots\leq s_{k-1} \leq t} e^{(t-s_{k-1})A}Be^{(s_{k-1}-s_{k-2})A}\ldots  B e^{(s_2-s_1)A}\psi_0
 \mathrm{d}u(s_1)\ldots\mathrm{d}u(s_{k-1}),
\]
the $s$ dependence being implicit, we write
\begin{align*}
\lefteqn{\|Y^{u_l}_{(n)}(t,s)\psi_0-Y^{u}_{(n)}(t,s)\psi_0\|}\\
 &\leq\sum_{k=1}^n \Big\|\int_{s<s_{k+1}\leq s_{k+2}\leq \ldots \leq s_{n}\leq t}
 e^{(t-s_n)A}Be^{(s_{n}-s_{n-1}A}\ldots \ldots  B e^{(s_{k+2}-s_{k+1})A}\times\\
& \Bigg(\int_{s<s_{k} \leq s_{k+1}} e^{(s_{k+1}-s_{k})A}F(s_k) \mathrm{d}u_l(s_k)
-\int_{s< s_{k} \leq s_{k+1}} e^{(s_{k+1}-s_{k})A}F(s_k)\mathrm{d}u(s_k)\Bigg)\times\\
&\times\mathrm{d}u_l(s_{k+1})\ldots \mathrm{d}u_l(s_n) \Big\|.
\end{align*}
As $F$ is regulated and right-continuous, we conclude on the convergence
using the previous consideration and
the Lebesgues theorem.

The differentiability properties in the bounded variation case are due to \cite[Theorem 1]{Kato1953} as
$A-\omega$ is the generator of a contraction semigroup and as $B$ in $\Bc(\Hc)$
then $B$ is also bounded for the norm  $N$. So that $A-\omega+u(t)B-R\|B\|_N$
for any $R>|u|_\infty$
satisfies the assumptions of \cite[theorem 1]{Kato1953} in the Banach space
structure associated with $N$.

The properties
\[
 \Upsilon^u_{t,s}=\Upsilon^u_{t,r}\Upsilon^u_{r,s}\quad\mbox{ for }  s<r<t, \quad \Upsilon^u_{t,t}=I_{\Hc}
\]
can be obtained by density and continuity from \cite[Theorem 1]{Kato1953} for $A-\omega+u(t)B-R\|B\|_N$ as well.
\end{proof}

Somehow $t\mapsto \Upsilon^u_{t,s}\phi_0$ can be considered as a ``mild'' solution in $\Hc$ of
\eqref{Eq:NonAutonomousCauchy} with $A(t)=A+u(t)B$.

\begin{corollary}\label{COR_bound_DA_controlBV_B_borne}
Let $A$ be with domain $D(A)$ be the generator of a strongly continuous
semigroup
and let $B$ be bounded. Then for any $u\in BV([0,T],\R)$
$\Upsilon^u_{T,0} D(A)\subset D(A)$ and  there exists $m$ (depending only on $A$, $B$
and $\|u\|_{L^\infty([0,T])}$) such that
 \[
  \|\Upsilon^u_{T,0}(\psi_0)\|_{2}\leq
me^{m\|u\|_{BV([0,T],\R)}}e^{\left|\int_0^t u\right|}
\|\psi_0\|_{2}
 \]
\end{corollary}
\begin{proof}
The proof is a consequence of estimates ~\cite[\S 3.8]{Kato1953} applied to $A-\omega+u(t)B-R\|B\|_N$ which is the generator of a contraction semigroup for the norm $N$.

\end{proof}

\begin{proposition}\label{Prop:Compactness}
 Let $A$ be the generator of a strongly continuous semi-group
and let  $B$ be bounded. Let $(u_l)_{l\in \mathbf{N}}$ be a
sequence of Radon measures $(0,T]$ such that for every $t$ in $[0,T]$, {

$\sup_{l \in \mathbf{N}}|u_l|((0,T])<+ \infty$.
Let $(t_l)_{l\in\N}\in [0,T]$.
Then $(\Upsilon^{u_l}_{t_l})_{l\in \N}$ has a strongly convergent subsequence.}
\end{proposition}
\begin{proof} Let us first notice that $(t\mapsto |u_l|((0,t]))_l$ defines a sequence of increasing bounded functions. So that we can assume
up to extracting a subsequence it is a pointwise convergent by Helly's selection theorem. Using Hahn decomposition, we have that $(t\mapsto |u_l|((0,t])-u_l((0,t]))_l$ is also a sequence of increasing functions so that one can extract a converging subsequence as well. For the rest the proof we assume that both sequences are pointwise convergent.

Let $\epsilon >0$. We make several reductions to the formula \eqref{Eq:UpsilonBoundedExpansion}. First for $S=\sup_{l \in \mathbf{N}}|u_l|((0,T])$, we have
\[
\left\|  \Upsilon^{u_l}_{t,s}- \sum_{n=0}^N Y^{u_l}_{(n)}(t,s)\right\|\leq C_A e^{\omega T}\left(\sum_{k>N}
\frac{(C_A\|B\|S)^n}{n!}\right)\|\psi_0\|.
\]
Therefore for some integer $N_0$,  this is smaller than $\epsilon$ whenever $N\geq N_0$.

Then recall that
$$
Y^{u_l}_{(n)}(t,s)\psi_0:=\int_{s<s_1\leq s_2 \leq \ldots \leq s_n\leq  t}\!\!\!\!\!\!\!
e^{(t-s_n)A}Be^{(s_{n}-s_{n-1})A}\ldots Be^{(s_2-s_1)A}B e^{(s_1-s)A} \psi_0\mathrm{d}{u_l}(s_1) \mathrm{d}{u_l}(s_2)\ldots
\mathrm{d}{u_l}(s_n).
$$
For $n<N_0$, the integrand is a continuous of all its variables in $[0,T]^{n+2}$, this can be replaced up to an error of size $\frac1{N_0}\epsilon T^{-N_0}$ by a piecewise constant function which is a finite linear combination of products of left-continuous characteristic functions. So the proof reduces to the convergence of
$$
\int_{0<s_l\leq s_1\leq s_2 \leq \ldots \leq s_n\leq  t_l}\prod_{\ell=0}^{n+1} 1_{(a_\ell,b_\ell]}(s_{\ell})\mathrm{d}{u_l}(s_1) \mathrm{d}{u_l}(s_2)\ldots
\mathrm{d}{u_l}(s_n).
$$
where $s_{n+1}=t_l$ also depends on $l$.
Since these $N_0$ sequences are all bounded we can extract convergent subsequences.
\end{proof}

{
\begin{corollary}\label{COR_int_ens-attain_B_borne} Let $A$ be the generator of a strongly continuous semi-group
and let  $B$ be bounded. Then for every $\psi_0$ in $\Hc$, the set
$$\bigcup_{T\geq 0} \bigcup_{\alpha >0} \bigcup_{u \in \mathcal{R}((0,T])}
\{ \alpha \Upsilon^u_{t,0}\psi_0| t\in [0,T]\}$$ is a meager set in $\Hc$.
\end{corollary}
\begin{proof}
Due to Proposition~\ref{Prop:Compactness}, the proof is similar to the one of Corollary \ref{Cor:NoExactControllability}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{PRO_introduction_BMS_Bborne}]
As already mentioned, the well-posedness result is classical (see \cite{BMS} for instance), while the emptiness of the attainable set with $L^1$ controls follows from Corollary
\ref{COR_int_ens-attain_B_borne} proved for Radon controls.
\end{proof}

}

Theorem~\ref{Thm:ExistenceBoundedPotential} can be generalized to include
a time dependent principal term as follows.
\begin{definition}\label{def:propagator-pcc-2}
Let  $(A,B_1)$ be a couple of operators
and $u_1:[0,T]\rightarrow \R$ be piecewise constant such that $A+u_1(t)B_1$ defined on $D(A)\cap D(B_1)$
 has maximal dissipative closure on $\Hc$ for any $t\in [0,T]$.
Let $B_2$ be a bounded operator and $u_2 : [0,T]\rightarrow \R$ be piecewise
constant.
Let  $\pro^{u_1,u_2}:[0,T]\rightarrow L(\Hc,\Hc)$
be the operator that associates, with every $t$ in
$[0,T]$ and every $u$ piecewise constant, the unitary linear transformation
 $$
\pro^{u}_{t} =    e^{(t-t_{j-1})(A +u_{1,j-1} B_1+u_{2,j-1} B_2)}
e^{(t_{j-1}-t_{j-2})(A
+u_{1,j-2} B_1+u_{2,j-2} B_2)}\cdots e^{t_1(A+u_{1,1} B_1+u_{2,1} B_2)},
$$
where $t_{j-1} \leq t<t_j$, $u_1:=\sum_{j \in \N} u_{1,j-1}
\chi_{[t_{j-1},t_{j})}$ and $u_2:=\sum_{j \in \N} u_{2,j-1}
\chi_{[t_{j-1},t_{j})}$.
\end{definition}

We consider the map $\psi(t)=\pro^{u_1,u_2}_{t} (\psi_{0})$
as a ``mild'' \emph{solution} of \eqref{Eq:NonAutonomousCauchy} with
$A(t)=A+u_1(t)B_1+u_2(t)B_2$ with initial condition $\psi_0 \in {\cal
H}$.

\begin{proposition}\label{Prop:ExistenceBoundedPotentialRadon}
Let $T>0$. Let $(A,B_1)$ satisfy Assumption~\ref{ass:StrongInteraction} and $B_2$ be a
bounded operator. Then
for every $\psi_{0} \in \Hc, t\in I$ the map $\pro_{t}(\psi_{0})$
admits a unique continuous extension on $\radon{(0,T]}\times\radon{(0,T]}$.

Moreover, if $\psi_0$ is in $D(A)$, for every $u_1\in
\radon{(0,T]}$ and $u_2\in \radon{(0,T]}$, the
map $t \mapsto
\pro^{u_1,u_2}_{t}(\psi_{0})$ is a weak solution of \eqref{Eq:NonAutonomousCauchy}
with $A(t):=A+u_1(t)B_1+u_2(t)B_2$ in the sense that it satisfies
\begin{eqnarray*}
\int_{I} \langle f'(t),  \pro^{u_1,u_2}_{t}(\psi_{0}) \rangle dt &=&
\int_{I}\langle A
f(t),   \pro^{u_1,u_2}_{t}(\psi_{0}) \rangle dt
+  \int_{I}\langle B_1 f(t),
\pro^{u_1,u_2}_{t}(\psi_{0})   \rangle du_1(t)\\
& & \quad \quad  +\int_{I}\langle B_2 f(t),
\pro^{u_1,u_2}_{t}(\psi_{0})   \rangle du_2(t)\,,
\end{eqnarray*}
for every $f \in C_{c}^{1}(I,\Hc)$.
\end{proposition}
\begin{proof}
First notice that from Proposition~\ref{prop:propagator-radon} there is a map
$\Upsilon^{u_1}_{t,s}$ such $t \mapsto \Upsilon^{u_1}(t,0)\psi_0$ is a weak solution of
\eqref{eq:main} with $B_1$ replacing $B$ and $u_1$ replacing $u$.
Then the proof mimics the proof of Theorem~\ref{Thm:ExistenceBoundedPotential}
using Duhamel's expansion, Corollary~\ref{Cor:Duhamel-Radon}, with respect to $\Upsilon_t^{u_1}\psi_0$ and the fact that
$\Upsilon^{u_1}_{t,s}$ is a contraction.
\end{proof}
\begin{remark}
 In the cases considered later $u_1$ and $u_2$ will be the same control $u$.
\end{remark}

\paragraph{A comment on the case of bounded operators}
We take advantage of this part of our analysis devoted to bounded  control potential
to make some general remarks on propagators associated with bounded operators.
\begin{proposition}\label{Prop:PropagatorBoundedPotential}
Let $t\in [0,T]\mapsto B\in \Bc(\Hc)$ be in $L_{s,loc}^1([0,T],\Bc(\Hc))$, that is
strongly measurable and strongly locally integrable.
Then there exists a unique family of operators
$\Upsilon_B:\Delta_{[0,T]}\to L(\Hc)$ strongly
continuous in $t$ and $s$  such that
\begin{enumerate}[label=(\roman*)]
\item for any $s<r<t$, $\Upsilon_B(t,s)=\Upsilon_B(t,r)\Upsilon_B(r,s)$,
\item $\Upsilon_B(t,t)=I_{\Hc}$,
\end{enumerate}
and for any $\psi_0\in\Hc$, $(s,t)\in\Delta_{[0,T]}\mapsto \Upsilon_B(t,s)\psi_0$
is almost everywhere
strongly left
differentiable in $t$ and  right differentiable in $s$ with
derivative (when $t=s$) $B(t)\psi_0$ and $-B(t)\psi_0$ respectively.

The map
\[
B\in L_{s,loc}^1([0,T],\Bc(\Hc))\mapsto \Upsilon_B\in L^\infty(\Delta_{[0,T]},\Bc(\Hc))
\]
is continuous.

The family $\Upsilon_B$ is called propagator associated with $B$.
\end{proposition}
\begin{proof}
The family $\Upsilon_B$ is built in the strong sense by generalizing the Dyson expansion as in Theorem~\ref{Thm:ExistenceBoundedPotential} :
\[
\Upsilon_B(t,s) = \sum_{n=0}^\infty Y^B_{(n)}(t,s)
\]
with
\[
Y^B_{(n)}(t,s)\psi_0:=\int_{s\leq s_1< s_2 < \ldots < s_n < t}
B(s_n)B(s_{n-1})\ldots B(s_2)B(s_1) \psi_0\mathrm{d}s_1 \mathrm{d}s_2\ldots
\mathrm{d}us_n.
\]
The convergence follows exactly from similar arguments and provides the continuity. The differentiability properties are a consequence of Lebesgue's differentiation Theorem. Notice that the uniqueness of absolutely continuous function $t\in[0,T]\mapsto U(t)\in\Hc$ with
Lebesgue derivative  $t\mapsto B(t)U(t)$ and such that $U(0)=\psi_0$ follows from Lebesgue's differentiation Theorem and Gronwall's Lemma.

The fact that $\Upsilon_B(t,t)$ is the identity can be checked immediately while
\[
\Upsilon_B(t,s)\psi_0=\Upsilon_B(t,r)\Upsilon_B(r,s)\psi_0,
\]
for any $s<r<t$, is a consequence of the uniqueness statement just mentioned.
\end{proof}

\section{Higher order norm estimates for \FEPS\
systems}\label{Sec:HigherOrderFEPS}

In the following we will restrict our analysis to the skew-adjoint case. This
are the objects that appear in quantum mechanics. Moreover the restriction to
self-adjoint operators makes the analysis simpler.

Our aim in this section is to analyze under which conditions the solution built
previously are smoother in the scale of $A$. This is indeed the case if the
assumptions of Assumptions~\ref{ass:ass},
\ref{ass:Interacting} or \ref{ass:StrongInteraction} are stated in $D(|A|^{k/2})$
instead of $\Hc$. Our aim is to provide a somewhat simpler
criteria, we show that the extension of assumptions on $B$ will be sufficient. { In
this respect, the $A$-boundedness of $B$ as operators acting on $D(|A|^{k/2})$ is crucial
and is stated in Lemma~\ref{lem:KatoRellichBoundWeakCoupling} which is the cornerstone of
the analysis of this section. This is especially important if we want to obtain the
regularity of propagators in the scale of $A$ up to the order $k/2$. For lower orders, a
simple interpolation provides the results.}
The criteria will be used in a perturbative framework (Kato-Rellich type argument) and we
will not consider
the whole of $K$ for the values of $u$, unless we assume that the domain of powers of $A+uB$
are the same for any $u\in K$. We recall that in the dissipative framework in order to use
Kato-Relich criterion $u$ has to be non negative when $B$ is dissipative, below we assume
that both $B$ and $-B$ have dissipativity properties (up to a shift by a constant) so
that the sign of $u$ does not have to be considered.
\subsection{The weak coupling}

\begin{definition}[\feps]\label{Def:WeakCoupling}
Let $k$ be a non negative real. A couple of \emph{skew-adjoint}
operators $(A,B)$
is \emph{$k$-\FEPS\ } if
\begin{enumerate}[label=(\roman*)]
 \item $A$ is invertible with bounded
inverse from $D(A)$ to $\Hc$\label{Assumption:FEPSOnA},
 \item for any real $t$, $e^{tB}D(|A|^{k/2})\subset D(|A|^{k/2})$,
 \item there exists $c\geq 0$ and $c'\geq 0$ such that $B-c$
and $-B-c'$ generate contraction semigroups on $D(|A|^{k/2})$ for the norm
$\psi
\mapsto \||A|^{k/2}u\|$.
 \end{enumerate}
\end{definition}

We set, for every positive
real $k$,
$$
\|\psi\|_{k/2} = \sqrt{   \langle  | A|^{k} \psi, \psi \rangle}.
$$
The optimal exponential growth is defined by
\[
 c_k(A,B):=\sup_{t\in \R}\frac{\log\|e^{tB}\|_{L(D(|A|^{k/2}),
D(|A|^{k/2})}}{|t|}.
\]

\begin{remark}
As in Section~\ref{Sec:BoundedPotential}, if
\begin{equation*}
t \mapsto  e^{tB}
\end{equation*}
is a strongly continuous semigroup on $D(|A|^{k/2})$ then there
exists
$\omega>0$ and $C>0$ such that
\[
 \|e^{tB}\|_{L(D(|A|^{k/2}),D(|A|^{k/2})}\leq C e^{\omega t}, \forall
t>0.
\]
Up to a change of norm, namely consider the equivalent norm
\[
 \sup_{t>0}\|e^{t(B-\omega)}\psi\|_{k/2}
\]
then one can assume $C=1$ but may loose the hilbertian structure and the skew
adjoint character of $A$.

\begin{remark}\label{rem:DomainB}
Notice that the content of Definition~\ref{Def:WeakCoupling}.(iii)
is that the restriction to $D(|A|^{k/2})$ of the group generated by $B-c$ originally
defined on $\Hc$ defines a contraction semigroup. We thus have two generators and due to
Definition~\ref{Def:WeakCoupling}.(i)-(ii) the domain of the second is included in the first.
Hence again Definition~\ref{Def:WeakCoupling}.(i) gives that the smallest domain is dense in $\Hc$.
The same comment can be made for $-B-c'$. Forgeting about $c$ and $c'$ and minus sign,
we identify these three operators (which are closed in $\Hc$ and $D(|A|^{k/2})$ respectively)
and with an abuse of notation
we denote them by the same $B$ as they are restrictions of $B$. The domains of $B-c$ and $-B-c'$ acting on $D(|A|^{k/2})$ are
actually both equal to $\{\phi \in D(|A|^{k/2})\cap D(B),\,B\phi \in D(|A|^{k/2})\}$. They both contain this set
as if $\phi$ is in this set then
\[
 e^{\pm tB}\phi-\phi\mp tB\phi=\int_0^t (e^{\pm tB}-1)B\phi ds
\]
is a $o(|t|)$ in $\Hc$ or $D(|A|^{k/2})$. They are obviously contained in this set as if
$t\mapsto e^{\pm tB}\phi$ is differentiable in $D(|A|^{k/2})$ it is also differentiable in $\Hc$.
\end{remark}

The invertibility of $A$ is needed to ensure that $\|\cdot\|_{k/2}$ is a norm
equivalent to the graph norm of $D(|A|^{k/2})$. The use of the associated norm
is due to the interpolation criterion we use below.

\end{remark}

For the systems encountered in the physics literature, the operator $A$ is skew-adjoint
with at least two isolated eigenvalues. Hence the invertibility of $A$ can be obtained by
replacing $A$ by $A-\lambda \mathrm{i}$ for a suitable $\lambda$ in $\R$. Notice that this
translation on $A$ only induces a global phase shift on the propagator that is physically
irrelevant (i.e., undetectable  by observations).

The following proposition gives another characterization of the weak-coupling
using Hille-Yosida Theorem.
\begin{proposition}\label{Prop:ThmHilleYosida}
Let $k$ be a non negative real. A couple of skew-adjoint operators
$(A,B)$ { with $A$ invertible}
is \emph{$k$\-{-}\FEPS\ } if
and only if $B$ is closed in $D(|A|^{k/2})$,
and there exists $\omega$ such that for every real $\lambda$, $|\lambda|>\omega$
belongs to the resolvent set of $B$ and for such $\lambda$
\[
\|(\lambda
I- B)^{-1}\|_{L(D(|A|^{k/2}),D(|A|^{k/2})}\leq\frac{1}{|\lambda|-\omega}.
\]
The smallest $\omega$ satisfying the above property is
$c_k(A,B)$.
\end{proposition}
\begin{proof}
If $(A,B)$
be \emph{$k$-\FEPS\ } then $B-c_k(A,B)$ is the generator of a contraction
semigroup in $D(|A|^{k/2})$. From Hille-Yosida Theorem, we deduce the stated
equivalence.
\end{proof}

The following proposition gives an equivalent defintion which may be easier to
check in practice.
\begin{proposition}\label{Prop:ThmLumerPhillips}
Let $k$ be a non negative real. A couple of skew-adjoint operators $(A,B)$
{ with $A$ invertible} is $k$-\FEPS\,
if and only if for some $\omega>0$,
\[
 (\omega\mp B)^{-1}D(|A|^{k/2})\subset D(|A|^{k/2})
\]
and for any $\psi \in (\omega- B)^{-1}D(|A|^{k/2})=(\omega+ B)^{-1}D(|A|^{k/2})$
\[
| \Re \langle |A|^k\psi,B\psi\rangle | \leq \omega
\|\psi\|_{D(|A|^{k/2})}^2.
\]
The smallest $\omega$ satisfying this property is
$c_k(A,B)$.
\end{proposition}
\begin{proof}
We first notice that  from the resolvent identity, the assumptions imply
\[
 (\omega- B)^{-1}D(|A|^{k/2})
\subset (\omega+ B)^{-1}D(|A|^{k/2})
\]
for any $\omega$ in the resolvent sets of $B$ and $-B$.

Assume $(A,B)$ is $k$-\FEPS\,
then as $B-c_k(A,B)$ and $-B-c_k(A,B)$ are generator of contraction semigroups
on $D(|A|^{k/2})$,
they are closed and maximal dissipative on $D(|A|^{k/2})$,
their respective resolvent sets contains positive half lines (by means of
Hille-Yosida theorem) and their domain is, by definition of the resolvent,
$(\omega\pm B)^{-1}D(|A|^{k/2})$ for any $\omega>c_k(A,B)$. Since they are maximal dissipative,
we have
\[
\pm \Re \langle |A|^k\psi,B\psi\rangle  \leq c_k(A,B) \|\psi\|_{D(|A|^{k/2})}^2.
\]
for any $\psi \in (\omega- B)^{-1}D(|A|^{k/2})=(\omega+ B)^{-1}D(|A|^{k/2})$.

Reciprocally, $\pm B+\omega$ are closed as operators on
$\Hc$ so they are closed on $D(|A|^{k/2})$. As $\pm B+\omega$ are
dissipative on $D(|A|^{k/2})$, they are generators of a
contractions semigroups if they are surjective. As for any $f\in D(|A|^{k/2})$,
$(\pm B +\omega)^{-1}f \in D(|A|^{k/2})$, they are indeed surjective.

\end{proof}
The notion of weak coupling was introduced
in~\cite{BoussaidCaponigroChambrion} but in a somewhat simpler form the
relation
between
these two definitions can be obtained by means of the following
lemma.
\begin{lemma}\label{LEM_lien_WC_nouvelle_definition_ancienne_definition}
Let $A$ be  an invertible skew-adjoint operator and let $B$ be a skew-symmetric
operators such that
 for every $u$ in $\mathbf{R}$, $A+uB$ is essentially
skew-adjoint on the domain $D(A)$ of $A$
and
 for some $k\geq 1$, for any real $u$,
$D(|A+uB|^{k/2})=D(|A|^{k/2})$ and
 there exists
a constant $C$ such that for every $\psi$ in $D(|A|^k)$,
\[
|\Re \langle |A|^k
\psi,B\psi \rangle |\leq C |\langle |A|^k \psi, \psi \rangle|.
\]
Then $(A,B)$ is $k$-\FEPS\ .
\end{lemma}
\begin{proof}

Since for every $u$ in $\mathbf{R}$, $A+uB$ is essentially
skew-adjoint on the domain $D(A)$, this implicitly imply that $D(A)\subset D(B)$.

The assumption that there exist $k\geq 1$ and
a constant $C$ such that for every $\psi$ in $D(|A|^k)$,
\[
|\Re \langle |A|^k
\psi,B\psi \rangle |\leq C |\langle |A|^k \psi, \psi \rangle|
\]
and the Nelson commutator theorem, see Section~\ref{Sec:Nelson}, imply that
$B$ with domain $D(|A|^{k/2})$ is essentially skew-adjoint so $B$ is essentially
skew-adjoint on $D(A)$. Then Trotter product formula, see~\cite[Theorem VIII.31]{reed_simon}, implies that :
\[
 \left(e^{\frac{t}{n}(A+uB)}e^{-\frac{t}{n}A}\right)^n
\]
tends to $e^{t u B}$ in the strong sense as $n$ goes to infinity. As each of
the term of the above  sequence is bounded on $D(|A|^{k/2})$ with a bound
$e^{C|t||u|}$, see~\cite[proposition 2]{BoussaidCaponigroChambrion}, we conclude
that $e^{tB}$ is bounded on $D(|A|^{k/2})$ with the same bound : $e^{C|t||u|}$.
This provides $(A,B)$ is $k$-\FEPS\ .
\end{proof}

Let us state state an interpolation result.
\begin{lemma}\label{Lem:kfeps-sfeps}
Let $k$ be a positive real. If $(A,B)$
is $k$-\FEPS\  then $(A,B)$
is $s$-\FEPS\  for any $s\in [0,k]$
and
\[
 c_s(A,B)\leq \frac{s}{k}c_k(A,B).
\]
\end{lemma}
\begin{proof}
We will consider $s\in(0,k)$ as there is noting to prove for $s=k$ and $s=0$ is
immediate since $B$ is skew adjoint.

Notice that as $B$ is skew-adjoint
\[
 \|e^{tB}u\|_{D(|A|^{k/2})}=\||A|^{k/2}e^{tB}u\|=\||{\cal A}(t)|^{k/2}u\|.
\]
where ${\cal A}(t)=e^{-tB}Ae^{tB}$.

As $(A,B)$ is \emph{$k$-\FEPS\ } we deduce
\[
\frac1{\|A^{-1}\|^k}\leq |{\cal A}(t)|^k\leq e^{2c |t|} |A|^k.
\]
which from Proposition~\ref{Prop:Interpolation} yields
\[
 |{\cal A}(t)|^s\leq e^{2cs|t|/k} |A|^s
\]
which concludes the proof.
\end{proof}

A corollary of this interpolation result is the following
result which is crucial in our analysis.
It shows
that  if $(A,B)$ is $k$-\FEPS\ the $A$-boundedness of $B$
extends naturally to $\Hc=D(|A|^{k/2})$.
\begin{lemma}\label{lem:KatoRellichBoundWeakCoupling}
 Let $k$ be a non-negative real. Let $(A,B)$ be $k$-\FEPS\ and such that $B$ is $A$-bounded.
Then
\[
\inf_{\lambda >0}
\|B(A-\lambda)^{-1}\|_{L(D(|A|^{\frac{k}{2}}),D(|A|^{\frac{k}{2}}))}
\leq \|B\|_A
\]
\end{lemma}
\begin{proof}
Let $\lambda$ be a real.
As $(A,B)$ is $k$-\FEPS,
 for any $s$ in $[0,k]$ due to Lemma~\ref{Lem:kfeps-sfeps}, and
Proposition~\ref{Prop:ThmLumerPhillips}
\[
|\langle \psi, [|A|^s,B]\psi\rangle| \leq \frac{s}{k}c_k(A,B)
\langle \psi,|A|^{s}\psi\rangle
\]
for any $\psi\in D(|A|^{s/2})\cap D(B)$ with $B\psi\in D(|A|^{s/2})$ so that that
\[
|A|^{-s/2} [|A|^s,B]|A|^{-s/2}
\]
extends to a bounded operator in $\Hc$ with bound lower than $\frac{s}{k}c_k(A,B)$. Then for $\phi\in D(|A|^{k/2})\cap D(B)$ with $B\phi\in D(|A|^{k/2})$,
we notice
\begin{align*}
 \||A|^{k/2}B\phi\|^2&=\langle B\phi,|A|^{k}B\phi\rangle\\
 &=\langle B\phi,[|A|^{k},B]\phi\rangle+\langle B\phi,B|A|^{k}\phi\rangle\\
 &=\langle |A|^{k/2}B\phi,|A|^{-k/2}[|A|^{k},B]|A|^{-k/2}|A|^{k/2}\phi\rangle+\langle B\phi,B|A|^{k}\phi\rangle\\
&\leq c_k(A,B)\||A|^{k/2}B\phi\|\||A|^{k/2}\phi\| + |\langle B\phi,B|A|^{k}\phi\rangle|
\end{align*}
and for $s\geq 0$

\[
 \langle B\phi,B|A|^{k}\phi\rangle=\langle |A|^{k/2}B\phi,|A|^{s-k/2}B|A|^{k-s}\phi\rangle+\langle |A|^{s/2}B\phi,|A|^{-s/2}[|A|^{s},B]|A|^{-s/2}|A|^{k-s/2}\phi\rangle
\]
so that if we impose
\[
0\leq s-k/2<k/2,\quad s/2<k/2,\quad k-s/2 < k/2+1
\]
or
\[
k/2\leq s<k,\quad k \leq s+2
\]
or if $k\in(s,\min\{s+2,2s\})$ for $s\geq 0$ this leads to
\[
 |\langle B\phi,B|A|^{k}\phi\rangle|\leq \||A|^{k/2}B\phi\|\||A|^{s-k/2}B|A|^{k-s}\phi\|+\frac{s}{k}c_k(A,B)\||A|^{s/2}B\phi\|\||A|^{k-s/2}\phi\|
\]
and as $s<k$ to
\[
  \||A|^{k/2}B\phi\|\leq c_k(A,B)\||A|^{k/2}\phi\| + \||A|^{s-k/2}B|A|^{k-s}\phi\|+\frac{s}{k}c_k(A,B)\||A|^{\frac{s-k}{2}}\|\||A|^{k-s/2}\phi\|.
\]

Since $s-k/2\in [0,k/2)$ a recurrence\footnote{This can be replaced by a connectedness argument on the set of $k'\in [0,k/2]$ for which the result holds} emerges.  First the value $s=k/2$ is allowed when $0<k<k/2+2$ that is $k<4$ and thus from the above inequality, we deduce, for every $\lambda>0$,
\begin{align*}
\lefteqn{   \||A|^{k/2}B(A-\lambda)^{-1}\phi\|}\\
&\leq c_k(A,B)\||A|^{k/2}(A-\lambda)^{-1}\phi\| + \|B|A|^{k/2}(A-\lambda)^{-1}\phi\|+\frac{1}{2}c_k(A,B)\||A|^{\frac{-k}{4}}\|\||A|^{3k/4}(A-\lambda)^{-1}\phi\|
\end{align*}
and since $3/4 k< k/2+1$ here, we deduce
\[
\lim_{\lambda \to \infty}  \||A|^{k/2}B(A-\lambda)^{-1}\phi\|\leq \|B\|_A\||A|^{k/2}\phi\|.
\]
This initialize the recurrence. Then as $k-s/2<k/2+1$
\[
\lim_{\lambda \to \infty}    \||A|^{k/2}B(A-\lambda)^{-1}\phi\|\leq \lim_{\lambda \to \infty}\||A|^{s-k/2}B(A-\lambda)^{-1}|A|^{-s+k/2}\|\||A|^{k/2}\phi\|
\]
As the recurrence provides
\[
 \lim_{\lambda \to \infty}\||A|^{s-k/2}B(A-\lambda)^{-1}|A|^{-s+k/2}\|\leq \|B\|_A
\]
this concludes the proof due to the density of the set
\[\{\phi \in D(|A|^{k/2})\cap D(B),\,B\phi \in D(|A|^{k/2})\}\]
in $D(|A|^{k/2})$.
\end{proof}

The following corollary shows that $D(|A+uB|^{k/2})=D(|A|^{k/2})$.
It seems a difficult issue otherwise.
Recall that as $D(A)\subset D(B)$ and $A+uB$ denotes a self-addjoint extension of $A+uB$,
we have $D(A)=D(A+u B)$ by Kato-Rellich theorem.
\begin{lemma}\label{lem_domaines_egaux}\label{lem:NormesEquivalentes}
 Let $k$ be a positive real, $(A,B)$ be $k$-\FEPS\ and $u$ be a real number such that
$|u|<\|B\|_A$. Then $D(|A|^s)=D(|A+u B|^s)$ for
every $s\in[0, k/2 +1]$.
\end{lemma}
\begin{proof}
We proceed by induction on $j$ to prove $D(|A|^{k/2-\lfloor k/2\rfloor +j})=D(|A+uB|^{k/2-\lfloor k/2\rfloor +j})$ for $j\leq \lfloor k/2\rfloor$.
By Kato-Rellich theorem, $D(A)=D(A+uB)$ for every $u$ in $\mathbf{R}$. By interpolation, see Lemma~\ref{Lem:DomainInterpolation}, $D(|A|^s)=D(|A+uB|^s)$ for $0\leq s \leq 1$, in particular for $s=\frac{k}{2}-\lfloor \frac{k}{2} \rfloor$. This initializes the induction for $j=0$.

Let us assume that
 $D(A^\ell)=D((A+u B)^\ell)$ for some $\ell \leq k/2$.
By definition,
$$D(A^{\ell+1})=\{f \in D(A^\ell)|Af \in D(A^\ell)\},$$
and, using the
induction hypothesis,
$$D(|A+u B|^{\ell+1})=\{f \!\in\! D(|A+u B|^\ell)||A+u B| f \! \in D(|A+u B|^\ell)\}=
\{f \!\in D(A^j)|A+u B)f \in D(|A|^\ell)\}.$$

So that $D(|A+u B|^{\ell+1})$ is the domain of $A+uB$ as an operator acting on $D(|A|^\ell)$. The domain of $A$
as an operator acting on $D(|A|^\ell)$ is $D(|A|^{\ell+1})$, as $A$ is skew adjoint on $D(|A|^\ell)$ and $B-c$ or
$-B-c'$ is dissipative in $D(|A|^\ell)$ due to Proposition~\ref{Prop:ThmLumerPhillips}, using Lemma~\ref{lem:KatoRellichBoundWeakCoupling}
and Kato-Rellich theorem we conclude that $A+uB$ with domain $D(|A|^{\ell+1})$ is maximal dissipative in $D(|A|^\ell)$.

\end{proof}
Notice that if $D(|A+uB|^s)=D(|A|^s)$ for some positive real $s$ then the associated norms are equivalent
as the operators are closed.

\subsection{Higher regularity}
\label{SEC_Higher_regularity}
From
Remark~\ref{Rem:NonInteractingCase} and Proposition~\ref{prop:continuity}, we
deduce the following statement.
\begin{proposition}\label{PRO_cont_entree_sortie_Hk-BV}
Let $k$ be a non negative real.
Let $(A,B)$ be $k$-\FEPS\ {and $B$ be $A$-bounded}.

For any $u\in BV([0,T],\R)\cap B_{L^\infty([0,T])}(0,1/\|B\|_A)$, there exists
a family of contraction propagators in $\Hc$ that extends uniquely as contraction propagators
to $D(|A|^{k/2})$ : $\Upsilon^u:\Delta_{[0,T]}\to L(D(|A|^{k/2}))$ such that
\begin{enumerate}[label=(\roman*)]
 \item for any $t\in [0,T]$, for any $\psi_0\in D(|A|^{k/2})$
 \[
  \|\pro_{t}(\psi_0)\|_{k/2}\leq e^{c_k(A,B) \int_0^t|u|}  \|\psi_0\|_{k/2}
 \]\label{Contraction}
 \item for any $t\in [0,T]$, for any $\psi_0\in D(|A|^{1+k/2})$ for any $u\in BV([0,T],\R)\cap B_{L^\infty([0,T])}(0,1/\|B\|_A))$,
 there exists $m$ (depending only on $A$, $B$ and $\|u\|_{L^\infty([0,T])}$)
 \[
  \|\pro_{t}(\psi_0)\|_{1+k/2}\leq me^{m\|u\|_{BV([0,T],\R)}}e^{c_k(A,B)\int_0^t \left|u\right|}  \|\psi_0\|_{1+k/2}
 \]\label{Domain}
\end{enumerate}
Moreover, for every
$\psi_{0}$ in $D(|A|^{k/2})$, the end-point
mapping
\begin{align*}
\pro(\psi_{0}):BV([0,T],K)&\rightarrow  D(|A|^{k/2})\\
u&\mapsto \Upsilon^u(0,T)(\psi_{0})
\end{align*}
is continuous.
\end{proposition}
\begin{proof}
The first part of the statement is a corollary of Theorem~\ref{thm:kato}
after replacing $B$ by $B-c_k(A,B)$ to ensure the dissipativity. Indeed
consider $D(|A|^{\frac{k}{2}})$ in place of $\Hc$. Notice first
that from Lemma~\ref{lem:KatoRellichBoundWeakCoupling},
if $a> 1/\|B\|_A$ such that $a\|u\|_\infty<1$ then there exists $b_a$ such
that for any $\psi\in D(|A|^{1+k/2})$
\[
 \|(1-A-uB) \psi\|_{k/2}\geq  \|(1-A)\psi\|_{k/2}- |u|\|B \psi\|_{k/2}\geq (1-a|u|)\|(1-A)\psi\|_{k/2}-b_a\|\psi\|_{k/2}
\]
or
\[
 \|(1-A-uB) \psi\|_{k/2}+b_a\|\psi\|\geq  (1-a|u|)\|(1-A)\psi\|_{k/2}
\]
leading to
\[
 \|B\psi\|_{k/2}\leq a\|(1-A)\psi\|_{k/2}\leq \frac{a}{1-a|u|}\|(1-A-uB) \psi\|_{k/2}+\frac{b_a}{1-a|u|}\|\psi\|_{k/2}.
\]
This gives
\begin{align*}
 \|(1-A(t))^{-1}\|_{L(D(|A|^{k/2}),D(|A|^{1+k/2}))}&=\|A(1-A-u(t)B)^{-1}\|_{L(D(|A|^{k/2}))}\\
 &\leq \|(A+u(t)B)(1-A-u(t)B)^{-1}\|_{L(D(|A|^{k/2}))}\\
 &\qquad+|u(t)|\|B(1-A-u(t)B)^{-1}\|_{L(D(|A|^{k/2}))}\\
 &\leq 2+|u(t)|\frac{a}{1-a|u(t)|}+\frac{b_a}{1-a|u(t)|}.
\end{align*}
Taking the supremum leads to
\[
\sup_{t\in[0,T]} \|(1-A(t))^{-1}\|_{L(D(|A|^{k/2}),D(|A|^{1+k/2}))}
\leq 2+\frac1{\|B\|_A}\frac{a}{1-\frac{a}{\|B\|_A}}+\frac{b_a}{1-\frac{a}{\|B\|_A}}.
\]
Then Remark~\ref{Rem:NonInteractingCase} gives the result using
$(A,B)$ to be $k$-\FEPS\, using Lemma~\ref{lem:KatoRellichBoundWeakCoupling} and
Kato-Rellich theorem for dissipative operators (see \cite[Corollary of Theorem X.50]{reed_simon}).

As moreover for $A_n(t)=A+u_n(t)B$ and $A(t)=A+u(t)B$ and $a$ a sufficiently large real
$\|A_n\|_{BV(I,L(D(A),\Hc))} \leq \|u_n\|_{BV(I,\R)}\|B\|_{L(D(A),\Hc)} $,
$\|A_n(0)\|_{L(D(A),\Hc))}\leq 1+|u_n(0)|\|B\|_{L(D(A),\Hc)}$.
and
\[
 (A_n(t)-a)^{-1}-(A(t)-a)^{-1}=(u_n(t)-u(t))(A_n(t)-a)^{-1}B(A(t)-a)^{-1}
\]
so that the strong resolvent convergence of $A_n$ to $A$ turns to be a
consequence of the convergence of $u_n$ to $u$ in $BV(I,\R)$. Proposition~\ref{prop:continuity}
applies as well.
\end{proof}

{
We now state another version of
Corollary~\ref{Cor:NoExactControllability}.
\begin{corollary}\label{Cor:NoExactControllability2}
Let $k$ be a non negative real. Let
$(A,B)$ be $k$-\FEPS\, $B$ $A$-bounded  and $\psi_0\in D(|A|^{k/2})$. Then
\[
\bigcup_{L,T,a >0}  \left\{ \alpha \Upsilon^{u}(\psi_{0}),
\|u\|_{BV([0,T],(-1/\|B\|_A,1/\|B\|_A))} \leq L, t\in [0,T], |\alpha|\leq a\right\}
\]
is  a meagre set (in the
sense of Baire) in
$D(|A|^{k/2})$ as a union of relatively compact subsets.
\end{corollary}
\begin{proof}
 This is similar to the one of Corollary~\ref{Cor:NoExactControllability}.
\end{proof}}

\paragraph{A comment on the exact controllability of time reversible system.}\label{sec:timereversibility}
Some comments on the exact controllability of time reversible system are in place.

Let $(A,B,K)$ satisfy Assumption~\ref{ass:ass} (or Assumptions~\ref{ass:Interacting}).
Let $\mathcal{C}$ be a bounded antilinear operator on $\Hc$ such that
$\mathcal{C}D(A)\subset D(A)$, $\mathcal{C}A=-A\mathcal{C}$, and
$\mathcal{C}B=-B\mathcal{C}$.
Let $u$ be piecewise constant with value in $K$
and let $\psi(t)$ be a strong solution of \eqref{eq:main} associated with the
control $t \mapsto u(t)$, then $ \varphi(t)=\mathcal{C} \psi(T-t)$ satisfies
\begin{align*}
\frac{d}{dt} \varphi(t) & = - \mathcal{C} (\psi'(T-t))\\
& = - \mathcal{C} (A+ u(T-t)B ) (\psi(T-t)) \\
& = A \varphi(t)  + u(T-t) B \varphi(t).
\end{align*}
and therefore
 is a strong solution of \eqref{eq:main} associated with the control $t \mapsto
u(T-t)$.

This well-known fact for the Schr\"odinger equation (see for
example~\cite{Nersy}) has been remarked for our abstract framework
in~\cite[Section~6.1]{Schrod2}. Using Proposition~\ref{prop:propagator-radon},
it can be extended for the case in which $u$ is a Radon measure as stated in the
following proposition.

\begin{proposition}
Let $(A,B,K)$ satisfy Assumption~\ref{ass:ass} (or Assumption~\ref{ass:Interacting}) with $A$ skew-adjoint and $B$ skew-symmetric.
Let $\mathcal{C}$ be a conjugation, that is a bounded antilinear operator on $\Hc$ with $\mathcal{C}\mathcal{C}=\Id$, such that
$\mathcal{C}D(A)\subset D(A)$, $\mathcal{C}A=-A\mathcal{C}$, and
$\mathcal{C}B=-B\mathcal{C}$.
Let $u$ be a bounded variation function (or a Radon measure) on $(0,T)$ with value in $K$
and let $\Upsilon_u$ be the associated contraction propagator.
For any $(t,s)\in \Delta_{[0,T]}$, $\Upsilon^u_{t,s}$ is unitary and its inverse coincides with
$\mathcal{C}\Upsilon_{u(T-\cdot)}(T-t,T-s)\mathcal{C}$ where $u(T-\cdot)$ denotes $t\in[0,T]\mapsto u(T-t)$ in the framework of Assumption~\ref{ass:ass} (or $t\in[0,T]\mapsto u((0,T])-u((0,t])=u([t,T))$ in the one of Assumption~\ref{ass:Interacting}).
\end{proposition}
\begin{proof}
The fact that the propagator is unitary is a consequence of the skew-adjointness.
The invertibility follows and to identify the inverse it is enough to differentiate
\[
 s\mapsto \Upsilon^u_{t,s}\mathcal{C}\Upsilon^{u(T-\cdot)}_{T-t,T-s}\mathcal{C}
\]
in $s$ when $u$ is continuous. The derivative is $0$ while
\[
 \Upsilon_u(t,t)\mathcal{C}\Upsilon_{u(T-\cdot)}(T-t,T-t)\mathcal{C}=\Id.
\]

Based on the continuity property, the identity
\[
\Upsilon^u_{t,s}\mathcal{C}\Upsilon_{u(T-\cdot)}(T-t,T-s)\mathcal{C}=\Id
\]
extends to all $u$ in the framework of Assumption~\ref{ass:ass} (or Assumption~\ref{ass:Interacting}).
\end{proof}
\begin{remark}
Proposition~\ref{PRO_cont_entree_sortie_Hk-BV} shows that for time reversible systems,
the input-output mapping does not change the regularity with
respect to $A$. As eigenvectors belong to any $D(|A|^k)$ this shows that exact
controllability clearly relies on the regularity of $B$ in the scale of $A$.
\end{remark}

\subsection{Extension to Radon measures } \label{SEC_weak_coupling_for_radon}

The conclusion of Proposition~\ref{rem:compactness} can be extended
to $D(|A|^{k/2})$ if Assumption~\ref{ass:commutator} is true in $D(|A|^{\frac{k}{2}})$
instead of $\Hc$.
This is indeed the only missing assumption to apply
Corollary~\ref{cor:continuity} with $D(|A|^{\frac{k}{2}})$
instead of $\Hc$. Without this assumption the following result together with
the interpolation result of Lemma~\ref{LEM_interpolation} gives an interesting
extension.

\begin{proposition}\label{prop:0987}
Let $k$ be a positive real. Let $(A,B,K)$ satisfy
Assumption~\ref{ass:Interacting} such that
$(A,B)$ be $k$-\FEPS. Then, $\Upsilon$ admits a unique continuous
extension to the set of Radon measures and, for every $s\in[0,k]$, $\psi_{0} \in
D({|A|}^{s/2})$, for every
$T\geq 0$, one has $\pro^{u}_{T}(\psi_{0})\in D({|A|}^{s/2})$ and
 \[
 \|\pro^{u}_{T}(\psi_{0})\|_{{s/2}} \leq
e^{\frac{s}{k}c_k(A,B)|u|([0,T]) }\|\psi_{0}\|_{{s/2}}
 \]
if $u\in \radon{(0,T]}$ with $u((0,t])\in K$  when $t\in [0,T]$.
\end{proposition}
\begin{proof}
We will make the proof for $s=k$, as from Lemma~\ref{Lem:kfeps-sfeps} the proof
will be the same.

Consider a sequence $v_n$ of piecewise constant functions converging to
$t\in[0,T]\mapsto u((0,t])$ pointwise with $\|v_n\|_{BV([0,T])}\leq K$. Then
$v_n$ is the cumulative function of $v_n'$, a discrete sum of Dirac delta
functions and, from \eqref{Eq:InteractionFramework}, $\pro^{v_n'}_t$ is a
product of unitary operators of the form
$$e^{vB} e^{-vB} e^{tA} e^{vB}=e^{t  A} e^{vB}.$$
So that
$$\|e^{vB} e^{-vB} e^{tA} e^{vB}\psi\|_{k/2}=\|e^{vB}\psi\|_{k/2}\leq
M(v)\|\psi\|_{k/2}$$
where $M(v):=\|e^{vB}\|_{L(D(|A|^{k/2}),D(|A|^{k/2}))}$. Noticing that
Definition~\ref{Def:WeakCoupling} and $ M(v_1+v_2)\leq M(v_1)M(v_2)$ for any
pair $(v_1,v_2)$ in $[0,\delta]^2$
imply
\[
 M(v)\leq e^{c_k(A,B)|v|}, \forall v \in \R.
\]
Hence, for every $n$,
$$
 \|\pro^{v'_n}_{t}(\psi_{0})\|_{{k/2}} < e^{c_k(A,B)K}.
 $$
For every $f$ in $D(|A|^k)$,
$$|\langle |A|^k f, \pro^{v'_n}_t \psi_0 \rangle| \leq \|f\| \|\psi_0\|_{k/2}
e^{c_k(A,B) K}.
$$
Because of the continuity result (Proposition \ref{prop:continuity}), the left
hand side tends to $|\langle |A|^k f, \pro^{u}_t \psi_0 \rangle|$ as $n $ tends
to infinity. The right hand side is bounded by $\|u\| \|\psi_0\|_{k/2} e^{
c_k(A,B)K}$.
Hence, for every $f$ in $D(|A|^k)$
$$|\langle |A|^k f, \pro^{u}_t \psi_0 \rangle|\leq \|f\| \|\psi_0\|_{k/2} e^{
c_k(A,B)
K}.$$
As a consequence, $\pro^{u}_t \psi_0 $ belongs to $D((|A|^k)^{\ast})=D(|A|^k)$
and
\[\| |A|^k  \pro^{u}_t \psi_0 \|\leq  \|\psi_0\|_{k/2} e^{ c_k(A,B)
K}.\qedhere\]
\end{proof}
\begin{remark}
{ Assumption~\ref{ass:commutator} is a regularity
assumption to make on top of $(A,B)$ $k$-\FEPS\ if one wants to extend the previous
lemma to
the limit case $s=k$.
It is also interesting to notice that this Assumption~\ref{ass:commutator} implies that
$(A,B)$ is $2$-\FEPS\ .
}
{ Indeed, if $(A,B)$ is a couple of skew-adjoint operators operators satisfying
Assumption~\ref{ass:StrongInteraction}, then Assumption~\ref{ass:commutator} implies for small $|t|$ that, for every $\psi$ in $D(A)$,
\begin{align*}
\||A|e^{-tB}\psi\|=\|Ae^{-tB}\psi\|=\|e^{tB}Ae^{-tB}\psi\|
&\leq \|e^{tB}Ae^{-tB}\psi-A\psi\| +\|A\psi\|\\
&\leq (1+L|t|)\|A\psi\|\leq e^{L|t|}\|A\psi\|=e^{L|t|}\||A|\psi\|
\end{align*}
as the map $t\in \R \mapsto e^{t B}Ae^{-t B} \in
L(D(A),\Hc)$ is locally Lipschitz with some constant $L$. Thus $(A,B)$ is $2$-\FEPS\ .
}
\end{remark}

As a consequence of Corollary~\ref{cor:continuity} and
Lemma~\ref{LEM_interpolation} we have the following proposition.
\begin{proposition}\label{PRO_cont_entree_sortie_Hk}
{Let $k$ be a positive real.
Let
$(A,B,K)$ satisfy Assumption~\ref{ass:Interacting} such that $(A,B)$ is $k$-\FEPS\ .
Let
\[
 \radon{(0,T],K}:=\{u\in \radon{(0,T]},\,u((0,t])\in K,\,\forall t\in [0,T]\}
\]

Then for any $s\in [0,k)$, for every $\psi_{0}$ in
$D(|A|^{s/2})$,
and the end-point
mapping
\begin{align*}
\pro(\psi_{0}):\radon{(0,T],K}&\rightarrow D(|A|^{s/2})\\
u&\mapsto \pro_T^u(\psi_{0})
\end{align*}
is continuous.}
\end{proposition}
\begin{proof}
 Let $(u_n)$ be a converging sequence in  $\radon{(0,T]}$ to some $u$. Then
 \[
  \pro_T^{u_n}(\psi_{0})-\pro_T^u(\psi_{0})
 \]
 is uniformly bounded in $D(|A|^{k/2})$ and converging to $0$ in $\Hc$. By
 Lemma~\ref{LEM_interpolation} it converges to $0$ in $D(|A|^{s/2})$ for $s<k$.
\end{proof}

\begin{remark}
 One can notice that under the assumptions of
Proposition~\ref{PRO_cont_entree_sortie_Hk}
both  Proposition~\ref{rem:compactness}
and Corollary~\ref{Cor:NoExactControllability}
extend to $D(|A|^{s/2})$ for $s \in [0,k)$.
\end{remark}

\paragraph{A remark on bounded control potentials}
If $B$ is bounded in $D(|A|^{k/2})$, the latter can be considered in place of
$\Hc$ in all the analysis of \S~\ref{Sec:BoundedPotential}. These leads to a simpler
version of the weak coupling theory.

\section{Galerkin approximations}\label{Sec:GGA}

In this section, again $A$ is skew-adjoint and $B$ skew-symmetric. This
assumption is introduced for convenience.

For every Hilbert basis $\Phi=(\phi_k)_{k \in \N}$ of $\Hc$,
 we define, for every $N$ in $\N$,
$$
\begin{array}{llcl}
 \pi^{\Phi}_N&:\Hc & \rightarrow & \Hc\\
      & \psi & \mapsto & \sum_{j\leq N} \langle \phi_j,\psi\rangle \phi_j\,.
\end{array}
$$

\begin{definition}
Let $(A,B)$ be a couple of (possibly unbounded) linear operators and
 $\Phi = (\phi_{n})_{n\in \N}$ be an Hilbert basis of $\Hc$.
Let $N \in \N$ and  denote ${\cal
L}_N^\Phi=\mathrm{span}(\phi_1,\ldots,\phi_N)$.
The \emph{Galerkin approximation} of order $N$ of system \eqref{eq:main}, when it makes sense, is the
system
\begin{equation}\label{eq:sigma}
\dot x = (A^{(\Phi,N)} + u B^{(\Phi,N)}) x
\tag{$\Sigma^{\Phi}_{N}$}
\end{equation}
where
$A^{(\Phi,N)}$ and $B^{(\Phi,N)}$, defined by
$$
A^{(\Phi,N)}=\pi^{\Phi}_N A_{\upharpoonright {\mathcal L}_N} \quad \mbox{ and }
\quad
B^{(\Phi,N)}=\pi^{\Phi}_N B_{\upharpoonright{\mathcal L}_N},
$$
are the \emph{compressions} of $A$ and $B$ (respectively) associated with ${\cal
L}_N$.
\end{definition}

Since $\mathcal{L}_N$ is a finite dimensional space by definition, for every integer $N$, every   $A$
 skew-adjoint and every  Hilbert basis $\Phi$   made of eigenvectors of $A$ then
$(A^{(\Phi,N)},B^{(\Phi,N)},\R)$ satisfies Assumptions~\ref{ass:ass}.
We can therefore define the contraction propagator $X^{u}_{(\Phi,N)}(t,0)$ of
\eqref{eq:sigma} associated with bounded variation control $u$. We can also write that
$(A^{(\Phi,N)},B^{(\Phi,N)})$ is $k$-\FEPS\ for any positive real $k$.

The weak coupling is actually invariant or at least does not deteriorate by compression with respect
to a basis of eigenvectors of $A$.
\begin{lemma}\label{LEM_uniformite_constante_fepsitudeBV}
Let $k$ be a non-negative real. Let $(A,B)$
is \emph{$k$-\FEPS\ } and let $\Phi$ be an Hilbert basis made of eigenvectors of $A$, $(A,B)$ be $k$-\FEPS\ .
Then, for every $N$ in $\N$, $(A ^{(\Phi,N)},B^{(\Phi,N)})$ is $k$-\FEPS\. .
Moreover,
\[
 \|e^{v B^{(\Phi,N)}}\psi\|_{D(|A
^{(\Phi,N)}|^{k/2})}\leq e^{c_{k}(A,B)|v|}
\|\psi\|_{D(|A
^{(\Phi,N)}|^{k/2})}
\]
for every $N$ in $\N$ and $\psi$ in
$\mathcal{L}_N^\Phi$.
\end{lemma}
\begin{proof}
From Proposition~\ref{Prop:ThmLumerPhillips}, one has
\[
| \Re \langle |A|^k\psi,B\psi\rangle | \leq c_k(A,B)
\|\psi\|_{D(|A|^{k/2})}\quad \forall \psi \in D(|A|^{k/2}).
\]
Hence restricted to ${\cal L}_N^\Phi$, we obtain  $(A^{(\Phi,N)},B^{(\Phi,N)})$
be $k$-\FEPS\ .
\end{proof}

In the sequel, we we will use some compactness properties for $B$ in the scale
of $A$. We make the following remark.
\begin{lemma}
Let  $A$ be a skew-adjoint operator acting on $\Hc$ such that there exists an Hilbert
basis of $\Hc$ made of eigenvectors of $A$. Let $B$ be a bounded operator on $\Hc$.
Then $B$ is compact from $D(|A|^{k/2})$ to $\Hc$ for some positive real $k$ if and only
it is compact for any positive real $k$.
\end{lemma}
\begin{proof}
We only have to prove the first implication in the following form : $B$ is compact
from $D(|A|^{k/2})$ to $\Hc$ for some positive real $k$ if and only
it is compact from $D(|A|^{s/2})$ to $\Hc$  for any $s$ in $(0,k)$.

Let $\Phi$ be an Hilbert basis made of eigenvectors of $A$. The associated
orthogonal projector $ \pi^{\Phi}_N$ tends strongly to the identity in $\Hc$ and in
$D(|A|^{k/2})$ as it commutes with $|A|^{k/2}$.

Hence if $B$ is compact from $D(|A|^{k/2})$ to $\Hc$, $B\pi^{\Phi}_N$ tends
to $B$ in $\Bc(D(|A|^{k/2}),\Hc)$ as $(B-B\pi^{\Phi}_N)_N$ is bounded in $\Bc(\Hc)$,
Lemma~\ref{LEM_interpolation} shows that $(B-B\pi^{\Phi}_N)_N$  tends to $0$ in
$\Bc(D(|A|^{s/2}),\Hc)$ for any $s$ in $(0,k)$. So that $B$ is compact from
$D(|A|^{s/2})$ to $\Hc$  for any $s$ in $(0,k)$.
\end{proof}

\begin{remark}
If $A$ has a discrete spectrum, which is equivalent to the
compactness of the resolvent of $A$, if $B$ is $A$-bounded,
$B(|A|+1)^{-1-\e}$ is compact for $\e>0$.
If moreover $B$ is bounded, then  $B(|A|+1)^{-\e}$ is compact for $\e>0$ that is $B$ is $A$-compact.
\end{remark}

\subsection{For bounded variation controls}

In this section is we consider the question of the Galerkin approximation
in the framework of Assumption~\ref{ass:ass}.

\begin{proposition}\label{Prop:GGAforBV}
Let $(A,B,K)$ satisfy Assumption~\ref{ass:ass} with $B(1-A)^{-1}$ compact. Let  $s$ be
non-negative  numbers with $0\leq s <1$.
Then
for every $\varepsilon > 0 $, $L\geq 0$, $n\in \mathbf{N}$, and
$(\psi_j)_{1\leq j \leq n}$ in $D(A)^n$
there exists $N \in \mathbf{N}$
such that
for any $u\in BV([0,T],K)$,
$$
\|u\|_{BV([0,T])}< L \Rightarrow \| \Upsilon^{u}_{t}(\psi_{j}) -
X^{u}_{(N)}(t,0)\pi_{N} \psi_{j}\|_{s/2} < \varepsilon,
$$
for every $t \geq 0$ and $j=1,\ldots,n$.
\end{proposition}
\begin{proof}
As the propagators are contractions in $\Hc$, the proof for can be restricted
to $(\psi_j)_{1\leq j\leq n}$ in a the dense subset $D(A)$.

For every $u$ in $BV([0,+\infty))$ such that $u(t)\in K$, for all $t\geq 0$, consider
$y^{u}: t \mapsto \pi^{\Phi}_{N} \pro^{u}_{t}(\psi)$ that is absolutely
continuous and, for almost every $t \geq 0$ satisfies for almost every $t \geq
0$,
$$
\dot y^{u}(t) = (A^{(\Phi,N)} + u B^{(\Phi,N)}) y^{u}(t) + u(t)
\pi^{\Phi}_{N} B(\Id - \pi^{\Phi}_{N}) \pro^{u}_{t}(\psi).
$$
Then, for every $t \geq 0$,
$$
y^{u}(t) = X^{u}_{(\Phi,N)}(t,0) \psi + \int_{0}^{t}
 X^{u}_{(\Phi,N)}(s,t) \pi^{\Phi}_{N} B(\Id -
\pi^{\Phi}_{N})
\pro^{u}_{s}(\psi) du(s)
$$

Since $\Id - \pi^{\Phi}_{N}$ tends strongly to $0$, as $N$ goes to infinity, and
$B(1- A)^{-1}$ is compact,
$B(1 - A)^{-1}(\Id - \pi^{\Phi}_{N})$ tends to $0$ in norm, as
$X^{u}_{(\Phi,N)}(t,0)$ is unitary and from Theorem~\ref{thm:kato},
\[
\|\pro^{u}_{s}(\psi)\|_{1/2}\leq M^2 e^{ M \|B(1-A)^{-1}\|\|u\|_{BV([0,+\infty))}}\|\psi\|_{2}
\]
with
\[
M=\sup_{t\in[0,T]} \|(1-A(t))^{-1}\|_{L(D(|A|^{k/2}),D(|A|^{1+k/2}))}
\leq 2+\frac1{\|B\|_A}\frac{a}{1-\frac{a}{\|B\|_A}}+\frac{b_a}{1-\frac{a}{\|B\|_A}}.
\]
where $a> 1/\|B\|_A$ is such that $a\|u\|_\infty\leq aK<1$ and $b_a$ some real which only depend on $a$ and thus $K$.
We have for any $\e>0$ the existence of $N$ such that for $n\geq N$
\begin{equation*}
\|y_{j}(t) - X^{u}_{(\Phi,N)}(t,0)(\psi)\| \leq \e
e^{c_k(A,B)M}\|u\|_{L^1([0,+\infty)}M^2 e^{ M \|B(1-A)^{-1}\|\|u\|_{BV([0,+\infty))}}\|\psi\|_{2}.
\end{equation*}
This achieves the proof for $s=0$.

Consider now the general case $s\in(0,1)$.
The sequence $\left ( \pi^{\Phi}_{N} \pro^{u}_t(\psi)-
X^{u}_{(\Phi,N)}(t,0)(\psi) \right )_{N \in \N}$ converges to zero in $\Hc$
as $N$ tends to infinity, uniformly with respect to $t\geq 0$ and $u$ with total
variation less than $L$ (this is the case treated previously). The same
sequence $\left ( \pi^{\Phi}_{N} \pro^{u}_t(\psi)-
X^{u}_{(\Phi,N)}(t,0)(\psi) \right )_{N \in \N}$
is bounded in $D(A)$ uniformly with respect to $t\geq 0$ and $u$ with
total variation less than $L$, since
\begin{eqnarray*}
 \| \pi^{\Phi}_{N} \pro^{u}_t(\psi)- X^{u}_{(\Phi,N)}(t,0)(\psi)\|_{1}
&\leq &
 \| \pi^{\Phi}_{N} \pro^{u}_t(\psi)\|_{1} +\|
X^{u}_{(\Phi,N)}(t,0)(\psi)\|_{1}
\end{eqnarray*}
and each of this therm is bounded, uniformly with respect to $N$, $u$ and $t$ by
Theorem~\ref{thm:kato} and since the projections $\pi^{\Phi}_{N}$ are bounded in $D(A)$ as they commute
to $A$.

The conclusion follows from interpolation see Lemma~\ref{LEM_interpolation}.
\end{proof}
\begin{remark}
The compactness of the operator $B(1-A)^{-1}$ was crucial to
control uniformly the norm of
\[
\{ \|(\Id - \pi^{\Phi}_{N}) B \pi^{\Phi}_{N} \psi(s)\|; s\geq 0,
|u|([0,+\infty)) \leq L  \}.
\]
One can notice that the compactness in Corollary~\ref{Cor:NoExactControllability2}
allows a weaker result where the $N$ will depend on $u$ or $t$.

One can also emphasize that this compactness was used only for the uniform smallness of the above
set.
\end{remark}

In case we have more regularity, we have the following result.
\begin{proposition}\label{PRO_regularite_supplementaire_Binv(A)compact}
Let $k$ be a positive real. Let
$(A,B)$ be $k$-\FEPS\ with $B(1-A)^{-1}$ is compact. Let  $s$ be
non-negative  numbers with
$0\leq s <k+2$.
Then
for every $\varepsilon > 0 $, $L\geq 0$, $n\in \mathbf{N}$, and
$(\psi_j)_{1\leq j \leq n}$ in $D(|A|^{1+k/2})^n$
there exists $N \in \mathbf{N}$
such that
for any $u\in BV([0,T],\R)\cap B_{L^\infty([0,T])}(0,1/\|B\|_A)$,
$$
\|u\|_{BV([0,T])}< L \Rightarrow \| \Upsilon^{u}_{t}(\psi_{j}) -
X^{u}_{(N)}(t,0)\pi_{N} \psi_{j}\|_{s/2} < \varepsilon,
$$
for every $t \geq 0$ and $j=1,\ldots,n$.
\end{proposition}
\begin{proof}
 The proof is exactly the same as this of Proposition~\ref{Prop:GGAforBV} using the results of
Section~\ref{Sec:HigherOrderFEPS}. Actually one only need to reconsider the step
prior to the interpolation in the proof of Proposition~\ref{Prop:GGAforBV}, namely the boundedness of
$\left ( \pi^{\Phi}_{N} \pro^{u}_t(\psi)-
X^{u}_{(\Phi,N)}(t,0)(\psi) \right )_{N \in \N}$
is bounded in $D(|A|^{1+k/2})$ which follows from Proposition~\ref{PRO_cont_entree_sortie_Hk-BV}.
\end{proof}

\subsection{For Radon measures}
In the case of Radon measure, for the interacting framework we already exploited
some regularity of $B$ in the scale of $A$. In order to have a Galerkin approximation
we need to assume further regularity.
\begin{proposition}[Good Galerkin
Approximation]\label{PRO_Good_Galerkin_approximation}
Let $k$ be a positive real. Let $(A,B)$  satisfy Assumptions \ref{ass:StrongInteraction} and be $k$-\FEPS\ with
$B$ $A$-compact. Let $\Phi$ be an Hilbert basis
made of eigenvectors of $A$.

Then,
for every $\e > 0 $, $L\geq 0$,   $l\in[0,k)$
there exists $N \in \N$ such that
for every $u \in \radon{(0,+\infty)}$ and  $\psi$ in ${\mathcal L}^\Phi_N$, $t
\geq 0$ and $n\geq N$
\begin{equation*}
|u|([0,+\infty)) \leq L \implies \| \pro^{u}_{t}(\psi) -
X^{u}_{(\Phi,n)}(t,0) \psi\|_{l/2} \leq
2^{\frac{l}{k}}L^{1-\frac{l}{k}}e^{c_k(A,B)M}\e\|\psi\|_{k/2}
\end{equation*}
\end{proposition}
\begin{proof}
First consider the case $l=0$. For every $u$ in $\radon{(0,+\infty)}$ consider
{ a sequence $(u_n)_n$ of piecewise constant functions that converges, as Radon measures, to $u$ and the sequence
$y^{u_n}: t \mapsto \pi^{\Phi}_{N} \pro^{u_n}_{t}(\psi)$ of absolutely
continuous mappings, that satisfies for almost every $t \geq
0$,}
$$
\dot y^{u_n}(t) = (A^{(\Phi,N)} + u_n{(t)} B^{(\Phi,N)}) y^{u_n}(t) + u_n(t)
\pi^{\Phi}_{N} B(\Id - \pi^{\Phi}_{N}) \pro^{u_n}_{t}(\psi).
$$
Then, for every $t \geq 0$, and $\alpha\in(0,k/2)$
$$
y^{u_n}(t) = X^{u_n}_{(\Phi,N)}(t,0) \psi + \int_{0}^{t}
X^{u_n}_{(\Phi,N)}(s,t) \pi^{\Phi}_{N} B(\Id + |A|)^{-\alpha}(\Id -
\pi^{\Phi}_{N})
(\Id + |A|)^{\alpha}\pro^{u_n}_{s}(\psi) du(s)
$$
Since $\Id - \pi^{\Phi}_{N}$ tends strongly to $0$, as $N$ goes to infinity, and
$B(\Id + |A|)^{-\alpha}$ is compact,
$B(\Id + |A|)^{-\alpha}(\Id - \pi^{\Phi}_{N})$ tends to $0$ in norm, as
$X^{u_n}_{(\Phi,N)}(t,0)$ is unitary
it gives as $(\Id + |A|)^{\alpha}\pro^{u_n}_{s}(\psi) $ is bounded in $\Hc$ by
$\|\psi\|_{\alpha}$
\begin{equation*}
\|y_{j}(t) - X^{u_n}_{(\Phi,N)}(t,0)(\psi)\| \leq \e
e^{c_k(A,B)M}|u_n|([0,+\infty))\|\psi\|_{\alpha} \leq L
e^{c_k(A,B)M}\e\|\psi\|_{\alpha}.
\end{equation*}
Letting $n$ tend to infinity  achieves the proof for $l=0$.

The rest of the proof is similar to the one of Proposition~\ref{Prop:GGAforBV}.
\end{proof}

\subsection{For bounded control potential}
In the case of bounded potentials we can state the following proposition. As we
need uniform in time estimates we cannot consider generators of continuous
semigroup as in Theorem~\ref{Thm:ExistenceBoundedPotential}. Below we consider
$B$ be bounded in $D(|A|^{k/2})$ which implies that $(A,B)$ is $k$-\FEPS\ .
\begin{proposition}
\label{Prop:GGAforBounded}
Let $k$ be a positive real. Let $A$ with
domain $D(A)$ be the generator of a contraction semigroup
and let $B$ be bounded in $\Hc$ and $D(|A|^{k/2})$ with $B(1-A)^{-1}$ compact. Let  $s$ be
non-negative  numbers with
$0\leq s <k$.
Then
for every $\varepsilon > 0 $, $L\geq 0$, $n\in \mathbf{N}$, and
$(\psi_j)_{1\leq j \leq n}$ in $D(|A|^{k/2})^n$
there exists $N \in \mathbf{N}$
such that
for any $u\in \radon{(0,T]}$,
$$
|u|([0,T])< L \Rightarrow \| \Upsilon^{u}_{t}(\psi_{j}) -
X^{u}_{(\Phi,N)}(t,0)\pi^\Phi_{N} \psi_{j}\|_{s/2} < \varepsilon,
$$
for every $t \geq 0$ and $j=1,\ldots,n$.
\end{proposition}
The proof is an adaptation of the previous ones and is omitted.

\subsection{The general case of operator with continuous spectrum}
{
A general extension of Propositions \ref{Prop:GGAforBV} or
\ref{PRO_Good_Galerkin_approximation}, with a
bound of the error $\| \Upsilon^u_t\psi -X^u_{(\Phi,n)}(t,0)\psi\|$ uniform in $t\geq 0$ is
not possible in the case where there is no Hilbert basis of the ambient space made of
eigenvectors of $A$. Indeed, the RAGE theorem (due to Ruelle \cite{MR0246603}, Armein and
Georgescu \cite{MR0363267} and Enss \cite{MR0523013}) states that if $A$ is a skew-adjoint
operator on the Hilbert space $\Hc$ and $\Hc^c$ is the  continuity subspace
of $\Hc$ relatively to $A$, that is the space $\{v| v \mbox{ eigenvector of } A \}^\perp$,
then for every $\psi$ in $\Hc^c$, for every compact mapping $C:\Hc\to \Hc$ (such as the
projection on a finite dimensional
subspace), $\liminf_{t\to \infty} \|C  e^{tA} \pi_c \psi \|=0$.
In other words: the part of the state which
enters $\Hc^c$ cannot be recovered by  any finite dimensional approximation.

In order to consider the extension of our result to the case where there is no Hilbert basis
made of eigenvectors for the free dynamics operator $A$, we recall a famous theorem by Weyl
and Von Neumann, {see~\cite[Theorem X.2.1]{Kato}}.

First we need the Schmidt class.
\begin{definition}
Let $\Phi=(\phi_n)_{n\in\N}$ be a complete orthonormal family of $\Hc$.
Let $T\in \Bc(\Hc)$. If the series
$\displaystyle{ \sum_{k\in\N}\|T\phi_k\|^2}$
converges, then $T$ is in the Schmidt class, denoted $\Bc_2(\Hc)$ and
the Schmidt norm is given by
$\displaystyle{
\|T\|_2:= \sqrt{\sum_{n\in\N}\|T\phi_n\|^2}}$.
\end{definition}

The previous definitions are independent of the choice of the basis $\Phi=(\phi_n)_{n\in\N}$
The Schmidt norm endows the Schmidt class with an Hilbert structure. We have that
\[
 \|T\|\leq  \|T\|_2
\]
so that the Schmidt class is formed by compact operators.

\begin{theorem}[Weyl-Von Neumann]
Let $A$ be a skew-adjoint operator on $\Hc$. For any $\varepsilon>0$, there
exists $C_\varepsilon\in \Bc_2(\Hc)$ skew-adjoint with $\|C_\varepsilon\|_2\leq \varepsilon$
and such that $A+C_\varepsilon$ has pure point spectrum.
\end{theorem}

\begin{proposition}
\label{Prop:GGAforContinuousSpectrum}
 Let $A$ be skew-adjoint,  $B$ be bounded with $B(1-A)^{-1}$ compact  and  $s$ be in $[0,1)$.
Then
for every $\varepsilon > 0 $, $L\geq 0$, $T>0$, $n\in \mathbf{N}$, and
$(\psi_j)_{1\leq j \leq n}$ in $D(|A|^{k/2})^n$
there exists a Hilbert basis $\Phi$ of $\Hc$ and $N \in \mathbf{N}$
such that
for any $u\in BV{(0,T]}$,
$$
|u|([0,T])< L \Rightarrow \| \Upsilon^{u}_{t}(\psi_{j}) -
X^{u}_{(\Phi,N)}(t,0)\pi^\Phi_{N} \psi_{j}\|_{s/2} < \varepsilon,
$$
for every $t$ in $[0,T]$ and $j=1,\ldots,n$.
\end{proposition}
\begin{proof} We do the proof for $s=0$, the rest of the proof is similar to the one of
Proposition~\ref{Prop:GGAforBV} using Corollary \ref{COR_bound_DA_controlBV_B_borne}.
By Weyl-Von Neumann theorem, there exists $C_\varepsilon\in \Bc_2(\Hc)$ skew-adjoint with
$\|C_\varepsilon\|_2\leq \varepsilon/T$
and such that $A_\varepsilon=A+C_\varepsilon$ has pure point spectrum. Let $\Phi$ be a Hilbert basis made of
eigenvectors of $A_\varepsilon$. For $u$ in $\radon{[0,T]}$, we denote with $\Upsilon^{\varepsilon,u}$ the propagator associated with
$\frac{d}{dt}x=A_\varepsilon x+u(t)Bx$.

 By Proposition  \ref{Prop:GGAforBounded} , there exists $N \in \mathbf{N}$ such
that
for any $u\in \radon{(0,T]}$,
$$
|u|([0,T])< L \Rightarrow \| \Upsilon^{\varepsilon, u}_{t}(\psi_{j}) -
X^{u}_{(\Phi,N)}(t,0)\pi^\Phi_{N} \psi_{j}\| < \frac{\varepsilon}{2},
$$
for every $t$ in $[0,T]$ and $j=1,\ldots,n$.
It is enough to notice that, for every $t\leq T$ and every $u$ in $\radon{(0,T]}$,
$$\| \Upsilon^{\varepsilon, u}_{t}(\psi_{j}) - \Upsilon^{u}_{t}(\psi_{j})\|\leq t\varepsilon \|C_\varepsilon\|$$
to achieve the proof.
\end{proof}
}

\subsection{Application: the equivalence theorem}\label{SEC_equivalence}

The notion of Good Galerkin Approximations has already been used in various contexts in
order to obtain error bounds for numerical simulations (\cite{ACC_QG}) or quantitative
estimates for approximate controllability (see an example with the controllability time in
\cite{time}).
We mention in this paragraph another application based on a finite dimensional tracking
result/.

\begin{lemma}\label{lem:tripoint}
Let $k$ be a positive real. Let $(A,B)$ satisfy Assumption \ref{ass:StrongInteraction}
such that $(A,B)$ is $k$-\FEPS\ . Let $\Phi$ be an Hilbert basis made
of eigenvectors of $A$,
 $a,b$ be two real numbers such that $a<0<b$,
$u^{\ast}$  be a Radon measure on $(0,T)$ and $\psi_{0}$ be in $\Hc$.
Then
for every $\e > 0 $ and every $l<k$, there exists a piecewise constant control
$u_{\e} : [0,T_{\e}] \to \{a,0,b\}$  such that
$$
\|{\pro^{u_{\e}}_{T_{\e}}(\psi_{0}) - \pro^{u^{\ast}}_{T}(\psi_{0})}\|_l < \e
\quad \mbox{ and } \quad \|u_{\e}\|_{L^{1}([0,T_\epsilon])} =|u^{\ast}|([0,T])\,.
$$

If $u^{\ast}$ is a positive Radon measure, then the conclusion holds true with
piecewise constant controls $u_{\varepsilon}$ taking value in $\{0,b\}$.
\end{lemma}

\begin{proof} For  locally integrable controls $u$, this is \cite{boussaid:hal-01006178}
[Lemma 12], based on the finite dimensional result \cite{boussaid:hal-01006178}[Lemma 7] and
an extension to infinite dimensional spaces thanks to the existence of Good Galerkin
Approximations.

The result for general Radon measures controls follows by continuity.
\end{proof}

\begin{lemma}\label{lem:bipoint}
Let $k$ be a positive real. If $(A,B)$ is $k$-\FEPS\  with $A$ have purely discrete
spectrum  and $B(1-A)^{-1}$ is a compact operator, then
 for any  $a,b$ be two real numbers such that $a<b$,
for any $u^{\ast}$  be a Radon measure on $(0,T)$
for any $\psi_{0}$  in $\Hc$, for every $\e > 0 $ and for every $l<k$, there exists a piecewise constant control
$u_{\e} : [0,T_{\e}] \to \{a,b\}$  such that
$$
\|{\pro^{u_{\e}}_{T_{\e}}(\psi_{0}) - \pro^{u^{\ast}}_{T}(\psi_{0})}\|_l < \e.
$$
\end{lemma}

\begin{proof}
First recall that
$D(|A+uB|^k)=D(|A|^k)$  for all $|u|<\|B\|_A$ from Lemma~\ref{lem:NormesEquivalentes}.

The proof goes in four steps:
\begin{enumerate}
 \item First notice that it is enough to prove the result for $a<0<b$.
In the case where $a>0$ or $b<0$, we translate $A$ by $\frac{a+b}{2} B$ and
write
$$A+uB=(A+ \frac{a+b}{2}B)+(u-\frac{a+b}{2})B.$$
The hypotheses of discrete spectrum on $A$ and  compactness of $B$ from $D(A)$
to $\Hc$ imply that $A+ \frac{a+b}{2}B$ still have discrete spectrum and
fulfills the other hypotheses of Lemma \ref{lem:bipoint}.

\item From Lemma \ref{lem:tripoint}, we can replace $u$ by any control with values in
$\{a,0,b\}$.
Again up to translating $A$ by $aB$
$$A+uB=(A+ aB)+(u-a)B$$
we can assume that $u$ has value in $\{0,-a,b-a\}$.

\item From the positive case in Lemma \ref{lem:tripoint}, we can replace $u$ by any control with values in
$\{0,b-a\}$.

\item Now we translate the system back to restore the original operators and the desired values for the control.\qedhere
\end{enumerate}
\end{proof}

\begin{definition}\label{DEF_attainable_set}
Let $\mathcal{K}$ be a set of Radon measures defined on $\R$,  $(A,B)$ satisfy Assumption \ref{ass:StrongInteraction}
and $\phi$ be a vector of $\Hc$. For $T>0$, we note $\mathcal{K}_T$ the set of the restrictions
 of the measures of $\mathcal{K}$ to $[0,T]$. The \emph{attainable set from $\phi$ by means of controls in $\mathcal{K}$} is the set
 $$\mathcal{A}tt_{\mathcal{K}}\phi=\bigcup_{T\geq 0}
 \bigcup_{u \in \mathcal{K}_T} \{ \Upsilon^u_{t,0}\phi | 0\leq t \leq T\}$$
\end{definition}

\begin{proposition}\label{PRO_egalite_cloture_ensembles_atteignables}
{Let $k$ be a positive real. Assume that $(A,B)$ satisfies Assumption \ref{ass:StrongInteraction}, is
$k$-\FEPS\  and $A$ has purely discrete spectrum,
$B(1-A)^{-1}$ is a compact operator and $D(|A+uB|^k)=D(|A|^k)$ for all $|u|<\|B\|_A$.
Let $\phi$ be an eigenvector of $A$.

Then, for every $s<k$, for every $a<b$ in $(-1/\|B\|_A, 1/\|B\|_A)$, the closures in $D(|A|^s)$ of the three following sets are the same:
\begin{enumerate}
\item the attainable set $X_1$ from $\phi$  by means of controls piecewise constant taking value in
$\{a,b\}$.
\item the attainable set $X_2$ from $\phi$  by means of controls piecewise constant taking value in $(a,b)$.
\item the attainable set $X_3$ from $\phi$  by means of Radon measures.
\end{enumerate}}
\end{proposition}
\begin{proof}
Clearly, $X_1 \subset X_2 \subset X_3$ hence the closures in $D(|A|^s)$ norm are the same.
It is enough to show that $X_3$ is a subset of the closure of $X_1$ in $D(|A|^s)$ norm, which is exactly the
content of Lemma \ref{lem:bipoint}.
\end{proof}

\begin{definition}\label{DEF_controllability_general}
Let $(X,\|\cdot\|_{X})$ be a Banach subspace included in $\Hc$ and let
$\mathcal{K}$ be a
set of Radon measures.
We say that  system \eqref{eq:main} is
\emph{approximately controllable} in norm $\|\cdot\|_{X}$ by means of control in
$\mathcal{K}$ if for every $\e>0$ and for every pair
$\psi_0,\psi_1$ in the intersection of the unit sphere of $L^2(\R,\C)$ with
$X$, there exists $T>0$ and  $u \in {\mathcal{K}}$ that steers the control
system
\eqref{eq:main} from $\psi_0$ to an $\e$-neighborhood of $\psi_1$ for the
norm $\|\cdot\|_{X}$.
\end{definition}

\begin{proposition}\label{PRO_equivalence_L^2}
Let $k$ be a positive real. If  $(A,B)$ satisfy Assumption \ref{ass:StrongInteraction}
and is $k$-\FEPS\  with $A$ have purely discrete spectrum,
$B(1-A)^{-1}$ be a compact operator and $D(|A+uB|^k)=D(|A|^k)$ for all $|u|<\|B\|_A$

Then, for every $s<k$, the four following statements are equivalent:
\begin{enumerate}
 \item For every $a<b$ in $\R$, the system \eqref{eq:main} is approximately
controllable in norm $|A|^s$ by means of piecewise constant functions taking
value in  $(a,b)$.
\item For every $a<b$ in $\R$, the system \eqref{eq:main} is approximately
controllable in norm $|A|^s$ by means of piecewise constant functions taking
value in  $\{a,b\}$.
\item There exist $a<b$ in $\R$ such that the system \eqref{eq:main} is
approximately controllable in norm $|A|^s$  by means of piecewise constant
functions taking value in  $\{a,b\}$.
\item The system \eqref{eq:main} is approximately controllable in norm $|A|^s$
by means  of Radon measures.
\end{enumerate}
\end{proposition}

\begin{proof}
Each of Assertion 1, 2 or 3 implies Assertion 4 as they consider a subclass.

The fact that Assertion 1 implies Assertion 2  is a consequence of the
continuity of the end point mapping (Proposition
\ref{PRO_cont_entree_sortie_Hk}) and density arguments.

Assertion 3 is obvious from Assertion 2 and Assertion 1 is obvious from Assertion 2.

At last, Assertion 4 implies Assertion
1 as a consequence of Lemma \ref{lem:bipoint}.
\end{proof}

\section{Examples}\label{sec:example}

Most of the examples of bilinear control systems (\ref{EQ_bilinear_abstrait}) encountered in the literature,
also without any relation to quantum control,  deal with bounded control operator $B$.
Proposition \ref{PRO_introduction_BMS_Bborne} applies and allows,
for instance, to complete the studies of the rod equation with clamped ends made in
\cite[Section 6, Example 4]{BMS} and \cite{BeauchardBeam}.

In the following, we concentrate on examples in relation with quantum control.

\subsection{Quantum systems with smooth potentials on compact manifolds}

This example motivated the present analysis because of its physical importance. We
consider $\Omega$ a compact Riemannian manifold endowed with the associated
Laplace-Beltrami operator $\Delta$ and the associated measure $\mu$, $r\geq 2$ an integer,
 $V,W:\Omega
\to \mathbf{R}$ two functions  of class $C^r$, and the bilinear quantum system
\begin{equation}\label{EQ-blse-compact-manifold}
 \mathrm{i}\frac{\partial \psi}{\partial t}=\Delta \psi + V \psi +u(t) W \psi.
\end{equation}
With the notations of Section \ref{sec:abstract-framework},
$\Hc=L^2(\Omega,\mathbf{C})$ endowed with the Hilbert product $\langle f,g \rangle
=\int_{\Omega} \bar{f} g \mathrm{d}\mu$, $A=-\mathrm{i}(\Delta+V)$ and
$B=-\mathrm{i}W$.
For every $s\geq 0$, $D(|A|^s)=H^r(\Omega,\mathbf{C})$. There exists a Hilbert
basis $(\phi_k)_{k\in \mathbf{N}}$ of $H$ made of eigenvectors of $A$. Each
eigenvalue of $A$ has finite multiplicity. For every $k$, there exists
$\lambda_k$ in $\mathbf{R}$ such that $A\phi_k=\mathrm{i} \lambda_k \phi_k$. The
sequence $(\lambda_k)_k$ tends to $+\infty$ and, up to a reordering, is non
decreasing.

Since $B$ is bounded from $D(|A|^{\frac{r}{2}})$ to $D(|A|^r)$, $(A,B,\R)$ satisfies
Assumption~\ref{ass:ass} and $(A,B)$ is $r$-\FEPS\ by Proposition \ref{Prop:ThmLumerPhillips}.

We thus obtain Theorem \ref{THE_main_result_equivalence} as a restatement of Proposition  \ref{PRO_equivalence_L^2}.

 \subsection{Potential well with dipolar interaction}\label{Sec:PotentialWell}

  In this example, $\Omega=(0,\pi)$ endowed with the standard Lebesgue
 measure, $V$ is the constant zero function and $W:x\mapsto x$ is the identity
 function. This academic
 example  is a simplification of the harmonic oscillator,  presented in
 Section \ref{SEC_harmonic_oscillator}, in the sense that $\Omega$ is bounded. It
 has been thoroughly studied by K. Beauchard in a serie of analyses (see among
 others \cite{beauchard,camillo}). These works give the first (and, at this
 time, almost the only one) satisfying description of the reachable set  with $L^2$ controls from the
 first eigenvector for systems of the type of \eqref{eq:blse}.
 Using Lyapounov techniques, V. Nersessyan gave practical algorithms  for
 approximate controllability in \cite{nersesyan}.

  Equation \eqref{eq:blse} writes
  \begin{equation}\label{EQ_potential_well}
  \rmi \frac{\p \psi}{\p t} = - \frac 1 2 \frac{\p^{2} \psi}{\p x^{2}} - u(t)x
 \psi
  \end{equation}
  with boundary conditions $\psi(0)=\psi(\pi)=0$.

   The linear operators $A=\frac{\rmi}{2} \Delta$ defined on $D(A)=(H^2\cap
 H^1_0)((0,\pi),\C)$ and $B:\psi\mapsto \rmi x \psi$ are skew symmetric in
the
 Hilbert space $H=L^2(\Omega,\C)$ endowed with the hermitian product
  $L^2(\Omega,\C),$
  $$\langle f,g \rangle =\int_{0}^{\pi}\overline{f(x)}g(x)
 \mathrm{d}x.$$

  Defining, for every $k$ in $\N$,
  $$\phi_k:x\mapsto  \sqrt{\frac{{2}}{\pi}} \sin (k\pi x)
   $$
  the family $\Phi=(\phi_k)_{k \in \N}$ is an orthonormal basis of $\Hc$ made of
 eigenvectors of $A$.
  $(A,B,\R)$ satisfies Assumption~\ref{ass:ass}.

For the sake of readability, we define for every $(j,k)$ in $\mathbf{N}^2$ $b_{jk}:=\langle
\phi_j,B\phi_k\rangle$. Since $B$ is skew-symmetric, $b_{jk}=-\overline{b}_{kj}$ for every
$j,k$. Easy computations show that, for every $j,k$ in $\mathbf{N}$,  $b_{jk}=0$ as soon as $j+k$ is even and, if $j+k$ is odd,
$$
|b_{jk}|\leq \frac{2jk}{(j-k)^2(j+k)^2}.
$$
We will use the following obvious technical remarks in the following.
\begin{lemma}
For every $j,k$ in $\mathbf{N}$ such that $j\neq k$, we have $\displaystyle{\frac{1}{|j-k|}\leq 2\frac{j}{k}}$.
\end{lemma}

\begin{lemma} \label{LEM_regularite_B_puits_potentiel}
Let $a$ in $\displaystyle{\left \lbrack 0, \frac{9}{8} \right )}$ and $b>\frac{1}{2}$.
Then (the restriction of) $B$ is bounded from $D(|A|^{a+b})$ to $D(|A|^a)$.

\end{lemma}
\begin{proof}
Let $\psi$ belong to $D(|A|^{a+b})$. Then $\displaystyle{\sum_{k\in \mathbf{N}} k^{4(a+b)} |
\langle  \phi_k, \psi \rangle|^2<+\infty}$. By hypothesis, $8a-1< \min\{8, 8(a+b)-5\}$. We chose $c \in (8a-1,\min \{8(a+b)-5, 8\})$. Notice that $4a-\frac{c}{2}-\frac{3}{2}<-1$ and $8(a+b)-4-c>1$. Then
\begin{eqnarray*}
\lefteqn{\||A|^{a} B\psi \|^2= \sum_{k\in \mathbf{N}} k^{4a} |\langle \phi_k, B \psi \rangle |^2
\leq  \sum_{k\in \mathbf{N}} k^{4a} \left | \sum_{j\in \mathbf{N}} b_{jk} \langle \phi_j, \psi \rangle  \right |^2 \leq  \sum_{k\in \mathbf{N}} k^{4a} \left ( \sum_{j\in \mathbf{N}} \left | b_{jk} \langle \phi_j, \psi \rangle  \right | \right )^2 }\\
&\leq & \sum_{k\in \mathbf{N}} k^{4a} \left \lbrack \left ( \sum_{j\in \mathbf{N}}   \frac{|b_{jk}|^2}{j^{4(a+b)}}  \right ) \left ( \sum_{j\in \mathbf{N}} j^{4(a+b)} |\langle \phi_j, \psi \rangle|^2    \right ) \right \rbrack \\
&\leq & \left ( \sum_{j\in \mathbf{N}} j^{4(a+b)}
|\langle \phi_j, \psi \rangle|^2    \right )
\sum_{k\in \mathbf{N}} k^{4a} \left \lbrack \left ( \sum_{j\in \mathbf{N},j\neq k}   \frac{4j^2 k^2}{j^{4(a+b)}(k-j)^4(k+j)^4}  \right )  \right \rbrack \\
&\leq & 4\left ( \sum_{j\in \mathbf{N}} j^{4(a+b)} |\langle \phi_j, \psi \rangle|^2    \right )
\sum_{k\in \mathbf{N}} k^{4a+2} \left \lbrack \left ( \sum_{j\in \mathbf{N},j\neq k}
\frac{1}{j^{8(a+b)-4}(k-j)^8} \right )^{\frac{1}{2}}   \left ( \sum_{j\in \mathbf{N},j\neq k}  \frac{1}{(k+j)^8}  \right )^{\frac{1}{2}}  \right \rbrack \\
&\leq & 4\left ( \sum_{j\in \mathbf{N}} j^{4(a+b)} |\langle \phi_j, \psi \rangle|^2    \right )
\sum_{k\in \mathbf{N}} k^{4a+2} \left \lbrack \left (2^c \sum_{j\in \mathbf{N},j\neq k}
\frac{j^c}{j^{8(a+b)-4}(k-j)^{4-c} k^c} \right )^{\frac{1}{2}}   \left ( \sum_{j>k}  \frac{1}{j^8}  \right )^{\frac{1}{2}}  \right \rbrack \\
&\leq & 2^{2+\frac{c}{2}}\left ( \sum_{j\in \mathbf{N}} j^{4(a+b)} |\langle \phi_j, \psi \rangle|^2    \right )
\sum_{k\in \mathbf{N}} k^{4a+2-\frac{c}{2}-\frac{7}{2}} \left \lbrack \left ( \sum_{j\in \mathbf{N}}
\frac{1}{j^{8(a+b)-4-c}} \right )^{\frac{1}{2}}    \right \rbrack .
\end{eqnarray*}
In other words, the restriction of $B$ is bounded from $D(|A|^{a+b})$ to $D(|A|^a)$ with bound less than
$\displaystyle{2^{\frac{1}{2}+\frac{c}{4}}\left (\sum_{j\in \mathbf{N}}
\frac{1}{j^{8(a+b)-4-c}} \right )^{\frac{1}{4}} \!\!\!\left (\sum_{k\in \mathbf{N}} k^{4a-\frac{c}{2}-\frac{3}{2}}  \right )^{\frac{1}{2}}\!\! \leq 2^{\frac{1}{2}+\frac{c}{4}} \left ( 1 +\frac{1}{8(a+b)-3-c}\right )^\frac{1}{4}\!\!\!
 \left ( 1 +\frac{2}{1+c-8a}\right )^\frac{1}{2} }$.
\end{proof}

We now present show the almost $5/2$-\FEPS\ character of the potential wall with dipolar interaction.
\begin{lemma}\label{LEM_puits_potentiel_borne}
Let $1<a<\frac{5}{4}$. Then $B$ is bounded from $D(|A|^a)$ to $D(|A|^a)$, and $(A,B)$ is $2a$ weakly coupled.
\end{lemma}
\begin{proof}
For every $j$ in $\N$, $B\phi_j=\sum_{k\in \N} b_{kj}\phi_k$. Noticing that $x\mapsto \frac{x^{2+4a}}{(j-x)^4(j+x)^4}$ is increasing on $[1,j-1]$ and decreasing on $[j+1,+\infty)$, and defining $y=x/j$, we get
\begin{eqnarray}
\||A|^a B\phi_j \|^2 & = & \sum_{k\in \N} k^{4a}|b_{jk}|^2\\
&\leq & \sum_{k\in \N, k\neq j} \frac{j^2 k^{2+4a}}{(j-k)^4(j+k)^4}\\
&\leq & j^2 \left ( \int_{1}^{j-1}  \frac{x^{2+4a}\mathrm{d}x }{(j-x)^4(j+x)^4} + \frac{1}{2^4}j^{4a-2} +\int_{j+1}^{+\infty} \frac{x^{2+4a}\mathrm{d}x }{(j-x)^4(j+x)^4} \right )\\
&\leq & j^2 \left ( j^{4a-5} \int_{\frac{1}{j}}^{1-\frac{1}{j}} \!\!\! \frac{y^{2+4a}\mathrm{d}y }{(1-y)^4(1+y)^4}
 +  \frac{1}{2^4}j^{4a-2} + j^{4a-5} \int_{1+\frac{1}{j}}^{+\infty}\!\!\! \frac{y^{2+4a}\mathrm{d}y }{(1-y)^4(1+y)^4} \right ) \label{EQ_B_borne_puits_potentiel}
\end{eqnarray}
Notice that the last integral in (\ref{EQ_B_borne_puits_potentiel}) converges in $+\infty$ if and only if
$4a<5$.
The two integrals in (\ref{EQ_B_borne_puits_potentiel}) are both equivalent to $\frac{j^3}{3.2^4}$ as $j$ tends to infinity. Hence there exists a positive  constant $c$ such that
$\||A|^a B\phi_j \|^2 \leq c j^{4a} = c \||A|^a \phi_j \|^2$.

We use  Lemma \ref{LEM_lien_WC_nouvelle_definition_ancienne_definition} to prove that $(A,B)$ is $2a$
weakly coupled.
Since $B$ is bounded from $D(A^a)$ to itself, as in Lemma~\ref{lem:NormesEquivalentes} we obtain the equality of the domains of $|A|^{s}$ and
$D(|A+uB|^s)$ for any real $u$ and $s\in[0,2]$.

For every $\psi$ in $D(|A|^{2a})$, by Cauchy-Schwarz inequality,
\begin{eqnarray}
|\langle |A|^{2a} \psi, B \psi \rangle |&=& |\langle |A|^{a} \psi, |A|^{a} B \psi \rangle |\\
& \leq & \||A|^a \psi \| \||A|^{\frac{a}{2}} B \psi \|\\
&\leq & \||A|^a \psi \|^2 \|B\|_{L(D(|A|^a,D(|A|^a))}\\
&\leq & |\langle |A|^{2a} \psi,\psi  \rangle | \|B\|_{L(D(|A|^a,D(|A|^a))}
\end{eqnarray}
Hence $(A,B)$ is $2a$ weakly coupled with coupling constant less than $\|B\|_{L(D(|A|^a,D(|A|^a))}$.
\end{proof}
\begin{remark}
Tedious but straightforward computations show that $(A,B)$ is $a$-weakly coupled with
$$c_a(A,B) \leq \left (\sum_{k\in \mathbf{N}} k^{2(1-a)}
\left ( \sum_{j\in \mathbf{N},j\neq k} \left |
\frac{(k^{2a}-j^{2a}) }{ j^{a-1}(k+j)^2(k-j)^2}\right |^2 \right )
 \right )^{\frac{1}{2}}$$
for every $2<a<\frac{5}{2}$.
\end{remark}

The fact that the present system is not more than $5/2$-\FEPS\ is the purpose of the following lemma.
\begin{lemma}
For every $a$ in $(0,+\infty)$, $\displaystyle{e^B\phi_1 \in D(|A|^a) \Leftrightarrow a<\frac{5}{4}}$.
\end{lemma}
\begin{proof}
Straightforward computations yield $\displaystyle{e^{B}\phi_1=\sum_{k\in \N}
2\mathrm{i}\frac{(-1)^k-1}{k^3-4k} \phi_k}$
hence $e^{B}\phi_1 \in D(|A|^a)$ if and only if $\displaystyle{ k^{4a}
\left |\frac{(-1)^k-1}{k^3-4k}\right |^2 \in \ell^1(\C)}$ that is $4a<5$.
\end{proof}

We sum up our results in the following
\begin{proposition}
Define $\displaystyle{\mathcal{A}tt(\phi_1)=\bigcup_{T\geq 0}
 \bigcup_{u \in \mathcal{R}([0,T])}\{\Upsilon^u_{t,0}\phi_1|0\leq t \leq T\}}$.
Then $\displaystyle{\mathcal{A}tt(\phi_1) \subset \bigcap_{s<\frac{5}{4}} D(|A|^s)}$ and
$\displaystyle{\mathcal{A}tt(\phi_1) \not \subset  D(|A|^{\frac{5}{4}})}$.
\end{proposition}

\begin{remark}
Notice that \cite[Theorem 2]{camillo} (exact controllability of (\ref{EQ_B_borne_puits_potentiel}) in
$D(|A|^{\frac{5}{2}})$ with $H^1_0$ controls)  is compatible with the result of
Proposition \ref{PRO_cont_entree_sortie_Hk-BV} (no exact controllability of
(\ref{EQ_B_borne_puits_potentiel}) in $D(|A|^{\frac{9}{4}})$ with BV controls).
\end{remark}

From \cite{Schrod2}, we know that $\{(k,k+1)|k\in \mathbf{N}\} $ is a non-degenerate chain of connectedness
for $(A+\eta B,B)$ for almost every real $\eta$.
Hence Proposition \ref{prop:controlHsFEPS} guarantees  the approximate controllability of the system
(\ref{EQ_potential_well}) from $\phi_1$ for the norm of $D(|A+\eta B|^{r})=
D(|A|^{r})$ for $\frac{3}{2}<r<\frac{5}{4}+1$. For practical exact control in
$D(|A|^\frac{3}{2})$, we can use the explicit
controls given by  Proposition \ref{prop:controlHsFEPS} to reach a neighborhood of the
target in $D(|A|^{2})$ (see for instance \cite{ACC_QG}), and then use the $L^2$
control given by \cite{camillo} for exact local controllability.

 \subsection{Quantum harmonic oscillator}\label{SEC_harmonic_oscillator}

 The quantum harmonic oscillator with angular frequency $\omega$ describes the
 oscilations of a particle of mass $m$ subject to the potential
 $V(x)=\frac{1}{2}m \omega x^2$.
 The corresponding uncontrolled Schr\"odinger equation is
 $$
 \rmi \frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m} \Delta \psi(x,t)
 + \frac{1}{2}m \omega x^2 \psi(x,t).
 $$
 With a suitable choice of units, it becomes
 $$
 \rmi \frac{\partial \psi}{\partial t}=-\frac{1}{2} \Delta \psi(x,t) +
 \frac{1}{2}
 x^2 \psi(x,t)
 $$
 The operator $A=\displaystyle{\frac{\rmi}{2}\Delta-\frac{x^2}{2}}$ is
 self-adjoint
 on $L^2(\mathbf{R},\mathbf{C})$,
 $A$ has
 a pure discrete spectrum.
 The $k^{th}$ eigenvalue (corresponding to the $k^{th}$ energy level)  is equal
 to $\displaystyle{\frac{2k+1}{2}i}$ and is associated with the eigenstate
 $$
 \phi_k:x\mapsto \frac{1}{\sqrt{2^k k!\sqrt{\pi}}} \exp \left ( -\frac{x^2}{2} \right
 ) H_k(x)
 $$
 where $H_k$ is the $k^{th}$ Hermite polynomial, namely
 $\displaystyle{ H_k(x)=(-1)^k e^{x^2} \frac{d^k}{dx^k} \left ( e^{-x^2} \right ).}$

 When considering the classical dipolar interaction, the control potential $W$
 takes the form $W(x)=x$ for every $x$ in $\R$. It is well known (see
 \cite{Rouchon} and references therein)  that the resulting control system
 \eqref{eq:blse} is not controllable in any reasonable sense. Indeed the system
 splits in two uncoupled subsystems. The first one is a finite dimensional
 classical harmonic oscillator which is controllable. The second one is a free
 (that is, without control) quantum harmonic oscillator, whose evolution does
not
 depend on the control and  is therefore not controllable.

 The proof given in  \cite{Rouchon, illner} (and especially the
 decomposition of the system in two decoupled systems) does not require more to
 the control than to be the derivative of a derivable function. Using the
 equivalence result (Proposition \ref{PRO_equivalence_L^2}), it can  be extended
 to Radon measures.

 \begin{proposition}
 The system~\eqref{eq:blse} with $\Omega=\mathbf{R}$, $V:x\mapsto x^2$ and
 $W:x\mapsto x$ is not approximately controllable by means of Radon measures.
 \end{proposition}

 Because of its non-controllability properties, the system $(A,B)$ does not
admit
 any non resonant chain of connectedness.
 To circumvent the resonances on the spectrum of $A$,
 we apply the classical techniques of analytical perturbations recalled in
 Appendix~\ref{SEC_analytic_perturbations}.
 For every real number $\nu$, we define the continuous real function
 $W_{\nu}:x\mapsto x+ \nu \exp(-a x^2)$.

 For every $\eta$ in $\mathbf{R}$, define the real
 function $V_{\eta,\nu}=V+\eta W_{\nu}$.
  Our aim is to apply Proposition~\ref{prop:controlHsFEPS}.
 We have, then, to check that for almost every $\nu \in \R$ that
 $\{(j,k)\,:\, |j-k|=1\}$ is a  nonresonant chain of connectedness for
 system~\ref{eq:blse}.

 The mapping $\eta \mapsto -\Delta + \eta
V_{\eta,\nu}$ is analytic.
Hence, there exists a family $(\lambda_k,\phi_k)_{k \in \mathbf{N}}$ of
analytic functions  $\lambda_k:\mathbf{R}\rightarrow \mathbf{C}$ and analytic
mappings $\phi_k: \mathbf{R}\rightarrow L^2(\mathbf{R}, \mathbf{C})$ such that
for every $\eta$, $-\Delta+V_{\eta,\nu}$ has pure point discrete spectrum
$(\lambda_k(\nu))_{k \in \mathbf{N}}$ associated to the family of eigenvectors
$(\phi_k(\nu))_{k \in \mathbf{N}}$.

We  first compute, for every $k$ in $\mathbf{N}$ and every $b>0$
\begin{equation}
 \int_{\mathbf{R}} x^{2k} \exp \left (-bx^2 \right ) dx=\sqrt{\frac{\pi}{b}}
\frac{(2k)!}{2^{2k}k!b^k}.
\end{equation}
Recalling that the $k^{th}$ Hermite polynomial $H_k$ is a polynomial of degree
exactly $k$ with integer coefficients and with leading coefficient $2^k$, one
sees that, for every $p$ in $\N$,
\begin{align*}
 F_{p}&:=\int_{\mathbf{R}} W_{\nu}(x) \phi_p(x)^{2} dx\\
 &=  \frac{1}{{2^p p!\sqrt{\pi}}} \int_{\mathbf{R}} (x+\nu \exp \left (-(a+1)
x^2 \right ) H_p(x)^{2} dx\\
 & =  \frac{\nu}{{2^p p!\sqrt{\pi}}} \int_{\mathbf{R}} \exp \left (-(a+1) x^2
\right)  H_p(x)^{2} dx
\end{align*}
can be written as $\nu\displaystyle{P_{p}\left ( \frac{1}{\sqrt{a+1}} \right )}$
where
$P_{p}$ is a polynomial with degree $2p+1$
with rational coefficients.
Moreover, we compute, for every $k$,
$$
\lambda'_k(0)=\int_{\mathbf{R}} \phi_k(0)^2  W_{\nu}=\nu F_{k}.
$$
If $a$ transcendental then the sequence $(F_{p})_{p\in \N}$ is rationally
independent, in particular,
$F_{p'} - F_{q'} = F_{p} - F_{q}$ implies $\{p',q'\} = \{p,q\}$.
Hence, for every $\nu\neq 0$, for almost every $\eta$ in $\mathbf{R}$, there
exists a nonresonant chain of connectedness for
$(\rmi(-\Delta+V_{\eta,\nu}),W_{\nu})$.

\begin{proposition}
For every $\eta,s>0$, the perturbed harmonic oscillator
$(\rmi(-\Delta+V_{\eta,\nu}),W_{\nu})$ is $s$-\FEPS
\end{proposition}

\begin{proof}
 Notice first that $\rmi(-\Delta+V),W)$ is a tridiagonal system satisfying the
conditions of \cite[Section IV]{BoussaidCaponigroChambrion} and is thus
$s$-weakly-coupled for every $s>0$.
Since all the derivatives of the smooth function $x\mapsto \nu \exp(-ax^2)$ are
bounded on $\mathbf{R}$, $(\rmi(-\Delta+V,W,(W_{\nu}-W)))$ is $s$-weakly-coupled
for every $s>0$.
\end{proof}

Proposition~\ref{prop:controlHsFEPS} can now be applied to $(\rmi(-\Delta+V_{\eta,\nu}),W_{\nu})$ and provides
the following proposition.
\begin{proposition}For every $\nu\neq 0$, for very $s>0$ and almost every $\eta\in\R$ the perturbed harmonic oscillator
$(\rmi(-\Delta+V_{\eta,\nu}),W_{\nu})$ is simultaneously approximately controllable in
$H^\eta(\R)$.
\end{proposition}

\subsection{A controllable system without good Galerkin approximation}

We present here a (non-physical) example which is simultaneously approximately
controllable and cannot be suitably represented by its finite dimensional
Galerkin approximations. It consists in Equation \eqref{eq:main}, in which
 the Hilbert space $\Hc$ is equal to $L^2(\R,\C)$ endowed with its standard
scalar product $\langle f,g\rangle=\int_{\R} \bar{f} g$, $A$ is equal to $\rmi
\left ( (\Delta +x^2) +  (\Delta +x^2)^{-1} \right )$ and $B$ is the
multiplication by $x^4$. The skew symmetric operator is skew adjoint, with
spectrum $Sp(A)=\rmi \{(2k-1)/2+ 2/(2k-1),k\in \N\}$. The $k^{th}$ Hermite
function $\phi_k$ is an eigenstate of $A$ associated with the eigenvalue $\rmi
(2k+1)/2+  \rmi 2/(2k-1)$. The family $\Phi=(\phi_k)_{k \in \N}$ is an Hilbert
basis of $\Hc$.

Notice that $B$ is not relatively bounded with respect to $A$, while, for every
(constant) $u$ in $[0,+\infty)$, $A+uB$ is the generator of group of unitary
transformations. We define the solutions of~\eqref{eq:main} only for controls
$u$ that are piecewise constant.

For parity reasons, $B$ leaves invariant the sets $\Phi^e=(\phi_{2k})_{k \in \N}$
and $\Phi^o=(\phi_{2k+1})_{k \in \N}$. Hence, no global controllability in $\Hc$
is to be expected.

Defining $\phi^o_k=\phi_{2k+1}$ for every $k$ in $\N$ and
$\Phi^o=(\phi_k^o)_{k\in \N}$, the closure $\Hc^o$ of $\mathrm{span} \Phi^o$ is
an Hilbert space stable by $A$ and $B$. The restriction of $A$ and $B$ to
$\Hc^o$ are denoted by $A^o$ and $B^o$ respectively.

The set $\{(j,k)\in \N^2,|j-k|=1\}$ is a non resonant chain of connectedness of
$(A^o,B^o,\Phi^o)$. By Proposition~\ref{prop:controlHs}, the control
system $(A^o,B^o)$ is approximately controllable in $\Hc^o$. Moreover, for every
$j$ in $\N$, for every $\e>0$, there exists a piecewise constant positive
control $u_{j,\e}$ with $L^1$ norm less than $\sum_{l=1}^{j-1} \frac{\pi}{2n^2}<
\frac{\pi^3}{12} $ that steers this system from the first eigenstate $\phi_1^o$
to an $\e$-neighborhood of $\phi_j^o$. The  $(A^o )^s$ norm of $\phi^o_j$ tends
to $+\infty$ as $j$ tends to infinity. Hence $(A^o,B^o)$ does not satisfy the
conclusions of Proposition \ref{PRO_Good_Galerkin_approximation}.

\paragraph*{Acknowledgments}
The financial support of the INRIA Color is acknowledge in this work. This long term
analysis was possible due to the support of our respective institutions (the universities of Franche-Comt\'e and
Lorraine and the CNAM) as well as the facilities offered by the CNRS, the PIMS and the University of Victoria during the stay
of the first author.

We are also grateful to many colleagues for the useful discussions that lead to many
improvements since our first result, among them we especially thank Farid Ammar-Khodja, Marcelo Laca or
Gilles Lancien.

\newpage

\appendix

\section{Analytical perturbations}\label{SEC_analytic_perturbations}
To apply our sufficient condition for approximate controllability (Proposition
\ref{prop:controlHsFEPS}), we need to find a non-resonant chain of connectedness,
which may require some work on practical examples. A classical idea we already
used in this study is to introduce a new control $\tilde{u}=u-\bar{u}$ and to
consider the system $x'=(A+\bar{u} B)+ (u-\bar{u})B$ for a suitably chosen
constant $\bar{u}$.

We have the following results by Kato~\cite[Section VII.2]{Kato}.
\begin{definition}
Let $D_0$ be a domain of the complex plane, a family $(T(z))_{z\in D_0}$ of
closed operators from a Banach space $X$ to a Banach space $Y$ is said to
be \emph{a holomorphic family of type (A)} if
\begin{enumerate}
 \item $D(T(z))=D$ is independent of $z$,
 \item $T(z)u$ is holomorphic for $z$ in $D_0$ for every $u$ in $D$.
\end{enumerate}
\end{definition}

\begin{theorem}[\protect{\cite[Theorem VII.3.9]{Kato}}]
 Let $T(z)$ be a \emph{selfadjoint} holomorphic family of type (A) defined for
$z$ in a neighborhood of an interval $I_0$ of the real axis such that
$T(z)^\ast=T(\bar z)$. Furthermore,
let $T(z)$ have a compact resolvent. Then all eigenvalues of $T(z)$ can be
represented by functions which are holomorphic in $I_0$\footnote{Each of them
is
holomophic in some neighborhood of $I_0$ but possibly different for each in
such a way that their intersection is just $I_0$.}.

More precisely, there is a sequence of scalar-valued functions $(z\mapsto
\mu_n(z))_{n \in \N}$ and operator-valued functions $(z \mapsto \phi_n(z))_{n
\in \N}$, all holomorphic on $I_0$, such that for
$z$ in $I_0$, the sequence $(\mu_n(z))_{n \in \N}$ represents all the repeated
eigenvalues of $T(z)$
and $(\phi_n(z))_{n \in \N}$ forms a complete orthonormal family of the
associated
eigenvectors of $T(z)$.
\end{theorem}

\begin{proposition}
If $(A,B,K)$ satisfies Assumptions~\ref{ass:ass} then the family $\rmi(A+ z B)_{z \in \C,|z|<1/\|B\|_A}$ is
holomorphic of type (A).
\end{proposition}

\begin{proof}
The question of domain is solved by the Kato-Rellich Theorem. The holomorphy is
immediate as the family $\rmi(A+ z B)$ is affine in $z$.
\end{proof}

\section{Interpolation}

\subsection{Convergence of sequences}
Through the present analysis, the following simple interpolation lemma was useful.
\begin{lemma}\label{LEM_interpolation}
  Let $A$ be a skew-adjoint operator, let $S$ be a set and $(u_n)_{n\in
\N}$ take value in the set of functions from $S$ to $D(|A|^k)$, such that
$(u_n)_{n \in N}$ is uniformly bounded in $S$ for the norm of $D(|A|^k)$, $k >
0$. If $(u_n)_{n\in \N}$ tends to zero in $\Hc$ uniformly in $S$, then
$(u_n)_{n\in \N}$ tends to zero in $D(|A|^l)$, uniformly in $S$ for every $l<k$.
\end{lemma}
\begin{proof}
The proof follows from the logarithmic convexity of $l\in[0,k]\mapsto
\||A|^lu\|$. Indeed
\[
 \||A|^{\frac{l+j}{2}}u\|=\sqrt{\langle
|A|^lu,|A|^ju\rangle} \leq \|
|A|^lu\|^{1/2}\||A|^ju\|^{1/2}.
\]
If $l<k$ then
$$
\||A|^l u_n\|\leq
\|u_n\|^{\frac{k-l}{k}}\||A|^{k} u_n \|^\frac{l}{k}.
$$
Let $C=\sup_{n\in\N}\||A|^{k} u_n \|^2$ and $N>0$ such that for any
$n>N$, $\|u_n\|^2\leq \varepsilon$ we obtain
\[
  n>N\Longrightarrow \||A|^l u_n\|^2 \leq
\varepsilon^{\frac{k-l}{k}}C^\frac{l}{k} , \]
which provides the lemma.
\end{proof}

\subsection{Interpolation of fractional powers of operators}
Let us now state a more sophisticated result. The following result can also be deduced from the content of \cite[Section 2.8]{amrein}.
\begin{proposition}\label{Prop:Interpolation}
Let $A$ and $B$ be two selfadjoint positive operators in $\Hc$ such that there
exists $c>0$ with
\[
 c\leq B \leq A
\]
in the form sense. Then  for any $\alpha\in (0,1)$, the following is true
\[
 c^\alpha\leq B^\alpha \leq A^\alpha.
\]
\end{proposition}
\begin{proof}
The proof follows from the following series of lemma.

For a selfadjoint operator $\Hc$ and $z\in \C\setminus\R$,
the functional calculus is the extension of the mapping:
\[
 \left\{x\in \R \mapsto (x-z)^{-1}\right\}\in B(\R) \to (A-z)^{-1} \in B(\Hc)
\]
as a  strong continuous
$\ast$-algebra homomorphisme map on bounded borelian functions on the real line
with the bounded pointwise topology : $B(\R)$.

Let us recall the following functional calculus identity based on the Poisson
formula.
\begin{lemma}\label{Lem:PoissonFormula}
Let $A$ be a selfadjoint operator in $\Hc$.
 Let $f$ be a bounded borelian function.
 Then
 \[
  f(A):=w-\lim_{\varepsilon\to 0+}\frac{1}{2i\pi}\int_{\R} f(\lambda)
\Im(A-\lambda-i \varepsilon)^{-1}\,d\lambda.
 \]
\end{lemma}

We also recall the formula for $\alpha\in (0,1)$ and $x>0$
\[
 x^{-\alpha} = \frac{\pi}{\sin(\pi\alpha)}\int_{\R^+}
\frac{w^{-\alpha}}{x+w}\,dw.
\]
and then the Fubini theorem with Lemma~\ref{Lem:PoissonFormula} we obtain the
\begin{lemma}
Let $A$ be a positive selfadjoint operator in $\Hc$.
Then for $\alpha\in (0,1)$
\[
 A^\alpha = \frac{\pi}{\sin(\pi\alpha)}\int_{\R^+} \frac{w^{-1+\alpha}
A}{A+w}\,dw.
\]
\end{lemma}

With this identity and the following lemma we have the proof of the proposition.
\begin{lemma}
Let $A$ and $B$ be two selfadjoint positive operators in $\Hc$ such that there
exists $c>0$ with
\[
 c\leq B \leq A.
\]
Then
\[
 A^{-1} \leq B^{-1}.
\]
\end{lemma}
\begin{proof}
First notice that both $A$ and $B$ are invertible from their domains to $\Hc$
as well as their square roots. Then from
\[
 \sqrt{c}\|u\|\leq \|\sqrt{B}u \| \leq \|\sqrt{A}u\|,
\]
we deduce that $\sqrt{B}\sqrt{A}^{-1}$ is a bounded operator with norm
at most $1$.

In the other hand the operator $\sqrt{A}^{-1}\sqrt{B}$ defined on $D(\sqrt{B})$
extends as the adjoint of $\sqrt{B}\sqrt{A}^{-1}$ to a closed operator on $\Hc$
and hence is bounded with norm at most $1$ and
\[
\|\sqrt{A}^{-1}\sqrt{B}u \| \leq \|u\|, \forall u \in D(\sqrt{B})
\]
and thus
\[
\|\sqrt{A}^{-1}u \| \leq \|\sqrt{B}^{-1}u\|.
\]
and the result follows.
\end{proof}

The proof of Proposition~\ref{Prop:Interpolation} then follows as
\[
 c\leq B \leq A
\]
implies for any $w>0$,
\[
 1-w(B+w)^{-1} \leq 1-w(A+w)^{-1}.
\]
and thus
\[
\frac{w^{-1+\alpha}
B}{B+w}\leq
 \frac{w^{-1+\alpha}
A}{A+w}
\]
integrating on $w>0$ gives the desired inequality.
\end{proof}

The above result can be extend to the case $c=0$ by replacing $A$ and $B$ by $A+\epsilon$ and $B+\epsilon$ as in \cite{pedersen},
we obtain
\[
0\leq B^\alpha\leq  (B+\epsilon)^\alpha\leq  (A+\epsilon)^\alpha.
\]
The second inequality is immediate. We obtain
\[
0\leq (A+\epsilon)^{-\alpha/2}B^\alpha(A+\epsilon)^{-\alpha/2}\leq  1
\]
so that taking $\epsilon$ to $0$ giving
\[
0\leq B^\alpha\leq (A)^\alpha.
\]
We immediately deduce the following corollary.
\begin{corollary}\label{Lem:DomainInterpolation}
 Let $A$ and $B$ be two positive self-adjoint operators sharing the
 same domains. For any $\alpha\in (0,1)$, we have :
 \[
  D(A^\alpha)=D(B^\alpha)
 \]
\end{corollary}
\begin{proof}
Let us first assume that $B$ has spectrum away from zero.
 As $B$ is closed it is a bounded operator from $D(A)$ to $\Hc$. Thus
 \[
\exists c>0,\forall \phi\in D(A),  \|B\phi\|\leq c\|A\phi\|.
 \]

 Hence
 \[
  B^2\leq c^2 A^2
 \]

from the above that $B^{2\alpha/2}$ is bounded from $D(A^{2\alpha/2})$ to $\Hc$.

The proof being now symmetric in $A$ and $B$ we can conclude
\end{proof}

\section{An extension of the commutator theorem by Nelson}\label{Sec:Nelson}
In this section only we will consider $A$ as self-adjoint instead of skew-adjoint
in order to follow the usual formulation of Nelson commutator theorem.

\begin{theorem}
 Let $N$ be a selfadjoint operator on a Hilbert space $H$ and $A$ be a
symmetric operator on $H$ defined on $D(N)$. Let $R$ be the resolvent of $N$ at
$i$ and assume that
\[
\exists C>0,\,\forall \psi \in D(A),\, \|[A,R]\psi\|\leq C \|R \psi\|
\]
then $A$ is essentially self-adjoint.
\end{theorem}

\begin{proof}
First we notice that $[A,R]$ extends uniquely to $\Hc$ and thus
\[
\exists C>0,\,\forall \psi \in \Hc,\, \|[A,R]\psi\|\leq C \|R \psi\|.
\]
Since the adjoint of $R$ is the resolvent of $N$ at $-i$,  the resolvent
identity provides
\[
\exists C>0,\,\forall \psi \in D(A),\, \|[A,R^\ast]\psi\|\leq C \|R^\ast \psi\|
\]
or
\[
\exists C>0,\,\forall \psi \in \Hc,\, \|[A,R^\ast]\psi\|\leq C \|R^\ast \psi\|.
\]
The adjoint of $[A,R^\ast]$ can be identified with an extension of $[A^\ast,
R]$ defined on $D(A^\ast)$ (which contains $D(A)$), thus
\[
\exists C>0,\,\forall \psi \in \Hc,\, \|[A^\ast,R]\psi\|\leq C \|R \psi\|.
\]

Let $c>C$ and $\psi \in D(A^\ast)$ such that $(A^\ast+ic)\psi=0$. Then
\[
 R(A^\ast+ic)\psi=0.
\]
As $R\psi \in D(N)\subset D(A)$,
\[
 (A^\ast+ic)R\psi=[A^\ast,R]\psi
\]
or
\[
 (A+ic)R\psi=[A^\ast,R]\psi.
\]

We have
\[
 c\|R\psi\|\leq \|(A+ic)R\psi\|=\|[A^\ast,R]\psi\|\leq C\|R\psi\|
\]
and
\[
 R\psi=0
\]
and thus $\psi=0$.
\end{proof}

Notice that in the form sense $[A,R]=-R[A,N]R$ and the assumption is actually
the boundedness of $R[A,N]$ or the boundedness of $[A,N]$ from $D(N)$ to $H$.
This is stronger than the usual assumption on the Nelson Commutator Theorem.

Actually the operator $N$ is not needed. The assumption can be replaced by
the existence of a bounded operator $R$ such that the range of $R$ is in the
domain of $A$ (or $R$ is bounded from $D(A^\ast)$ to $D(A)$).

\section{Some sufficient conditions for approximate controllability with bounded variation controls}
The aim of this Section is to recall approximate controllability results obtained in other
contexts and how this results may be adapted in our context.

We first recall the following definitions from~\cite{Schrod}.
\begin{definition}
 Let $(A,B,\R)$ satisfy Assumptions \ref{ass:ass}
 such that $A$ and $B$ are skew-symmetric. Let $\Phi=(\phi_k)_k$ be a
Hilbert basis of $\Hc$ made of eigenvectors of $A$, $A\phi_k=\mathrm{i}\lambda_k
\phi_k$ for every $k$ in $\mathbf{N}$. A pair $(j,k)$ of integers is a
\emph{non-degenerate transition} of $(A,B,\Phi)$ if (i)  $\langle \phi_j,B\phi_k \rangle \neq
 0$ and (ii) for every $(l,m)$ in $\mathbf{N}^2$, $|\lambda_j-\lambda_k|=|
\lambda_l-\lambda_m|$ implies $(j,k)=(l,m)$ or $\langle \phi_l, B \phi_m\rangle =0$ or
$\{j,k\}\cap \{l,m\}=\emptyset$.
\end{definition}
 \begin{definition}
 Let $(A,B,\R)$ satisfy Assumptions \ref{ass:ass}
 such that $A$ and $B$ are skew-symmetric. Let $\Phi=(\phi_k)_k$ be a
Hilbert basis of $\Hc$ made of eigenvectors of $A$, $A\phi_k=\mathrm{i}\lambda_k
\phi_k$ for every $k$ in $\mathbf{N}$. A subset $S$ of $\mathbf{N}^2$
is a \emph{non-degenerate chain of connectedness} of $(A,B,\Phi)$ if
(i) for every $(j,k)$ in $S$, $(j,k)$ is a non-degenerate transition of $(A,B)$ and (ii)
for every $r_a,r_b$ in $\mathbf{N}$, there exists a finite sequence $r_a=r_0,r_1,
\ldots,r_p=r_b$ in $\mathbf{N}$ such that, for every $j\leq p-1$, $(r_j,r_{j+1})$ belongs
to $S$.
\end{definition}
\begin{proposition}\label{prop:controlHs}
 Let $(A,B,\R)$ satisfy Assumptions \ref{ass:ass}
 such that $A$ and $B$ are skew-symmetric. Let $\Phi=(\phi_k)_k$ be a
Hilbert basis of $\Hc$ made of eigenvectors of $A$, $A\phi_k=\mathrm{i}\lambda_k
\phi_k$ for every $k$ in $\mathbf{N}$. Let $S$ be a non-degenerate chain of
connectedness of $(A,B)$.

 Then, for every
$\eta>0$, $(A,B)$ is simultaneously approximately controllable in
$D(|A|^{1-\eta})$.
\end{proposition}
\begin{proof}
 First of all, it is enough to prove the result for target propagators
$\hat{\Upsilon}$ leaving invariant the space of co-dimension 2 spanned by
$(\phi_j,\phi_k)$ for $(j,k)$ in $S$
$$
\hat{\Upsilon}=e^{\mathrm{i}\nu_l}  (\cos(\theta) \phi_l^\ast \phi_l +
\sin(\theta) \phi_l^\ast \phi_k) +
e^{\mathrm{i}\nu_k}  (-\sin(\theta) \phi_k^\ast \phi_l + \cos(\theta)
\phi_l^\ast \phi_k)
$$
The result in $\Hc$-norm is a consequence of \cite[Theorem 1]{periodic}: for every piecewise
constant $u^{\ast}:\R\rightarrow \R$,
$2\pi/|\lambda_j-\lambda_k|$-periodic such that
$$
\int_0^{\frac{2\pi}{|\lambda_j-\lambda_k|}}
u^{\ast}(\tau)e^{\mathrm{i}(\lambda_j-\lambda_k)\tau}\mathrm{d}\tau \neq 0
$$
and
$$
\int_0^{\frac{2\pi}{|\lambda_j-\lambda_k|}}
u^{\ast}(\tau)e^{\mathrm{i}(\lambda_l-\lambda_m)\tau}\mathrm{d}\tau =0
$$
for every $l,m$ such that $(\lambda_l-\lambda_m)\in \mathbf{Z}
(\lambda_j-\lambda_k)$ and $b_{l,m}\neq 0$, there exists $T^\ast$ such that
$\Upsilon^{u^\ast/n}(nT^\ast,0)$ tends to $\hat{\Upsilon}$ as $n$ tends to
infinity.

The conclusion follows using Lemma~\ref{LEM_interpolation} and the estimate in
$A$-norm of Theorem~\ref{thm:kato}.
\end{proof}

Let us just mention the following result in case of higher regularity.
\begin{proposition}\label{prop:controlHsFEPS}
Let $k$ be a positive real.  Let $(A,B,\R)$ satisfy Assumptions \ref{ass:ass}
 such that $(A,B)$ is $k$-\FEPS\ . Let $\Phi=(\phi_k)_k$ be a
Hilbert basis of $\Hc$ made of eigenvectors of $A$, $A\phi_k=\mathrm{i}\lambda_k
\phi_k$ for every $k$ in $\mathbf{N}$. Let $S$ be a non-degenerate chain of
connectedness of $(A,B)$ such that, for every $(j,k)$ in $S$, the set
$\{(l,m)\in \mathbf{N}^2| (\lambda_l-\lambda_m)\in \mathbf{Z}
(\lambda_j-\lambda_k) \mbox{ and } \langle \phi_l,B\phi_m\rangle\neq 0\}$ is finite. Then, for every
$\eta>0$, $(A,B)$ is simultaneously approximately controllable in
$D(|A|^{k/2+1-\eta})$.
\end{proposition}
\begin{proof}
The proof differs from the previous by the interpolation step by using
Proposition~\ref{PRO_cont_entree_sortie_Hk-BV}.
\end{proof}