text:
\begin{document}

\title{Existence of non-coercive Lyapunov functions is equivalent to integral uniform global asymptotic stability

}

\titlerunning{Integral UGAS and non-coercive Lyapunov functions
}

\author{Andrii Mironchenko  \and Fabian Wirth       }

\institute{Andrii Mironchenko (corresponding author)  \at
							Faculty of Computer Science and Mathematics, University of Passau, Innstra\ss e 33, 94032 Passau, Germany\\

              \email{andrii.mironchenko@uni-passau.de}

           \and
           Fabian Wirth \at
							Faculty of Computer Science and Mathematics, University of Passau, Innstra\ss e 33, 94032 Passau, Germany\\
              Tel.: +49-421-21863825\\
              Fax: +49-421-2189863825\\
              \email{fabian.(lastname)@uni-passau.de}
}

\date{Received: date / Accepted: date}

\maketitle

\begin{abstract}
In this paper a class of abstract dynamical systems is considered which
encompasses a wide range of nonlinear finite- and infinite-dimensional systems.
We show that the existence of a non-coercive Lyapunov function without any further requirements on the flow of the forward complete system ensures an integral version of uniform global asymptotic stability. We prove that also the converse statement holds without any further requirements on regularity of the system.

Furthermore, we give a characterization of uniform global asymptotic stability in terms of
the integral stability properties and analyze which stability properties
can be ensured by the existence of a non-coercive Lyapunov function, provided either the flow has a kind of uniform continuity near the equilibrium or the system is robustly forward complete.

\keywords{
nonlinear control systems \and infinite-dimensional systems \and Lyapunov methods \and global asymptotic stability
}

\end{abstract}

\section{Introduction}
\label{sec:introduction}

The theory of Lyapunov functions is one of the cornerstones in the
dynamical and control systems theory.  Numerous applications of Lyapunov
theory include characterization of stability properties of fixed points
and more complex attractors \cite{Yos66,Hah67,LSW96,Kel15}, conditions for
forward completeness of trajectories \cite{AnS99}, criteria for the
existence of a bounded absorbing ball \cite[Theorem 2.1.2]{Chu15} etc.  Some of these uses extend from finite-dimensional
applications to the infinite-dimensional case, while others rely on distinct
finite-dimensional arguments.

On the other hand numerous converse results have been obtained which prove
the existence of certain types of Lyapunov functions characterizing
different stability notions. Indeed, before starting to look for a Lyapunov function it is highly desirable to
know in advance that such a Lyapunov function for a given class of systems
exists.  The first results guaranteeing existence of Lyapunov functions
for asymptotically stable systems appeared in the works of Kurzweil
\cite{Kur56} and Massera \cite{Mas56}.  These have been
generalized in different directions, see \cite{Kel15,MiW17a} for references.

The standard definition of a Lyapunov
function $V$, found in many textbooks on finite-dimensional dynamical
systems, is that it should be a continuous (or more regular) positive
definite and proper function, i.e. a function for which there exist
$\Kinf$\footnote{An increasing, unbounded, continuous, positive definite
  function from $\R_+$ to itself that maps $0$ to $0$.} functions $\psi_1,\psi_2,\alpha$ such that
\begin{equation}
    \label{eq:2}
    \psi_1(\|x\|) \leq V(x) \leq \psi_2(\|x\|) \quad \forall x \in X,
\end{equation}
and such that
\begin{equation}
    \label{eq:3}
    \dot V(x) < - \alpha(\|x\|) \quad \forall x \in X,
\end{equation}
where $\dot V(x)$ is some sort of generalized derivative of $V$ along the
trajectories of the system.

If $V$ is as above with the exception that instead of \eqref{eq:2}, $V$
satisfies the weaker property
\begin{equation}
    \label{eq:2b}
    0 < V(x) \leq \psi_2(\|x\|)\,,\quad x\neq 0,
\end{equation}
then $V$ is called a non-coercive Lyapunov function.

Noncoercive Lyapunov functions are frequently used in the linear infinite-dimensional systems theory.
There are at least two reasons for this. On the one hand, using the
generalized Datko lemma \cite{Dat70,Lit89} one can show that the existence of noncoercive Lyapunov functions already proves exponential stability of a linear system (and thus it is not necessary to look for coercive Lyapunov functions). On the other hand, noncoercive Lyapunov functions are in a certain sense even more natural than coercive ones. For example, a classic type of Lyapunov functions for linear exponentially stable systems over Hilbert spaces are quadratic Lyapunov functions constructed by solving the operator Lyapunov equation \cite[Theorem 5.1.3]{CuZ95}. However, solutions of this equation are not coercive in general, and hence the corresponding Lyapunov functions are not coercive as well.

In spite of these advantages, the usage of non-coercive Lyapunov functions was limited to linear infinite-dimensional systems and to nonlinear time-delay systems, for which the efficient method of Lyapunov-Krasovskii functionals is widely used \cite{HaV93,PeJ06} (Lyapunov-Krasovskii functionals have, however, a different type of noncoercivity, see \cite{MiW17d} for a comparison and discussion).
Recently the situation has changed: in \cite{MiW17a} the authors have
shown that for a broad class of forward complete \textit{nonlinear}
infinite-dimensional systems existence of a non-coercive Lyapunov function
ensures uniform global asymptotic stability (UGAS) of a system, provided
the flow of the system has a certain uniform continuity at the origin and finite-time reachability sets of the system are bounded.
On the other hand, it was demonstrated in \cite{MiW17a} that without these additional assumptions uniform global asymptotic stability cannot be guaranteed, even for systems of ordinary differential equations (ODEs). In particular, the existence of a non-coercive Lyapunov function alone does not ensure forward completeness of the system (in contrast to coercive Lyapunov functions).
Hence, although non-coercive Lyapunov functions provide more flexibility
for the stability analysis of dynamical systems, further conditions have to be verified separately.
Another result of \cite{MiW17a} is a construction of a Lipschitz continuous non-coercive Lyapunov function by means of an integration of the solution along trajectories.

In this paper, we continue the investigations initiated in \cite{MiW17a}. In \textit{our first main result} (Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem}), we show that forward complete systems possessing non-coercive Lyapunov functions (even if they do not satisfy any further assumptions) enjoy an ``integral version'' of uniform global asymptotic stability (iUGAS),
which is a weaker notion than UGAS.
\textit{Our second result}
(Theorem~\ref{thm:converse_Non-coercive_Lyapunov_theorem}) is a converse
non-coercive Lyapunov theorem for the iUGAS property. Since iUGAS is weaker than UGAS, a coercive Lyapunov function does not exist for such systems in general.
However, we show (without requiring any further regularity of the flow!) that we can construct a non-coercive Lyapunov function for this system.
The construction is motivated by \cite{MiW17a} and based upon classic converse theorems and
Yoshizawa's method \cite[Theorem 19.3]{Yos66}, \cite[Theorem 4.2.1]{Hen81}.
A key tool for achieving our main results are the characterizations of the iUGAS property in terms of weaker stability notions,  developed in Theorem~\ref{thm:iUGAS_characterization}, which is a third notable result in this work.
In Figure~\ref{fig:Stability} we provide a graphical overview of the results obtained in this paper, in particular, the relationship between the introduced stability notions.

Relations between integral and ``classic'' stability notions have been studied in a number of papers.
In particular, in \cite{TPL02} uniform global asymptotic stability of finite-dimensional differential inclusions has been characterized via ``integral'' uniform attractivity.
A natural extension of the iUGAS notion to the case of systems with inputs leads to the nonlinear counterparts of $L_2$-stability (which was originally introduced in the context of linear systems in the seminal work \cite{Zam66}, see also \cite{Sch02}).
In \cite{Son98,KeD16} it was shown that these extensions are equivalent to input-to-state stability for the systems of ordinary differential equations with Lipschitz continuous nonlinearities.

\subsection{Notation}

The following notation will be used throughout. By $\R_+$ we
denote the set of nonnegative real numbers. For an arbitrary set $S$ and
$n\in\N$ the $n$-fold Cartesian product is $S^n:=S\times\ldots\times
S$.

The open ball in a normed linear space $X$ endowed with the norm $\|\cdot\|_X$ with radius $r$ and center in $y \in X$ is denoted by $B_r(y):=\{x\in X \ |\  \|x-y\|_X<r\}$ (the space $X$ in which the ball is taken, will always be clear from the context). For short, we denote $B_r:=B_r(0)$. The (norm)-closure of a set $S \subset X$ will be denoted by $\overline{S}$.

For the formulation of
stability properties the following classes of comparison functions are
useful, see \cite{Hah67,Kel14}. The set ${\K}$ is the set of functions $\gamma:\R_+
\to \R_+$ that are continuous, strictly increasing and with
$\gamma(0)=0$; $\K_{\infty}$ is the set of unbounded $\gamma\in\K$; $\KL$
  is the set of continuous $\beta: \R_+^2 \to \R_+$, such that
  $\beta(\cdot,t)\in{\K}$, for all $t \geq 0$ and $ \beta(r,\cdot)$ is
  decreasing to $0$ for all $r >0$.

\section{Problem statement}
\label{sec:problem-statement}

We consider abstract axiomatically defined time-invariant
and forward complete systems on the state space $X$ which are subject to a shift-invariant set of
disturbances $\Dc$.
\begin{definition}
\label{Steurungssystem}
Consider the triple $\Sigma=(X,\Dc,\phi)$, consisting of:
\begin{enumerate}[(i)]
	\item A normed linear space $(X,\|\cdot\|_X)$, called the {state space}, endowed with the norm $\|\cdot\|_X$.
	\item A set of  disturbance values $D$, which is a nonempty subset of a certain normed linear space.
	\item A {space of disturbances} $\Dc \subset \{d:\R_+ \to D\}$
          satisfying the following two axioms.

\textit{The axiom of shift invariance} states that for all $d \in \Dc$ and all $\tau\geq0$ the time
shift $d(\cdot + \tau)$ is in $\Dc$.

\textit{The axiom of concatenation} is defined by the requirement that for all $d_1,d_2 \in \Dc$ and for all $t>0$ the concatenation of $d_1$ and $d_2$ at time $t$
\begin{equation}
d(\tau) :=
\begin{cases}
d_1(\tau), & \text{ if } \tau \in [0,t], \\
d_2(\tau-t),  & \text{ otherwise},
\end{cases}
\label{eq:Composed_Input}
\end{equation}
belongs to $\Dc$.

	\item A map $\phi:\R_+ \times X \times \Dc \to X$, called the \textit{transition map}.
\end{enumerate}
The triple $\Sigma$ is called a (forward complete) system, if the following properties hold:

\begin{sysnum}
	\item forward completeness: for every $(x,d) \in X \times \Dc$ and
          for all $t \geq 0$ the value $\phi(t,x,d) \in X$ is well-defined.
	\item\label{axiom:Identity} The identity property: for every $(x,d) \in X \times \Dc$
          it holds that $\phi(0, x,d)=x$.
	\item Causality: for every $(t,x,d) \in \R_+ \times X \times
          \Dc$, for every $\tilde{d} \in \Dc$, such that $d(s) =
          \tilde{d}(s)$, $s \in [0,t]$ it holds that $\phi(t,x,d) = \phi(t,x,\tilde{d})$.
	\item \label{axiom:Continuity} Continuity: for each $(x,d) \in X \times \Dc$ the map $t \mapsto \phi(t,x,d)$ is continuous.
		\item \label{axiom:Cocycle} The cocycle property: for all $t,h \geq 0$, for all
                  $x \in X$, $d \in \Dc$ we have
$\phi(h,\phi(t,x,d),d(t+\cdot))=\phi(t+h,x,d)$.
\end{sysnum}
\end{definition}
Here  $\phi(t,x,d)$ denotes the state of the system at the moment $t \in
\R_+$ corresponding to the initial condition $x \in X$ and the disturbance $d \in \Dc$.

We require a stronger version of forward completeness.
\begin{definition}
\label{Def_RFC}
The system $\Sigma=(X,\Dc,\phi)$ is called robustly forward complete (RFC)
if for any $C>0$ and any $\tau>0$ it holds that
\begin{equation*}
 \sup\big\{ \|\phi(t,x,d)\|_X \ |\  \|x\|_X\leq C,\: t \in [0,\tau],\: d\in \Dc \big\} < \infty.
\end{equation*}
\end{definition}
In other words, a system $\Sigma$ is RFC iff its finite-time reachability sets (emanating from the bounded sets) are bounded.

The condition of robust forward completeness is satisfied by large classes of infinite-dimensional
systems.

\begin{definition}
\label{def:EqPoint}
We call $0 \in X$ an equilibrium point of the system $\Sigma=(X,\Dc,\phi)$, if
$\phi(t,0,d) = 0$ for all $t \geq 0$, $d \in \Dc$.
\end{definition}

Note that according to the above definition disturbances cannot
move the system out of the equilibrium position.
\begin{definition}
\label{def:RobustEqPoint}
We call $0 \in X$ a robust equilibrium point (REP) of the system
$\Sigma=(X,\Dc,\phi)$, if it is an equilibrium point such that
for every $\eps >0$ and for any $h>0$ there exists $\delta = \delta
(\eps,h)>0$, satisfying
\begin{equation}
\hspace{-7mm} t\in[0,h],\ \|x\|_X \leq \delta, \ d \in \Dc \quad \Rightarrow \quad  \|\phi(t,x,d)\|_X \leq \eps.
\label{eq:RobEqPoint}
\end{equation}
\end{definition}

In this paper we investigate the following stability properties of equilibria of
abstract systems.

\begin{definition}{}
    \label{d:stability_new}
Consider a system $\Sigma=(X,\Dc,\phi)$ with a fixed point $0$. The
equilibrium position $0$ is called
\begin{enumerate}[(i)]

  \item uniformly locally stable (ULS), if for every $\eps >0$ there is a $\delta>0$ so that
\begin{eqnarray}
\|x\|_X \leq \delta,\ d\in\Dc,\ t \geq 0 \quad\Rightarrow\quad\|\phi(t,x,d)\|_X \leq \eps.
\label{eq:Uniform_Stability}
\end{eqnarray}

\item uniformly globally asymptotically
stable (UGAS)
if there exists a $\beta \in \KL$ such that
\begin{equation}
\label{UGAS_wrt_D_estimate}
x \in X,\ d\in \Dc,\ t \geq 0  \srs \|\phi(t,x,d)\|_X \leq \beta(\|x\|_X,t).
\end{equation}
\item uniformly (locally) asymptotically stable (UAS)
if there exist a $\beta \in \KL$ and an $r>0$ such that
\begin{equation*}

\|x\|_X\leq r,\ d\in \Dc,\ t \geq 0  \srs \|\phi(t,x,d)\|_X \leq \beta(\|x\|_X,t).
\end{equation*}

\item \label{def:UniformGlobalWeakAttractivity} uniformly globally weakly attractive (UGWA), if for every $\eps>0$ and
for every $r>0$ there exists a $\tau = \tau(\eps,r)$ such that for all $\|x\|_X\leq r,\ d\in\Dc$
\begin{equation*}
 \exists t = t(x,d,\eps) \leq \tau:\ \|\phi(t,x,d)\|_X \leq \eps.

\end{equation*}
\item \label{def:UniformGlobalAttractivity} uniformly globally attractive (UGATT), if for any $r,\eps >0$ there exists $\tau=\tau(r,\eps)$ so that
\begin{equation*}
\|x\|_X \leq r,\ d\in\Dc,\ t \geq \tau(r,\eps) \quad \Rightarrow \quad \|\phi(t,x,d)\|_X \leq \eps.

\end{equation*}
\end{enumerate}
\end{definition}
It is clear, that UGAS of $0$ implies UGATT of $0$, which in turn implies UGWA of $0$.

As we will see, in the study of non-coercive Lyapunov functions one
arrives very naturally at ``integral'' versions of the notions stated above:
\begin{definition}
\label{def:iREP}
We call $0 \in X$ an integrally robust equilibrium point (iREP) of the system
$\Sigma=(X,\Dc,\phi)$, if it is an equilibrium point and there is $\alpha\in\K$ such that
for every $\eps >0$ and for any $h>0$ there exists $\delta = \delta
(\eps,h)>0$, satisfying
\begin{equation}
\hspace{-7mm} \|x\|_X \leq \delta, \ d \in \Dc \quad \Rightarrow \quad \int_0^h \alpha(\|\phi(s,x,d)\|_X)ds \leq \eps.
\label{eq:iREP}
\end{equation}
\end{definition}

\begin{definition}
\label{Def_iRFC}
For a given $\alpha\in\K$, system $\Sigma=(X,\Dc,\phi)$ is called $\alpha$-integrally robustly forward complete ($\alpha$-iRFC),
if for any $C>0$ and any $\tau>0$ it holds that
\[
\sup_{x\in\overline{B_C},\ d\in\Dc }    \int_0^\tau \alpha\big(\|\phi(t,x,d)\|_X\big) dt < \infty.
\]
\end{definition}

\begin{remark}
\label{rem:RFC_and_iRFC}
Note that every forward-complete system is automatically $\alpha$-iRFC for
any bounded $\alpha\in\K$, since
$\int_0^\tau \alpha\big(\|\phi(t,x,d)\|_X\big) dt <\tau \sup_s \alpha(s)$.
On the other hand, if $\Sigma$ is RFC, then $\Sigma$ is also $\alpha$-iRFC for any $\alpha\in\K$.
\end{remark}

\begin{definition}
\label{def:Integral_stability_notions}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ with a fixed
point at $0$.
The equilibrium position $0$ is called
\begin{enumerate}[(i)]
\item integrally uniformly locally stable (iULS) provided there are $\alpha\in\K$, $\psi\in\Kinf$ and $r>0$ so that
\begin{equation}
\|x\|_X\leq r,\ d\in\Dc \srs \int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds  \leq \psi(\|x\|_X).
\label{eq:Integral_boundedness}
\end{equation}
\item integrally uniformly globally stable (iUGS) provided there are $\alpha\in\K$, $\psi\in\Kinf$ so that \eqref{eq:Integral_boundedness} is valid for $r:=\infty$.
\item integrally uniformly globally attractive (iUGATT)  provided there is $\alpha\in\K$ so that
\begin{eqnarray}
\forall \ r>0 \,:\, \lim_{t\to\infty} \sup_{x\in \overline{B_r},\;d\in\Dc} \int_t^\infty \alpha(\|\phi(s,x,d)\|_X)ds  = 0.
\label{eq:integral_UGATT}
\end{eqnarray}
\item integrally uniformly globally asymptotically stable (iUGAS)
  provided there are $\alpha\in\K$ and $\beta\in\KL$ so that for all $x\in
  X,\ d\in\Dc,\ t\geq 0$ we have
\begin{equation}

\int_t^\infty \alpha(\|\phi(s,x,d)\|_X)ds  \leq \beta(\|x\|_X,t).
\label{eq:integral_UGAS}
\end{equation}
\end{enumerate}
\end{definition}

\smallskip{}

Properties \eqref{eq:Integral_boundedness} and \eqref{eq:integral_UGATT} resemble a kind of uniform attractivity.
This similarity becomes even more apparent if we rewrite the definition of UGATT in an equivalent form:
\begin{lemma}
\label{lem:UGATT_restatement}
Let $\Sigma=(X,\Dc,\phi)$ be a forward complete system with fixed point $\Sigmafp$. Then $\Sigmafp$ is UGATT iff there is $\alpha\in\K$ so that
\begin{eqnarray}
\label{eq:UGATT_restatement}
\lim_{t\to\infty }\sup_{x\in\overline{B_r},\; d\in\Dc}  \alpha\Big(\big\|\phi(\cdot + t,x,d)\big\|_{C(X)}\Big) =0\quad \forall r>0,
\end{eqnarray}
where $\|\phi(\cdot + t,x,d)\|_{C(X)}$ is the $\sup$-norm of the ``tail'' of the trajectory $\phi$ after the time $t$.
\end{lemma}

\begin{proof}
If $\Sigmafp$ is UGATT, then for any $\alpha\in\Kinf$ and any $r,\eps >0$ there exists $\tau=\tau(r,\eps)$ so that
\begin{eqnarray*}
\|x\|_X \leq r,\ d\in\Dc,\ t \geq \tau(r,\eps) \quad \Rightarrow \quad \|\phi(t,x,d)\|_X \leq \alpha^{-1}(\eps).
\end{eqnarray*}

Equivalently, the left hand side implies $\alpha\big(\|\phi(t,x,d)\|_X\big) \leq \eps$
and taking the limit $\eps\to+0$ we arrive at \eqref{eq:UGATT_restatement}. The proof of the converse implication is analogous.
\end{proof}

\begin{remark}
\label{rem:with_PD_alpha_we_do_not_get_UGATT}
Note that merely choosing a positive definite $\alpha$ in
\eqref{eq:UGATT_restatement} (i.e. $\alpha\in C(\R_+,\R_+)$: $\alpha(0)=0$ and $\alpha(r)>0$ for $r>0$) we do not arrive at any kind of attractivity, since
the trajectory may grow to infinity, and $\alpha(\|\phi(t,x,d)\|_X)$ may converge to zero at the same time.
E.g. consider $\dot{x}(t)=x(t)$, $x(t)\in\R$, $\alpha(r):=\frac{r}{r^2+1}$.
\end{remark}

Analogously, it is possible to restate the UGS property.
In Theorem~\ref{thm:UGAS_characterization} we will show that UGAS implies iUGAS.

Finally, it is easy to see that
\begin{lemma}
\label{lem:REP_implies_iREP}
Let $\Sigma=(X,\Dc,\phi)$ be a system. If $\Sigmafp$ is a REP, then
$\Sigmafp$ is an iREP with arbitrary $\alpha\in\Kinf$.
\end{lemma}

\begin{proof}
Fix $\alpha\in\Kinf$. Since $\Sigmafp$ is a REP of
$\Sigma=(X,\Dc,\phi)$, for every $\eps >0$, $h>0$ there is $\delta =
\delta(\eps,h)>0$ such that
\begin{equation*}
\hspace{-7mm} \|x\|_X \leq \delta, \ d \in \Dc \quad \Rightarrow \quad \sup_{t\in[0,h]} \|\phi(s,x,d)\|_X \leq \alpha^{-1}\big(\frac{\varepsilon}{h}\big).
\end{equation*}
Hence it holds that
\begin{equation*}
\hspace{-7mm} \|x\|_X \leq \delta, \ d \in \Dc \quad \Rightarrow \quad \int_0^h \alpha\big(\|\phi(s,x,d)\|_X\big) ds \leq \varepsilon,
\end{equation*}
which shows $\Sigmafp$ is an iREP with the above $\alpha$.
\end{proof}

We now introduce Lyapunov functions which will help in characterizing the
UGAS and iUGAS concepts. To this end we first recall the notion of Dini derivative.
For
$h: \R \to \R$
the right-hand lower Dini derivative $D_+$ and the right-hand upper Dini
derivative $D^+$ at a point $t\in \R$ are
defined by, see \cite{Sza65},
\begin{equation}
    \label{eq:4}
    \begin{aligned}
       D_+h(t)&:=\mathop{\underline{\lim}} \limits_{\tau \rightarrow +0}
       {\frac{1}{\tau}\big(h(t+\tau)-h(t)\big) },
\\
		D^+h(t)&:=\mathop{\overline{\lim}} \limits_{\tau \rightarrow +0} {\frac{1}{\tau}\big(h(t+\tau)-h(t)\big) }.
    \end{aligned}
\end{equation}

Consider a system $\Sigma= (X,\Dc,\phi)$ and let $V:X \to\R$ be a map.

Given $x\in X,d\in \mathcal{D}$, we consider the (right-hand lower) Dini
derivative of the function $t \mapsto V(\phi(t,x,d))$ at $t=0$ denoted by:
\begin{equation}
\label{UGAS_wrt_D_LyapAbleitung}
\dot{V}_d(x):=\mathop{\underline{\lim}} \limits_{t \rightarrow +0} {\frac{1}{t}\Big(V\big(\phi(t,x,d)\big)-V(x)\Big) }.
\end{equation}
We call this the Dini derivative of $V$ along the trajectories of
$\Sigma$. We stress that at this point no continuity assumption has been
placed on $V$.

Having introduced the main stability properties, we introduce now a predominant tool for their study, which is a Lyapunov function.
\begin{definition}
\label{def:UGAS_LF_With_Disturbances}
Consider a system $\Sigma= (X,\Dc,\phi)$ and a function $V:X \to \R_+$, satisfying for each $y\in X$, each $s> 0$ and each $d\in\Dc$ the inequalities

\begin{eqnarray}
\label{eq:V_estimate_along_trajectory_final}
\mathop{\underline{\lim}} \limits_{h \rightarrow +0} V\big(\phi(s-h,y,d)\big) \geq V\big(\phi(s,y,d)\big)
\geq
\mathop{\underline{\lim}} \limits_{h \rightarrow +0}
V\big(\phi(s+h,y,d)\big).
\nonumber
\end{eqnarray}
Assume also that the right inequality in \eqref{eq:V_estimate_along_trajectory_final} is satisfied for $s:=0$ as well.
The map $V$ is called:
\begin{enumerate}[(i)]
	\item a \textit{non-coercive Lyapunov function} for the system
          $\Sigma=(X,\Dc,\phi)$,  if $V(0)=0$ and if there exist $\psi_2 \in \Kinf$ and $\alpha \in \K$ such that
\begin{equation}
    \label{eq:1}
    0 < V(x) \leq \psi_2(\|x\|_X) \quad \forall x \in X \backslash\{0\}.
\end{equation}
holds and the Dini derivative of $V$  along the trajectories of $\Sigma$ satisfies
\begin{equation}
\label{DissipationIneq_UGAS_With_Disturbances}
\dot{V}_d(x) \leq -\alpha(\|x\|_X)
\end{equation}
for all $x \in X$ and all $d \in \Dc$.

	 \item a \textit{(coercive) Lyapunov function} if in addition
           there is $\psi_1\in\Kinf$ satisfying \\
					$\psi_1(\|x\|_X) \leq V(x)$ for all $x\in X$.

\end{enumerate}

\end{definition}
The inequalities \eqref{eq:V_estimate_along_trajectory_final}
 say that if a Lyapunov function is not continuous along a trajectory at some point, then its value jumps down at this point.

The following result is known:
\begin{proposition}
\label{Direct_LT_0-UGAS_maxType}
Let $\Sigma=(X,\Dc,\phi)$ be a system. Then:
\begin{itemize}
	\item[(i)] If there exists a coercive continuous Lyapunov function for $\Sigma$, then $\Sigmafp$ is UGAS.
	\item[(ii)] If there exists a non-coercive continuous Lyapunov function for $\Sigma$, and if $\Sigma$ is RFC and $\Sigmafp$ is a robust equilibrium, then $\Sigmafp$ is UGAS.
\end{itemize}
\end{proposition}
Proposition~\ref{Direct_LT_0-UGAS_maxType}\,(i) is a classic result, and item (ii) has been shown in
\cite{MiW17a}, where the concept of a non-coercive Lyapunov function for nonlinear systems has been introduced and analyzed.

\textit{There is an apparent distinction in the results (i) and (ii)}, in that
in item (ii) the existence of a non-coercive Lyapunov function implies
UGAS, provided that REP and RFC hold. In case that either REP or RFC do not hold, non-coercive Lyapunov functions do not imply UGAS, as demonstrated by examples in \cite{MiW17a}.

This difference in the formulations of items (i) and (ii) of Proposition~\ref{Direct_LT_0-UGAS_maxType} motivates the first question:
\begin{center}
\textit{What are the stability properties, which can be inferred from the existence of a non-coercive Lyapunov function, without requiring any further assumptions on $\Sigma$?}
\end{center}
On the other hand, it is well-known, that UGAS implies existence of a coercive Lyapunov function, at least under certain regularity assumptions on the flow of $\Sigma$. This leads to the second problem which we analyze in this paper:
\begin{center}
\textit{What property, which is weaker than UGAS, implies existence of a non-coercive Lyapunov function (and at the same time does not imply the existence of a coercive Lyapunov function)?}
\end{center}
In Section~\ref{sec:Non-coercive_LFs} we resolve both these questions by
showing that \emph{existence of a non-coercive Lyapunov function is equivalent
to the iUGAS property}. Moreover, in
Section~\ref{sec:iUGAS_criterion_and_Noncoercivity} we show several useful
criteria for iUGAS and iUGATT, we give ``atomic decompositions'' of the UGAS
property in Section~\ref{sec:UGAS_characterizations}. Furthermore, in
Section~\ref{sec:Non-coercive_LFs} we analyze which stability properties
can be ensured by the existence of a non-coercive Lyapunov function, if
it is only assumed that either $\Sigmafp$ is a REP or that the RFC property of $\Sigma$ holds.

In Figure~\ref{fig:Stability} a reader can find a graphical overview of the results obtained in this paper, in particular, the relationship between the introduced stability notions.

\section{Criteria for iUGATT and iUGAS}
\label{sec:iUGAS_criterion_and_Noncoercivity}

In this section we study ``integral'' stability properties starting with
criteria for integral UGATT and then for iUGAS.

\subsection{Criteria for integral UGATT}

First we would like to give a criterion for iUGATT in terms of UGWA.
To this end we need one more concept:
\begin{definition}
\label{def:integral_Ultimate_uniform_stability}
Let $\Sigma$ be a forward complete system. We say that the fixed point $\Sigmafp$ is ultimately (locally) integrally stable (ultimately iULS) if there is $\alpha\in\K$
so that for any $\eps>0$ there exist $T=T(\eps)>0$ and $\delta = \delta (\eps)>0$ so that
\begin{eqnarray}
\|x\|_X \leq \delta,\ d \in \Dc \quad \Rightarrow \quad \int_{T(\eps)}^\infty \alpha\big(\|\phi(t,x,d)\|_X\big)ds \leq \eps.
\label{eq:integral_UltimateLyapunovStability_epsdelta}
\end{eqnarray}
\end{definition}

Now we are in a position to characterize iUGATT.
\begin{proposition}
\label{prop:iUGATT_equiv_UGWA_Ult_iULS}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ with fixed point $\Sigmafp$.
Then $\Sigmafp$ is iUGATT with some $\alpha\in\K$ if and only if $\Sigmafp$ is UGWA and ultimately iULS with the same $\alpha$.
\end{proposition}

\begin{proof}
$\Rightarrow$. Assume $\Sigmafp$ is iUGATT for a given $\alpha\in\K$. Ultimate iULS of $\Sigmafp$ (with the same weight function $\alpha$) easily follows from iUGATT. Let us show that $\Sigmafp$ is UGWA.

Pick any $R>0$ and any $\eps>0$. Since $\Sigmafp$ is integrally UGATT, there is a time $\tilde\tau=\tilde\tau(R,\eps)$ so that
\begin{eqnarray*}
\sup_{x\in \overline{B_r},\;d\in\Dc} \int_{\tilde{\tau}(R,\eps)}^\infty \alpha(\|\phi(s,x,d)\|_X)ds  \leq \frac{1}{2}\alpha(\eps).
\end{eqnarray*}

Assume that for some $x\in \overline{B_r}$, some $d\in\Dc$ and all $s\in[\tilde{\tau}(R,\eps),\tilde{\tau}(R,\eps)+1]$ it holds that $\|\phi(s,x,d)\|_X\geq \eps$. Then
\begin{eqnarray*}
\frac{1}{2}\alpha(\eps) \geq \int_{\tilde{\tau}(R,\eps)}^\infty
\alpha(\|\phi(s,x,d)\|_X)ds

\geq \alpha(\eps),
\end{eqnarray*}
a contradiction.
This shows that $\Sigmafp$ is uniformly globally weakly attractive with $\tau(R,\eps):=\tilde{\tau}(R,\eps) + 1$.

$\Leftarrow$. Since $\Sigmafp$ is ultimately iULS, there exists $\alpha\in\K$ so that for all $\eps>0$ there are $\delta(\eps)>0$ and $T(\eps)>0$ so that \eqref{eq:integral_UltimateLyapunovStability_epsdelta} holds.

Pick any $\varepsilon>0$ and $r>0$. Since $\Sigmafp$ is uniformly globally weakly attractive, there is a time $\tilde\tau=\tilde\tau(r,\eps)$ so that for any $x\in B_r$ and any $d\in \Dc$ there is a time $\bar{t}\in [0,\tilde\tau(r,\eps))$ so that $\|\phi(\bar{t},x,d)\|_X\leq \delta(\eps)$.

In view of the ultimate iULS property we have that
\[
t\geq T(\eps) \srs \int_t^\infty \alpha\big(\|\phi(s,\phi(\bar{t},x,d),d(\bar{t}+\cdot))\|_X\big)ds \leq \eps.
\]
Due to the cocycle property

it holds that
\begin{multline*}
 \int_t^\infty \alpha\big(\|\phi(s,\phi(\bar{t},x,d),d(\bar{t}+\cdot))\|_X\big)ds \\
=
\int_t^\infty \alpha\big(\|\phi(s+\bar{t},x,d)\|_X\big)ds
=
\int_{t+\bar{t}}^\infty \alpha\big(\|\phi(s,x,d)\|_X\big)ds.
\end{multline*}

Considering $t\geq \bar{t} + T(\eps)$, it is now easy to see that $\Sigmafp$ is iUGATT (with the same $\alpha$).
\end{proof}

Analogously to Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS} one can
characterize the UGATT property. We define

\begin{definition}
\label{def:Ultimate_uniform_stability}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ with fixed point $\Sigmafp$.
The fixed point $\Sigmafp$ is called ultimately uniformly stable if for any $\eps>0$ there exist $T=T(\eps)>0$ and $\delta = \delta (\eps)>0$ so that
\begin{eqnarray}
t \geq T,\ \|x\| \leq \delta,\ d \in \Dc \quad \Rightarrow \quad \|\phi(t,x,d)\| \leq \eps.
\label{eq:UltimateLyapunovStability_epsdelta}
\end{eqnarray}
\end{definition}

\begin{proposition}
\label{prop:UGATT_equiv_UGWA_Ult_ULS}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ with fixed point $\Sigmafp$.

Then $\Sigmafp$ is UGATT if and only if $\Sigmafp$ is ultimately uniformly stable and UGWA.
\end{proposition}

\begin{proof}
"$\Rightarrow$". Clear.

"$\Leftarrow$". Fix $\eps>0$. Since $\A$ is ultimately uniformly stable,
there are positive $\delta(\eps)$ and $T(\eps)$ so that
\eqref{eq:UltimateLyapunovStability_epsdelta} holds.
Now pick any $r>0$.

By uniform global weak attractivity of
$\Sigmafp$

there is a time $\tilde\tau=\tilde\tau(r,\eps)$ so that for any $x\in B_r(\A)$ and any $d\in \Dc$ there is a time $\bar{t}\in [0, \tilde\tau(r,\eps))$ so that $\|\phi(\bar{t},x,d)\|\leq \delta(\eps)$.

Due to the cocycle property
\begin{eqnarray*}
\phi(t+\bar{t},x,d)= \phi(t,\phi(\bar{t},x,d),d(\bar{t} +\cdot)),
\end{eqnarray*}
and in view of ultimate uniform stability we have that
\[
t\geq \bar{t}+T(\eps),\ x\in B_r(\A),\ d\in\Dc \srs \|\phi(t,x,d)\|\leq \eps,
\]
Specializing this to $t\geq \tilde\tau(r,\eps)+T(\eps)$, we see that $\A$ is UGATT.
\end{proof}

\subsection{Characterization of iUGAS}

In \cite[Proposition 3.7]{MiW17a} the following result has been obtained (the statement in \cite{MiW17a} was somewhat different, but the proof is exactly the same):
\begin{proposition}
\label{prop:From_ncLF_to_UGWA}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$  with fixed point $\Sigmafp$. If $\Sigmafp$ is iUGS, then $\Sigmafp$ is UGWA.
\end{proposition}

We also note:
\begin{lemma}
\label{lem:iULS_criterion}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ with fixed point $\Sigmafp$.
Then $\Sigmafp$ is iULS if and only if $\Sigmafp$ is an iREP and ultimately iULS.
\end{lemma}

\begin{proof}
$\Rightarrow$. This is clear.

$\Leftarrow$. Since $\Sigmafp$ is ultimately iULS, there is $\alpha_1\in\K$ so that for any $\varepsilon>0$ there are
$r=r(\eps)>0$ and a time $\tau=\tau(\varepsilon)>0$ satisfying
\begin{eqnarray*}
\|x\|_X\leq r(\eps),\ d \in\Dc \srs \int_{\tau}^\infty \alpha_1\big(\|\phi(s,x,d)\|_X\big)ds \leq \frac{\varepsilon}{2}.
\end{eqnarray*}
Now since $\Sigmafp$ is an iREP,  there is $\alpha_2\in\K$ so that  for these $\varepsilon,\tau$ there is a $0<\tilde\delta=\tilde{\delta}(\varepsilon)\leq r(\eps)$ so that
\begin{eqnarray*}
\|x\|_X\leq \tilde\delta, \ d \in\Dc \srs \int_0^{\tau} \alpha_2\big(\|\phi(s,x,d)\|_X\big)ds \leq \frac{\varepsilon}{2}.
\end{eqnarray*}
Define $\alpha(s):=\min\{\alpha_1(s),\alpha_2(s)\}$, $s\geq0$. Clearly, $\alpha\in\K$ and it holds that
\begin{eqnarray*}
\|x\|_X\leq \tilde\delta, \ d \in\Dc \srs \int_0^{+\infty} \alpha\big(\|\phi(s,x,d)\|_X\big)ds \leq \varepsilon.
\end{eqnarray*}
Without loss of generality we can assume that $\tilde{\delta}$ is
non-decreasing as a function of $\eps$. Furthermore, by construction it
holds that $\tilde{\delta}$ can be continuously extended by $\tilde{\delta}(0)=0$. Then it can be lowerbounded by a certain $\delta\in\K$.

Now iULS of $\Sigmafp$ follows by choosing $\psi(s):=\delta^{-1}(s)$,
$s\in[0, \lim_{s\to\infty} \delta(s))$.
\end{proof}

The main result in this section is the characterization of the notion of iUGAS:
\begin{theorem}
\label{thm:iUGAS_characterization}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$. Then the following statements are equivalent:
\begin{enumerate}[(i)]
	\item $\Sigmafp$ is iUGAS.
	\item $\Sigmafp$ is iUGS.
	\item \label{item3} $\Sigmafp$ is iULS (with a certain $\alpha\in\K$) and $\Sigmafp$ is UGWA.
	\item \label{item4} $\Sigmafp$ is an iREP and $\Sigmafp$ is iUGATT.
	\item $\Sigmafp$ is iULS and $\Sigmafp$ is iUGATT.
	\item $\Sigmafp$ is iUGS and $\Sigmafp$ is iUGATT.
\end{enumerate}
Moreover, in item~(\ref{item4}) the function $\alpha$ can be chosen to be equal
to $\alpha$ from item~(\ref{item3}).
\end{theorem}

\begin{proof}
\textbf{(i) $\Rightarrow$ (ii).} Evident.

\textbf{(ii) $\Rightarrow$ (iii).} Follows by Proposition~\ref{prop:From_ncLF_to_UGWA}.

\textbf{(iii) $\Rightarrow$ (iv).}
As $\Sigmafp$ is iULS it follows that it is an iREP.
Furthermore, since $0$ is UGWA and ultimately iULS with $\alpha\in\K$, then, by means of
Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS}, $0$ is iUGATT with the same $\alpha$.

\textbf{(iv) $\Rightarrow$ (v).} This follows directly from
Lemma~\ref{lem:iULS_criterion} and Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS}.

\textbf{(v) $\Rightarrow$ (vi).}
Note that if $\Sigmafp$ is iULS or iUGATT for a certain $\alpha\in\K$,
then the same is true for any $\tilde{\alpha} \in {\cal K}$ with
$\tilde{\alpha}\leq\alpha$ (here $\leq$ is the pointwise ordering).
Thus without loss of generality $\Sigmafp$ is both iULS and iUGATT for same $\alpha\in\K\backslash\Kinf$.

As $\Sigmafp$ is iUGATT we know that for each $R>0$ and $\eps>0$ there is $\tau:=\tau(R,\eps)$ so that
\begin{eqnarray*}
\sup_{x\in \overline{B_R},\;d\in\Dc} \int_\tau^\infty \alpha(\|\phi(s,x,d)\|_X)ds \leq \eps.
\end{eqnarray*}
Denoting $\alpha(\infty):=\lim_{s\to\infty}\alpha(s) <\infty$ we see that
\begin{eqnarray*}
\sup_{x\in \overline{B_R},\;d\in\Dc} \int_0^\tau \alpha(\|\phi(s,x,d)\|_X)ds \leq \tau \alpha(\infty),
\end{eqnarray*}

and hence, for all $R>0$,
\begin{eqnarray*}
\tilde{\sigma}(R):=  \sup_{x\in \overline{B_R},\;d\in\Dc} \int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds <\infty.
\end{eqnarray*}
Clearly, $\tilde{\sigma}$ is a non-decreasing function of $R$, and so
there exist  $\overline{\sigma}\in\Kinf$ and $c>0$ so that
$\tilde{\sigma}(r)\leq \overline{\sigma}(r) + c$ for all $r\in\R_+$.
Consequently, for any $x\in X$ and any $d\in\Dc$ we obtain
\begin{eqnarray}
\label{eq:practical_iUGS_estimate_tmp}
\int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds   \leq \overline{\sigma}(\|x\|_X) + c.
\end{eqnarray}
As $\Sigmafp$ is iULS, there exist $\psi\in\Kinf$ and $r>0$ so that
\begin{eqnarray}
\label{eq:_iULS_estimate_tmp}
\|x\|_X\leq r,\ d\in\Dc \srs \int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds   \leq \psi(\|x\|_X).
\end{eqnarray}
Using \eqref{eq:practical_iUGS_estimate_tmp},
\eqref{eq:_iULS_estimate_tmp} and standard manipulations of
$\Kinf$-functions (see e.g. \cite[proof of Lemma I.2, p. 1287]{SoW96}), it
may be seen that there is a $\sigma\in\Kinf$ so that
\begin{eqnarray*}
x \in X,\ d\in\Dc \srs \int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds  \leq \sigma(\|x\|_X).
\end{eqnarray*}
This shows that $\Sigmafp$ is iUGS with $\alpha\in\K\backslash\Kinf$.

 \textbf{(vi) $\Rightarrow$ (i).}
 As in the previous step, without loss of generality we may assume that the function $\alpha$
 is the same in the definitions of iUGS and iUGATT. We now consider a
 fixed, suitable
 $\alpha$.

 Since $\Sigmafp$ is a iUGS fixed point, there exists $\psi\in\Kinf$ so that for all
 $t\geq0$, $\delta\geq0$, $\|x\|_X \leq \delta$, $d \in \Dc$ we have
 \begin{eqnarray}

 \int_0^\infty \alpha(\|\phi(t,x,d)\|_X) dt \leq \psi(\delta).
 \label{eq:TmpEstimate}
 \end{eqnarray}
 For a fixed $\delta\geq0$, define
 $\eps_n:=\frac{1}{2^n} \psi(\delta)$, $n \in \N$.
 Let $\tau_0:=0$.
 As $\Sigmafp$ is iUGATT there exist times $\tau_n:=\tau(\eps_n,\delta)$,
 $n\geq1$, which we assume without loss of generality to be strictly increasing, such that
 \[
 t \geq \tau_n,\ \|x\|_X \leq \delta,\ d \in \Dc \srs \int_t^\infty \alpha(\|\phi(s,x,d)\|_X) ds \leq \eps_n.
 \]
 Define $\omega(\delta,0):= 2 \psi(\delta)$ and $\omega(\delta,\tau_n):=\eps_{n-1}$, for $n \in \N$, $n \neq 0$. Extend the function $\omega(\delta,\cdot)$ to $\R_+$ so that $\omega(\delta,\cdot) \in \LL$.
 Note that for any $n\in\N$ and for all $t \in (\tau_n,\tau_{n+1})$ it holds that
 \begin{eqnarray}
 \int_t^\infty \alpha(\|\phi(s,x,d)\|_X) ds \leq \eps_n < \omega(\delta,t).
 \label{eq:Tmp_Estimate}
 \end{eqnarray}
 Doing this for all $\delta \in \R_+$ we obtain a function $\omega: \R_+^2\to\R_+$.

 Now define $\tilde\beta(r,t):=\sup_{0 \leq s \leq r}\omega(s,t) \geq \omega(r,t)$. Obviously, $\tilde\beta$ is non-decreasing in the first argument and decreasing in the second.
 Moreover, for each fixed $t\geq0$, $\tilde \beta(r,t) \leq \sup_{0 \leq s \leq r}\omega(s,0)=2\psi(r)$, which implies that $\tilde\beta$ is continuous in the first argument at $r=0$ for any fixed $t\geq0$.
 Now Proposition~\ref{prop:Upper_estimate_as_KL_function} implies that $(r,t) \mapsto \tilde\beta(r,t) + |r|e^{-t}$ may be upper bounded by $\beta\in \KL$ and the estimate
 \eqref{eq:integral_UGAS}
 is satisfied with such a $\beta$.
\end{proof}

\begin{remark}
\label{rem:integral_notions_with_Kinf_functions}
Note that in all the integral notions we have assumed that the corresponding function $\alpha$ belongs to the class $\K$.
If we require in the definitions that $\alpha$ must belong to the class
$\Kinf$, we obtain stronger versions of the corresponding concepts.
The difference is that every forward-complete system is automatically
$\alpha$-integrally RFC with $\alpha\in\K\backslash\Kinf$ (see Remark~\ref{rem:RFC_and_iRFC}),
but it need not be $\alpha$-integrally RFC for all $\alpha\in\Kinf$.

For the stronger concepts the proof of (iv) $\Rightarrow$ (i) in
Theorem~\ref{thm:iUGAS_characterization} does not work as described.
In order to close the gap in the proof the stronger version of this implication, we need to
strengthen the assumptions in items
(iii), (iv) by assuming in addition
that the system is  $\alpha$-iRFC with a certain $\alpha\in\Kinf$.
Then after some minor modifications we recover the characterization of
iUGAS with $\alpha\in\Kinf$.

We leave the details to the reader.
\end{remark}

\section{``Integral'' characterization of the UGAS property}
\label{sec:UGAS_characterizations}

Until now we have worked nearly completely on the level of the ``integral''
notions, which is almost parallel to the world of classic notions of stability. Now we are going to relate ``integral'' and ``classic'' worlds.

The next proposition shows that classic stability properties can be
recovered from the ``integral'' version combined with the REP property.

\begin{proposition}
\label{prop:Influence_of_REP_and_RFC}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$. Then the following holds:
\begin{itemize}
	\item[(i)] \label{prop:From_ncLF_to_LS} If $\Sigmafp$ is a REP and iULS, then $\Sigmafp$ is ULS.
	\item[(ii)] \label{prop:NewProp_and_REP_imply_UGATT}
	If $\Sigmafp$ is a REP and iUGATT, then $\Sigmafp$ is UGATT and UAS.

\end{itemize}
\end{proposition}

\begin{remark}
    We note that in \cite[Theorem 3.1]{Mir17a} it is shown that if
    $\Sigma$ is RFC and $\Sigmafp$ is iUGATT or iUGS or iUGAS, then
    $\Sigma$ satisfies a property that is termed practically UGAS in
    \cite{Mir17a} and which amounts to saying that not the fixed point
    $\Sigmafp$ but a certain neighborhood of it has a stability property.
\end{remark}

\begin{proof} (of Proposition~\ref{prop:Influence_of_REP_and_RFC}).

\textbf{(i).}
Seeking a contradiction, assume that $\Sigma$ is not uniformly stable in $x^* = 0$. Then there exist an
$\varepsilon >0$ and sequences $\{ x_k \}_{k\in\N}$ in $X$, $\{ d_k
\}_{k\in\N}$ in $\mathcal{D}$, and $t_k \geq 0$ such that $x_k \to 0$ as $k \to \infty$ and
\begin{equation*}
    \| \phi(t_k,x_k,d_k) \|_X = \varepsilon \quad \forall k \geq 1.
\end{equation*}

Since $\Sigmafp$ is iULS, there are $\alpha\in\K$ and $\psi\in\Kinf$ so that for the above $\eps$ there is a $\delta_1=\delta_1(\varepsilon)>0$ satisfying
\begin{eqnarray}
\|x\|_X\leq \delta_1,\ d\in\Dc \srs \int_0^\infty \alpha\big(\|\phi(s,x,d\|_X\big)ds \leq \psi(\|x\|_X).
\label{eq:iULS_tmp}
\end{eqnarray}
Without loss of generality we assume that $\|x_k\|_X\leq \delta_1$ for all $k\in\N$ (otherwise we can pick a subsequence of $\{x_k\}$ with this property).

Since $0$ is a REP, for the above $\eps$ there is a $\delta = \delta(\eps,1)$ so that
\begin{eqnarray}
\|x\|_X\leq \delta,\ t\in [0,1],\ d\in\Dc \qrq \|\phi(t,x,d)\|_X\leq \frac{\eps}{2}.
\label{eq:ncLF_REP_application}
\end{eqnarray}
Define for this $\delta$ the following quantities:
\[
\tilde t_k := \sup\{t\in[0,t_k]: \|\phi(t,x_k,d_k)\|_X \leq \delta\},
\]
provided the supremum is taken over a nonempty set, and $\tilde{t}_k:=0$ otherwise.
Denote also $\eta_k:=t_k - \tilde t_k$, $k\in\N$. There are two possibilities.

First assume that $\{\eta_k\}_{k\in\N}$ does not converge to 0 as $k\to\infty$.
Then there is a $\eta^*>0$ and a subsequence of $\{\eta_{k_m}\}$ so that $\eta_{k_m} \geq \eta^*$ for all $m\geq 1$.

Using \eqref{eq:iULS_tmp} for $x:=x_{k_m}$, $d:=d_{k_m}$ and $t:=t_{k_m}$, we see that
\begin{eqnarray*}
\eta^* \alpha(\delta) \leq \eta_{k_m} \alpha(\delta)

\leq \psi(\|x_{k_m}\|_X).
\end{eqnarray*}
Since $\psi(\|x_{k_m}\|_X)\to0$ as $m\to\infty$, we obtain a contradiction.

Now assume that $\eta_k\to 0$ as $k\to\infty$. Then there is a $k_1>0$ so that $\eta_{k_1}<1$.
In view of a cocycle property, we have that
\begin{eqnarray*}
\phi(t_k,x_k,d_k) = \phi(\eta_k,\phi(\tilde t_k,x_k,d_k),d_k(\cdot + \tilde t_k)).
\end{eqnarray*}
Since $\|\phi(\tilde t_k,x_k,d_k)\|_X \leq \delta$, by
\eqref{eq:ncLF_REP_application} we obtain
$\|\phi(t_k,x_k,d_k)\|_X \leq  \frac{\eps}{2}$,

which contradicts to the assumption that $\|\phi(t_k,x_k,d_k)\|_X = \eps$.
This shows uniform stability of $\Sigmafp$.

\textbf{(ii).}
It is easy to see that iUGATT implies ultimate iULS. According to
Lemma~\ref{lem:REP_implies_iREP}, $\Sigmafp$ is an iREP.
Using Lemma~\ref{lem:iULS_criterion}

and Proposition~\ref{prop:Influence_of_REP_and_RFC}~(i) we have that $\Sigmafp$ is ULS.

Furthermore, by Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS}  the equilibrium point $\Sigmafp$ is UGWA, and Proposition~\ref{prop:UGATT_equiv_UGWA_Ult_ULS}
shows that $\Sigmafp$ is UGATT.
Finally, since $\Sigmafp$ is UGATT and ULS, then $\Sigmafp$ is UAS as well.

\end{proof}

Next we show criteria for UGAS in terms of integral stability
notions. To this end we need two technical results. The first one is
Sontag's well-known $\KL$-lemma \cite[Proposition 7]{Son98}:
\begin{lemma}
\label{Sontags_KL_Lemma}
For all $\beta \in \KL$ there exist $\alpha_1,\alpha_2 \in \Kinf$ with
\begin{equation}
\beta(r,t) \leq \alpha_2(\alpha_1(r)e^{-t}) \quad \forall r \geq 0, \; \forall t \geq 0.
\label{eq:KL-Lemma_Estimate}
\end{equation}
\end{lemma}
The second one is a characterization of UGAS in terms of the UGATT property from \cite[Theorem 2.2]{KaJ11b}:
\begin{proposition}
\label{prop:UGAS_Characterization}
Consider $\Sigma=(X,\Dc,\phi)$.
Then $\Sigmafp$ is UGAS if and only if $\Sigma$ is robustly forward complete and
$0$ is a UGATT robust equilibrium point for $\Sigma$.
\end{proposition}

The main result of this section is:
\begin{theorem}
\label{thm:UGAS_characterization}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$. Then the following statements are equivalent:
\begin{enumerate}[(i)]
	\item $\Sigmafp$ is UGAS.
	\item $\Sigma$ is RFC and $\Sigmafp$ is a REP $\wedge$ iUGAS.
	\item $\Sigma$ is RFC and $\Sigmafp$ is a REP $\wedge$ iUGATT.
	\item $\Sigma$ is RFC and $\Sigmafp$ is a REP $\wedge$ UGWA $\wedge$ ultimately iULS.
	\item $\Sigma$ is RFC and $\Sigmafp$ is a REP $\wedge$ UGWA $\wedge$ ultimately ULS.
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(i) $\Rightarrow$ (ii).}
Since $\Sigmafp$ is UGAS, there is a $\beta\in\KL$ so that \eqref{UGAS_wrt_D_estimate} holds.
In view of Lemma~\ref{Sontags_KL_Lemma} there are $\alpha_1,\alpha_2\in\Kinf$ so that
\eqref{eq:KL-Lemma_Estimate} holds.
Set $\alpha:=\alpha^{-1}_2$. Then we have for any $r>0$ and any $t>0$ it holds that:
\begin{multline*}
\hspace*{-0.3cm}\sup_{x\in \overline{B_r},\ d\in\Dc} \int_t^\infty \alpha(\|\phi(s,x,d)\|_X)ds  \leq
\sup_{x\in \overline{B_r}} \int_t^\infty \alpha(\beta(\|x\|_X,s))ds  \\
\leq  \int_t^\infty \alpha_1(r)e^{-s}ds
= \alpha_1(r)e^{-t}
\end{multline*}
and $\Sigmafp$ is iUGAS with $\psi:=\alpha_1 \in\Kinf$ and $\alpha\in\Kinf$.

\textbf{(ii) $\Rightarrow$ (iii).} Clear.

\textbf{(iii) $\Iff$ (iv).} Follows from Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS}

\textbf{(iii) $\Rightarrow$ (v).} Follows from Proposition~\ref{prop:Influence_of_REP_and_RFC}, item (ii).

\textbf{(v) $\Rightarrow$ (i).} Follows from Propositions~\ref{prop:UGATT_equiv_UGWA_Ult_ULS} and \ref{prop:UGAS_Characterization}.
\end{proof}

\begin{remark}[``Atomic decompositions'']
\label{rem:}
Items (iv) and (v) of Theorem~\ref{thm:UGAS_characterization} give a decomposition of UGAS into elementary stability notions. In some sense the notions of UGWA, REP, RFC and ultimate ULS and their integral counterparts iREP, ultimate iULS and (possibly) iRFC are the ``atoms'' by combinations of which the other stability notions can be constructed.

Comparing items (iv) and (v) of Theorem~\ref{thm:UGAS_characterization} to
the analogous ``atomic'' decompositions of iUGAS shown in
Theorem~\ref{thm:iUGAS_characterization}, we see that the notion of UGWA plays a remarkable role in such characterizations, supported by the integral variants of REP and ultimate ULS.
Uniform global weak attractivity is the common point of the worlds of classic and integral notions, which are otherwise largely parallel.
\end{remark}

\begin{remark}

It is worth mentioning that for the special case of linear systems over Banach spaces without disturbances the notions of UGAS, iUGAS and UGWA  coincide, as can be seen from \cite[Proposition 5.1]{Mir17a}.
\end{remark}

\section{Non-coercive Lyapunov theorems}
\label{sec:Non-coercive_LFs}

In this section we relate the existence of noncoercive Lyapunov functions
to the integral stability concepts we have introduced. It is shown that
for forward complete systems the existence of noncoercive Lyapunov
functions implies iUGAS.

In the next step we treat a converse
result.

\subsection{Direct Lyapunov theorems}

For the proof of direct Lyapunov theorems we need the generalized Newton-Leibniz formula (see \cite[Theorem 7.3, p. 204-205]{Sak47} and the comments directly after that result):
\begin{proposition}
\label{prop:Generalized_Newton-Leibniz_from_Saks_p204}
Suppose that $F:\R \to \R$ is a function\footnote{In the formulation of \cite[Theorem 7.3, p. 204-205]{Sak47} the terminology that $F$ is a finite function is used, which means that $F(x) \in\R$ for any $x\in\R$ (see \cite[p. 6]{Sak47}).} such that for all $x\in\R$ we have
\begin{eqnarray}
\mathop{\overline{\lim}} \limits_{h \rightarrow +0} F(x-h) \leq F(x) \leq
\mathop{\overline{\lim}} \limits_{h \rightarrow +0} F(x+h).
\label{eq:Nice_Discontinuity}
\end{eqnarray}
Let $g$ be a Perron-integrable\footnote{For a definition of Perron
  integrability see e.g. \cite[p. 201]{Sak47}. The (P) in front of the
  integral in \eqref{eq:Generalized_Newton-Leibniz} indicates that this is
a Perron integral.} function of a real variable satisfying
$D^+F(x) \geq g(x)$ for all $x\in I$.

Then for all $a,b >0$: $a<b$ it holds that
\begin{eqnarray}
F(b) - F(a) \geq (P)\int_a^b g(x) dx.
\label{eq:Generalized_Newton-Leibniz}
\end{eqnarray}
\end{proposition}

Using Theorem~\ref{thm:iUGAS_characterization} and Proposition~\ref{prop:Generalized_Newton-Leibniz_from_Saks_p204},
we can show that the existence of a non-coercive Lyapunov function implies iUGAS without any
further requirements on the flow of the system. If we
additionally assume either the REP or the RFC property, we obtain additional stability properties.

\begin{theorem}
\label{thm:direct_Non-coercive_Lyapunov_theorem}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$.
Assume that $V$ is a non-coercive Lyapunov function for $\Sigma$ with  corresponding functions $\alpha\in\K$ and $\psi_2\in\Kinf$. Then:
\begin{enumerate}[(i)]
	\item The following statements hold:
\begin{itemize}
	\item[(i-a)] $\Sigmafp$ is iUGS with this $\alpha$ and with $\psi:=\psi_2$.
	\item[(i-b)] $\Sigmafp$ is iUGATT with this $\alpha$.
	\item[(i-c)] $\Sigmafp$ is iUGAS.
\end{itemize}

	\item If additionally $\Sigmafp$ is a REP, then $\Sigmafp$ is UGATT and UAS.
	\item If additionally $\Sigmafp$ is a REP and $\Sigma$ is RFC, then $\Sigmafp$ is UGAS.
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(i-a).}
Since $V$ is a non-coercive Lyapunov function (with a corresponding $\alpha\in\K$), we have the decay estimate \eqref{DissipationIneq_UGAS_With_Disturbances}.
Pick any $x\in X$ and any $d\in\Dc$ and define $\xi :\R_+ \to\R$ via $\xi(t):=-V(\phi(t,x,d))$.

Along the trajectory $\phi$ of $\Sigma$ we have the inequality
\begin{equation}
\dot V_{d(t+\cdot)}(\phi(t,x,d)) \leq -\alpha(\|\phi(t,x,d)\|_X), \quad
\forall t\geq 0.
\label{eq:JustEq1}
\end{equation}

Due to the cocycle property we have
\begin{align*}
\dot V_{d(t+\cdot)}(&\phi(t,x,d)) \\
=& \mathop{\underline{\lim}}  \limits_{h \rightarrow +0} \frac{1}{h}\big(V(h,\phi(t,x,d),d(t+\cdot)) - V(\phi(t,x,d))\big)\\
=& \mathop{\underline{\lim}}  \limits_{h \rightarrow +0} \frac{1}{h}\big(V(\phi(t+h,x,d)) - V(\phi(t,x,d))\big)\\
=& \mathop{\underline{\lim}}  \limits_{h \rightarrow +0} \frac{1}{h}\big(-\xi(t+h) + \xi(t)\big)
=D_+(-\xi(t))
=-D^+ \xi(t).
\end{align*}
With this new notation, equation \eqref{eq:JustEq1} can be rewritten  as
\begin{eqnarray}
D^+ \xi(t) \geq \alpha(\|\phi(t,x,d)\|_X), \quad \forall t\geq 0.
\label{eq:JustEq2}
\end{eqnarray}

In view of \eqref{eq:V_estimate_along_trajectory_final} and since

$\mathop{\underline{\lim}} \limits_{h \rightarrow +0}-\xi(t+h)=-\mathop{\overline{\lim}} \limits_{h \rightarrow +0}\xi(t+h)$,

we see that the inequality
\begin{eqnarray}
\mathop{\overline{\lim}} \limits_{h \rightarrow +0} \xi(t-h) \leq \xi(t) \leq
\mathop{\overline{\lim}} \limits_{h \rightarrow +0} \xi(t+h)
\label{eq:LF_Nice_Discontinuity}
\end{eqnarray}
is satisfied for all $t>0$, and the right inequality is satisfied for $t=0$ as well.

Now we can apply
Proposition~\ref{prop:Generalized_Newton-Leibniz_from_Saks_p204} to the
above inequality. Since $t\mapsto \phi(t,x,d)$ is continuous due to the
continuity axiom $\Sigma$\ref{axiom:Continuity}, the function $g: t\mapsto
\alpha(\|\phi(t,x,d)\|_X)$ is continuous as well, and thus it is Riemann
integrable on any compact interval in $\R_+$.  As $g$ is a positive
function, the Riemann and the Perron integral
coincide (see \cite[p. 203]{Sak47}). Thus in our case the Perron integral in the
formula \eqref{eq:Generalized_Newton-Leibniz} is merely a Riemann
integral.

Applying Proposition~\ref{prop:Generalized_Newton-Leibniz_from_Saks_p204}, we obtain:
\begin{eqnarray}
\xi(t) - \xi(0) \geq \int_0^t \alpha(\|\phi(s,x,d)\|_X) ds, \quad \forall t\geq 0.
\label{eq:JustEq3}
\end{eqnarray}
Since $\xi(0) = - V(\phi(0,x,d)) = -V(x)$ due to the identity axiom of
$\Sigma$, the above inequality immediately implies that
\begin{equation*}
V(\phi(t,x,d)) - V(x) \leq -\int_0^t \alpha(\|\phi(s,x,d)\|_X)ds,

\end{equation*}
which in turn shows that for all $t\geq 0$ we have
\begin{eqnarray}
\int_0^t \alpha(\|\phi(s,x,d)\|_X)ds
\leq
V(x)\leq \psi_2(\|x\|_X).
\label{eq:Vintbound}
\end{eqnarray}
Taking the limit $t\to\infty$, we see that $\Sigmafp$ is iUGS.

\textbf{(i-b).} By Proposition~\ref{prop:From_ncLF_to_UGWA} and item (i) we see that $\Sigmafp$ is UGWA. Checking the proof of the Theorem~\ref{thm:iUGAS_characterization} (implication (iii) $\Rightarrow$ (iv)) we see that $\Sigmafp$ is iUGATT with the same $\alpha$.

\textbf{(i-c).} Follows from items (i), (ii) and Theorem~\ref{thm:iUGAS_characterization}.

\textbf{(ii).} By the item (i-b) of this theorem, $\Sigmafp$ is iUGATT. Now Corollary~\ref{prop:NewProp_and_REP_imply_UGATT} implies that $\Sigmafp$ is UGATT and UAS.

\textbf{(iii).} By the item (i-b) of this theorem, $\Sigmafp$ is iUGATT. The rest follows from Theorem~\ref{thm:UGAS_characterization}.
\end{proof}

\begin{remark}
\label{rem:Discontinuity_of_V}
Condition \eqref{eq:V_estimate_along_trajectory_final} means that for each $x\in X$ and $d\in \Dc$ the map
$t\mapsto V(\phi(t,x,d))$ is either continuous at $t^*$ or the function
jumps down at $t^*$, for arbitrary $t^* \in \R_+$.
\end{remark}

\begin{remark}
\label{rem:Relations_to_classic_results}
The crucial difference of Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem} from classic Lyapunov theorems is that  we do not assume the coercivity of a Lyapunov function. This makes it impossible to use any kind of a comparison principle to derive the desired UGAS stability property.

On the other hand, in contrast to the non-coercive direct Lyapunov theorem
shown in \cite{MiW17a} we assume for item (i) of
Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem} neither robustness
of the trivial equilibrium, nor the RFC property of the system $\Sigma$
(however, we still assume in advance the forward completeness of system
$\Sigma$). Even under such mild assumptions (and with very mild regularity
assumptions on $V$) we are able to infer the iUGAS property.

We note that it is also possible to show a practical UGAS property if in
addition to the existence of $V$ we assume RFC.
Item (ii) of Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem} is a variation of \cite[Corollary 3.10]{Mir17a} and is given here for completeness.
Item (iii) of Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem} is
slightly stronger than \cite[Theorem 4.5]{MiW17a}, where a more direct proof of this result was given.
\end{remark}

\subsection{Converse non-coercive Lyapunov theorem}
\label{sec:Converse_non-coercive_Lyapunov_theorem}

We proceed to the converse Lyapunov theorem.

\begin{theorem}
\label{thm:converse_Non-coercive_Lyapunov_theorem}
Consider a forward complete system $\Sigma=(X,\Dc,\phi)$ and let $0$ be an equilibrium of $\Sigma$. Assume that $\Sigma$ is iUGS with $\alpha\in\K$ and $\psi\in\Kinf$.

Then for any $\rho\in\K\backslash\Kinf$ so that $\rho(r)\leq \alpha(r)$ for all $r\in\R_+$ it holds that
\begin{eqnarray}
V(x):= \sup_{d\in\Dc} \int_0^\infty \rho(\|\phi(s,x,d)\|_X)ds
\label{eq:ncLF_discontinuous_construction}
\end{eqnarray}

is a (possibly non-coercive) Lyapunov function for $\Sigma$, satisfying \eqref{eq:1} with $\psi_2$ as above and so that
\eqref{eq:V_estimate_along_trajectory_final} holds.
\end{theorem}

Before we proceed to the proof of
Theorem~\ref{thm:converse_Non-coercive_Lyapunov_theorem}, we would like to
stress, that in contrast to most of the converse Lyapunov theorems for
infinite-dimensional nonlinear systems (as \cite{Mir17a},
\cite[Section 3.4]{KaJ11b}), we do not impose any additional regularity
assumptions on the flow of the system, in particular, we assume neither
continuous dependence on data, nor robustness of the equilibrium point,
nor the RFC property.

Theorems~\ref{thm:direct_Non-coercive_Lyapunov_theorem} and~\ref{thm:converse_Non-coercive_Lyapunov_theorem} together show that noncoercive Lyapunov functions are a natural tool for analysis of integral stability properties.

In the proof we follow ideas from \cite[Section 3.4]{KaJ11b},
\cite[Theorem 5.6]{Mir17a}.

\begin{proof}(of Theorem~\ref{thm:converse_Non-coercive_Lyapunov_theorem}).

Pick any $\rho\in\K\backslash\Kinf$ so that $\rho(r)\leq \alpha(r)$ for all $r\in\R_+$.

\textbf{(i).} Since $\Sigmafp$ is iUGS with $\alpha,\psi$, it follows that
\begin{eqnarray*}
0\leq V(x) \leq \sup_{d\in\Dc} \int_0^\infty \alpha(\|\phi(s,x,d)\|_X)ds\leq \psi_2(\|x\|_X).
\end{eqnarray*}

Since $0$ is an equilibrium of $\Sigma$, $\phi(s,0,d)\equiv 0$ for all
$s\geq 0$ and all $d\in\Dc$, which immediately implies that $V(0)=0$. If
$x \neq0$, then by continuity of solutions, we have for every $d
\in D$ a $T>0$ such that $\phi(t,x,d) \neq0$ for all $t\in [0,T]$. This
implies that $V(x) > 0$ and in summary \eqref{eq:1} is satisfied.

\textbf{(ii).} To compute the Dini derivative of $V$, fix $x \in X$ and $v \in \Dc$. In view of the cocycle property we have for any $h>0$:
\begin{eqnarray*}
V\big(\phi(h,x,v)\big) &=& \sup_{d \in \Dc} \int_0^{\infty} \rho(\|\phi(t,\phi(h,x,v),d)\|_X) dt\\
&=& \sup_{d \in \Dc} \int_0^{\infty} \rho(\|\phi(t+h,x,\tilde{d})\|_X) dt \\
&=& \sup_{d \in \Dc} \int_h^{\infty} \rho(\|\phi(t,x,\tilde{d})\|_X) dt,
\end{eqnarray*}
where the disturbance function $\tilde{d}$ is defined as
\[
\tilde{d}(t) :=
\begin{cases}
v(t), & \text{ if } t \in [0,h] \\
d(t-h)  & \text{ otherwise}.
\end{cases}
\]
Note that $\tilde{d}\in \Dc$ due to the axiom of concatenation.
Since $\tilde{d}(t) = v(t)$ for $t\in[0,h]$, it holds that
\begin{align*}
\int_0^h &\rho(\|\phi(t,x,v)\|_X) dt + V\big(\phi(h,x,v)\big) \\
&= \sup_{d \in \Dc} \Big(\int_0^h \rho(\|\phi(t,x,v)\|_X) dt +  \int_h^{\infty} \rho(\|\phi(t,x,\tilde{d})\|_X) dt\Big)\\
&= \sup_{d \in \Dc} \Big(\int_0^h \rho(\|\phi(t,x,\tilde{d})\|_X) dt +  \int_h^{\infty} \rho(\|\phi(t,x,\tilde{d})\|_X) dt\Big)\\
&= \sup_{d \in \Dc} \int_0^{\infty} \rho(\|\phi(t,x,\tilde{d})\|_X) dt.
\end{align*}

Since the supremum cannot decrease, if we allow a larger class of
disturbances, it may be seen that for all $h>0$ we have
\begin{align}
\label{eq:Bellman-principle}
\int_0^h \rho(\|\phi(t,x&,v)\|_X) dt + V\big(\phi(h,x,v)\big) \nonumber\\
&\leq \sup_{d \in \Dc} \int_0^{\infty} \rho(\|\phi(t,x,d)\|_X) dt
= V(x).
\end{align}
The obtained inequality may be interpreted as an instance of Bellman's principle.

To compute the Dini derivative of $V$ along trajectories we note that
rearranging the inequality \eqref{eq:Bellman-principle} we obtain for all
$h>0$ that
\begin{equation}
\frac{1}{h}\big( V\big(\phi(h,x,v)\big) - V(x) \big) \leq -\frac{1}{h}\int_0^h \rho(\|\phi(t,x,v)\|_X) dt.
\label{eq:Auxiliary_Estimate_LF}
\end{equation}
As the map $t\mapsto \rho(\|\phi(t,x,v)\|_X)$ is continuous by the axiom
of continuity, it follows that

 \begin{equation*}
 \mathop{\lim} \limits_{h \rightarrow +0} \frac{1}{h}\int_0^{h} \rho(\|\phi(t,x,v)\|_X) dt
 =  \rho(\|x\|_X),
 \end{equation*}
 and due to \eqref{eq:Auxiliary_Estimate_LF} we obtain

$\dot{V}_v(x) \leq -  \rho(\|x\|_X)$ and so
\eqref{DissipationIneq_UGAS_With_Disturbances} holds for the given $\rho \in {\cal K}$.

 \textbf{(iii).} It remains to show that
 \eqref{eq:V_estimate_along_trajectory_final} holds. The right hand side
 in this inequality holds (even in $s=0$) by a direct application of
 \eqref{eq:Bellman-principle} to the case $x=\phi(s,y,d)$ and arbitrary
 $s\geq0,  y\in X, d\in{\cal D}$, because the integral on the left hand
 side of \eqref{eq:Bellman-principle} is always nonnegative.

To show the left hand side, fix  $d\in\Dc$,  $s>0$,  $y\in X$ and $h\in (0,s)$.  Substitute $x:=\phi(s-h,y,d)$ and $v:=d(s-h+\cdot)$ into Bellman's inequality \eqref{eq:Bellman-principle}
to obtain
 \begin{align*}

 \int_0^h \rho\big(\|\phi(t,\phi(s-h,y,d),d(s-h+\cdot))\|_X\big) dt
&+ V\big(\phi(h,\phi(s-h,y,d),d(s-h+\cdot))\big) \\
&\leq V(\phi(s-h,y,d)).
 \end{align*}
 Using again the cocycle property and rearranging the terms of the above
 inequality, we conclude for all $h\in (0,s)$ that
 \begin{eqnarray*}
  V(\phi(s-h,y,d)) \geq  V\big(\phi(s,y,d)\big) + \int_0^h \rho(\|\phi(t+s-h,y,d)\|_X) dt.
 \end{eqnarray*}
 Arguing as above, we obtain that:
 \begin{eqnarray}
 \label{eq:V_estimate_along_trajectory_2}
 \mathop{\underline{\lim}} \limits_{h \rightarrow +0} V\big(\phi(s-h,y,d)\big) \geq V\big(\phi(s,y,d)\big).
 \end{eqnarray}

 This shows \eqref{eq:V_estimate_along_trajectory_final} and the proof is complete.
\end{proof}

\section{Conclusions}

In order to understand the implications of the existence of non-coercive
Lyapunov functions we have introduced several integral notions of
stability, which do not measure the pointwise distance to the equilibrium but
rather a weighted average along trajectories. It has been shown that in a
quite general setting noncoercive Lyapunov functions characterize these
integral notions. Also the relation to standard stability notions are
discussed, see also Figure~\ref{fig:Stability}.
It will be of interest to investigate how the results obtained
here carry over to questions of input-to-state stability (ISS).
Some results in this direction have been recently developed in \cite{MiW18b,JMP18}.

We point out that in \cite{Mir17a} the relation between uniform weak attractivity and closely related concepts of weak
attractivity and recurrence are discussed for systems without inputs.

 \begin{figure*}[tbh]
 \centering
 \begin{tikzpicture}[>=implies,thick]

 \node (UGAS) at (3,8) {UGAS};
 \node (iUGATT_REP_RFC) at (7,8) {iUGATT $\wedge$ 0-REP $\wedge$ RFC};
 \node (UGATT_REP_RFC) at (12.5,8) {UGATT $\wedge$ 0-REP $\wedge$ RFC};

 \draw [rounded corners] (2.5,8.8) rectangle (14.9,7.6);
 \node (Thm_UGAS_Charact) at (8.6,8.5) {\footnotesize Theorem~\ref{thm:UGAS_characterization}};

 \node (ncLF) at (0.4,6) {$\exists$ a noncoercive LF};
 \node (iUGAS) at (3.4,4) {iUGAS};

 \node (iUGATT_iREP) at (6.3,4) {iUGATT $\wedge$ iREP};
 \node (UGWA_iULS) at (-1,4) {UGWA $\wedge$ iULS};
 \node (iUGS) at (1.6,4) {iUGS};
 \node (iULS) at (-2,1.8) {iULS};
 \node (iUGATT) at (7,1.8) {iUGATT};
 \node (UGWA_Ult_iULS) at (4,1.8) {UGWA $\wedge$ Ult-iULS};
 \node (iREP_Ult_iULS) at (0.5,1.8) {iREP $\wedge$ Ult-iULS};

 \draw [rounded corners] (5.2,6.8) rectangle (12.2,5.6);

 \node (iUGATT_REP) at (7,6) {iUGATT $\wedge$ 0-REP};
 \node (iUGATT_ULS) at (10.5,6) {iUGATT $\wedge$ ULS};
 \node (UGATT_REP) at (14,4) {UGATT $\wedge$ 0-REP};
 \node (UGATT_ULS) at (10.5,4) {UGATT $\wedge$ ULS};
 \node (UGATT) at (10.5,1.8) {UGATT};
 \node (UGWA) at (7,0.5) {UGWA};

 \draw [rounded corners] (-2.5,4.8) rectangle (7.8,3.5);
 \node (Thm5) at (3,4.5) {\footnotesize Theorem~\ref{thm:iUGAS_characterization}};

 \node (Lemma_REP_iREP) at (5.5,5.25) {\footnotesize Lemma~\ref{lem:REP_implies_iREP} };

 \node (Prop_iUGATT_REP) at (8.6,6.5) {\footnotesize Proposition~\ref{prop:NewProp_and_REP_imply_UGATT}};
 \node (Prop_iUGATT_REP_2) at (11.8,5.2) {\footnotesize Proposition~\ref{prop:NewProp_and_REP_imply_UGATT}};

 \node (Prop_NC_LF) at (-1.25,5.35) {\footnotesize Theorem~\ref{thm:direct_Non-coercive_Lyapunov_theorem}};
 \node (Prop_NC_LF_Conv) at (2.2,5.35) {\footnotesize Theorem~\ref{thm:converse_Non-coercive_Lyapunov_theorem}};

 \node (REP) at (8.6,0.5) {REP};

 \draw[->,double equal sign distance] (7,5.5) to (7,4.9);
 \draw[->,double equal sign distance] (8.6,5.5) to (REP);

 \draw[->,double equal sign distance] (9.5,3.4) to (REP);

 \draw[<->,double equal sign distance] (UGAS) to (iUGATT_REP_RFC);
 \draw[<->,double equal sign distance] (iUGATT_REP_RFC) to (UGATT_REP_RFC);

 \draw[->,double equal sign distance] (-0.2,5.7) to (-0.2,4.9);

 \draw[<-,double equal sign distance] (1.2,5.7) to (1.2,4.9);

 \draw[<->,double equal sign distance] (UGWA_iULS) to (iUGS);
 \draw[<->,double equal sign distance] (iUGS) to (iUGAS) ;
 \draw[<->,double equal sign distance] (iUGAS)  to (iUGATT_iREP);

 \draw[->,double equal sign distance] (8.6,7.5) to (8.6,6.9);

 \draw[<->,double equal sign distance] (iUGATT_REP) to (iUGATT_ULS);

 \draw [rounded corners] (9.1,4.8) rectangle (15.6,3.5);

 \draw[->,double equal sign distance] (10.4,5.5) to (10.4,4.9);
 \draw[->,double equal sign distance] (12.7,3.4) to (12.7,2.8);
 \draw[<->,double equal sign distance] (UGATT_ULS) to (UGATT_REP);

 \draw[<->,double equal sign distance] (iUGATT) to (UGWA_Ult_iULS);

 \draw[->,double equal sign distance] (7,1.3) to (7,0.7);
 \draw [rounded corners] (2.35,1.4) rectangle (7.8, 2.6);

 \node (UGWA_UltULS) at (14,1.8) {UGWA $\wedge$ UltULS};
 \node (UltULS) at (12.7,0.5) {UltULS};

 \draw[->,double equal sign distance] (5.1,3.4) to (5.1,2.7);
 \draw[->,double equal sign distance] (UGATT) to (UGWA);
 \draw[<->,double equal sign distance] (UGATT) to (UGWA_UltULS);
 \node (Prop_iUGATT_Crit) at (5.1,2.25) {\footnotesize Proposition~\ref{prop:iUGATT_equiv_UGWA_Ult_iULS}};
 \draw [rounded corners] (9.7,1.4) rectangle (15.6, 2.6);

 \node (Prop_UGATT_Crit) at (12.7,2.25) {\footnotesize Proposition~\ref{prop:UGATT_equiv_UGWA_Ult_ULS}};

 \draw[->,double equal sign distance] (12.7,1.3) to (UltULS);

 \node (iREP) at (-0.25,0.5) {iREP};
 \node (Ult_iULS) at (2.2,0.5) {Ult-iULS};

 \draw[->,double equal sign distance] (-0.25,3.4) to (-0.25,2.7);

 \draw[->,double equal sign distance] (4,1.3) to (Ult_iULS);

 \node (Prop_iUGATT_Crit) at (-0.2,2.25) {\footnotesize Lemma~\ref{lem:iULS_criterion} };

 \draw[<->,double equal sign distance] (iULS) to (iREP_Ult_iULS);

 \draw[->,double equal sign distance] (0,1.3) to (Ult_iULS);
 \draw[->,double equal sign distance] (-0.25,1.3) to (iREP);
 \draw [rounded corners] (-2.5,1.4) rectangle (1.9, 2.6);

 \end{tikzpicture}

 \caption{Relations between stability notions}
 \label{fig:Stability}
 \end{figure*}

\appendix
\renewcommand\thesection{\Alph{section}}
  \section{Appendix}
\label{sec:appendix}
\renewcommand{\thetheorem}{\thesection.\arabic{theorem}}

  In this appendix, we show a result providing conditions for the
  existence of a $\KL$-bound for a function of two arguments that has been
  used in our proofs. Although the result may not be surprising for the
  expert, we have not found an explicit reference and so prefer to present
  the construction here.

  We start with an auxiliary statement:
  \begin{proposition}
  \label{prop:Partition_of_Reals}
  For any $z\in C(\R_+,\R)$ and for any $\eps>0$ there is a

sequence $\{R_k\}_{k\in\Z} \subset (0,+\infty)$ satisfying the following properties:
  \begin{itemize}
      \item[(i)] $R_k \to 0$ as $k\to -\infty$.
      \item[(ii)] $R_k\to+\infty$ as $k\to+\infty$.
      \item[(iii)] $R_k<R_{k+1}$ for all $k\in\Z$.

      \item[(iv)] $\max_{s\in[R_k,R_{k+1}]}z(s) - \min_{s\in[R_k,R_{k+1}]}z(s) <\eps$, for all $k\in\Z$.
  \end{itemize}
  \end{proposition}

We stress that by conditions (i), (ii), (iii), zero is the only finite accumulation point of $\{R_k\}_{k\in\Z}$.

  \begin{proof}
  It is easy to see that a sequence satisfying the properties (i)--(iii) always exists
  (and can be chosen independently on $z$). Pick any such sequence and denote it $\{S_k\}_{k\in\Z}$.
  Since $z\in C(\R_+,\R)$, by Cantor's theorem $z$ is uniformly continuous on $[S_k,S_{k+1}]$ for any $k\in\Z$.
  Thus, there exists a partition of $[S_k,S_{k+1}]$ into finitely many
  subintervals with boundary points $S_k=S_{k1} < S_{k2} \ldots < S_{km(k)}=S_{k+1}$ so that $\max_{s\in[S_{ki},S_{ki+1}]}z(s) - \min_{s\in[S_{ki},S_{ki+1}]}z(s) <\eps$, for each $i=1,\ldots,m(k)-1$.

  Now define the desired sequence $\{R_k\}_{k\in\Z}$ by inserting
  considering the ordered sequence $\{S_{kj}\}_{k\in\Z,
    j=1,\ldots,mk}$. Clearly, $\{R_k\}_{k\in\Z}$ satisfies (i)-(iv).
  \end{proof}

  The following estimation result is useful in our derivations.
  \begin{proposition}
  \label{prop:Upper_estimate_as_KL_function}
  Let $\psi:\R_+ \times \R_+ \to \R_+$ be any function which is
  nondecreasing and continuous at 0 in the first argument, nonincreasing
  in the second argument and so that $\lim_{t\to\infty}\psi(r,t)= 0$ for
  any $r\geq 0$. Let also $\psi(0,t)=0$ for any $t\geq 0$. Then there
  exists a $\beta\in\KL$ such that
  \[
  \psi(r,t)\leq \beta(r,t) \quad \forall r,t\geq \R_+.
  \]
  \end{proposition}

  \begin{proof}
  Pick a sequence $R:=\{R_k\}_{k\in\Z} \subset (0,+\infty)$ satisfying the properties (i)-(iii) of Proposition~\ref{prop:Partition_of_Reals} and
  another sequence $\tau:=\{\tau_m\}_{m\in\N} \subset [0,+\infty)$
  satisfying the properties (ii)-(iii) of
  Proposition~\ref{prop:Partition_of_Reals} and so that $\tau_0=0$. The Cartesian product $R\times\tau$ defines a mesh over $\R_+\times\R_+$.
  Let $\omega\in\KL$ be arbitrary so that $\omega(r,t) > 0$ for all $(r,t)
  \in (0,\infty) \times \R_+$.

  For each $k\in\Z$ and $m\in\N$, $m\neq 0$ define
  \[
  \beta(R_k,\tau_m):=\psi(R_{k+1},\tau_{m-1}) + \omega(R_{k+1},\tau_{m-1}).
  \]
  For $m=0$ define
  \[
  \beta(R_k,0)=\beta(R_k,\tau_0):=2\psi(R_{k+1},0) + \omega(R_{k+1},0).
  \]
  For each $k\in\Z$ and $m\in\N$ define $\beta(r,t)$ for all $(r,t)$ in
  the triangles with corner points ($(R_{k+1},\tau_{m})$, $(R_{k},\tau_{m+1})$, $(R_{k},\tau_{m})$) or
  ($(R_{k+1},\tau_{m})$, $(R_{k+1},\tau_{m+1})$, $(R_{k},\tau_{m+1})$)
  by linear interpolation of the values in the corner points (which have
  already been defined).

  This defines the values of $\beta$ in $(0,+\infty)\times[0,+\infty)=\bigcup_{k\in\Z}\bigcup_{m\in\N}[R_k,R_{k+1}]\times[\tau_k,\tau_{k+1}]$.

  Defining $\beta(0,t):=0, t\geq 0,$ we see that $\beta$ is defined over $\R_+\times\R_+$, is continuous, strictly increasing in the first argument and decreasing in the second argument. Since $\lim_{t\to\infty}\psi(r,t)= 0$ for any $r\geq 0$, it holds also that
  $\lim_{t\to\infty}\beta(r,t)= 0$ for any $r\geq 0$. Overall, $\beta\in\KL$.

  It remains to show that $\beta$ estimates $\psi$ from above.
  To see this, pick any $k\in\Z$ and $m\in\N$. For every $(r,s)\in [R_k,R_{k+1}]\times[\tau_m,\tau_{m+1}]$ we have:
  \begin{eqnarray*}
  \beta(r,s) - \psi(r,s) &\geq& \beta(R_k,\tau_{m+1}) - \psi(R_{k+1},\tau_{m}) \\
  &=& \psi(R_{k+1},\tau_{m}) + \omega(R_{k+1},\tau_{m}) - \psi(R_{k+1},\tau_{m})\\
  &=& \omega(R_{k+1},\tau_{m}).
  \end{eqnarray*}
  Hence, $\beta(r,s) \geq \psi(r,s)$ for all $r,s\geq0$.
  \end{proof}

\begin{acknowledgements}
This research has been supported by the German Research Foundation (DFG) within the project
``Input-to-state stability and stabilization of distributed parameter systems'' (grant Wi 1458/13-1).
\end{acknowledgements}


TITLE: What is the negation for "$A - B \subseteq C$?"
QUESTION [0 upvotes]: I was given an implication to prove, and I need to prove it using an indirect proof via contrapositive. I need to know the negation of this original statement: $A - B \subseteq C$. Presumably, I think it's $A - B \not\subseteq C$, but I'm not certain that $A - B \not\subseteq C$ is correct. The semantic of this statement is quite puzzling. Is it $\forall x \in A$, $\exists x \not\in B$, $\forall x \not\in C$? Does this mean $A - B$ and $C$ are disjoint sets when negated?

REPLY [0 votes]: Let $X$ and $Y$ be sets. The negation that involves a subset is simply $X \not\subseteq Y$. In this particular example, the negation of $A \setminus B \subseteq C$ is simply $A \setminus B \not\subseteq C$. At this point, we already did the negation. If we wanted to find another form of $A \setminus B \not\subseteq C$, we must consider the set $A \setminus B$. We know that there exist some element $x$, that is $x \in A$,  but $x \not\in B$, but we also know that $x \not\in C$ as well. Therefore, the negation of $A \setminus B \not\subseteq C$ can be simply rewritten to $A \not\subseteq B \cap C$.


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\title{A Unique Continuation Property for the Level Set Equation}
\author{Nick Strehlke\footnote{This research was partially supported by NSF Grant DMS 1404540.}}
\date{}

\maketitle

\begin{abstract}
We prove the following unique continuation result: if a solution to the level set equation for mean curvature flow in a mean-convex domain agrees to infinite order at the point where it attains its maximum with the solution for a ball, then it agrees everywhere and the domain is a ball.
\end{abstract}

\section{Introduction}

Let $\Omega$ be a mean-convex domain in $\mathbb R^{n+1}.$ The level set equation for mean curvature flow is a degenerate elliptic boundary value problem asking for a function $t\colon \Omega \to\mathbb R$ satisfying $t=0$ on $\partial \Omega$ and
\begin{align}
	|\nabla t| \on{div}\left(\frac{\nabla t}{|\nabla t|}\right) = -1. \label{lse}
\end{align}
The solution $t$ to this problem exists and is unique and twice differentiable on $\Omega.$ It is sometimes called the \emph{arrival time} for mean curvature flow on the domain $\Omega,$ for the following reason: Let $T = \max_{x\in\Omega} t(x)$ and for $\tau\in [0,T)$ define the hypersurface $M_\tau$ by
\begin{align*}
	M_\tau &= \{x\in \Omega\colon t(x) = \tau\}.
\end{align*}
Then the $1$-parameter family $\{M_\tau\}_{\tau\in [0,T)}$ of surfaces is a mean curvature flow: the position vector $x(\tau)$ of $M_\tau$ satisfies the equation
\begin{align*}
	\frac{\mathrm d x}{\mathrm d\tau} \bdot \bs N(x,\tau) &= - H(x,\tau),
\end{align*}
where $\bdot$ is the inner product in $\mathbb R^{n+1}$ and $H$ and $\bs N$ are the scalar mean curvature and outer unit normal, respectively, of the hypersurface $M_\tau.$\footnote{This point of view for studying mean curvature flow was first taken in a computational context by Osher and Sethian, \cite{OsSe88}, and the theory was then developed in \cite{ChGi91} and \cite{EvSp91}.}

When $\Omega = B_r(x_0)$ is the ball of radius $r$ centered at the point $x_0$ in $\mathbb R^{n+1},$ the solution $t_B$ to (\ref{lse}) is
\begin{align}
	t_B(x) &= \frac{r^2}{2n} - \frac{|x-x_0|^2}{2n} \label{lse-ball}
\end{align}
and the corresponding mean curvature flow is a family of homothetically shrinking spheres. The main result is that a solution to the level set equation (\ref{lse}) on a mean-convex domain $\Omega$ that attains its maximum at $x_0$ and agrees to infinite order at the point $x_0$ with the solution $t_B$ to the level set equation for a ball centered at $x_0$ must actually coincide with $t_B$ everywhere.\footnote{A different and more complicated \emph{parabolic} unique continuation property for self-shrinkers was recently proved by Jacob Bernstein in \cite{Be17}.}

\medskip
\begin{theorem}
	\label{unique-cont} Suppose $\Omega\subset\mathbb R^{n+1}$ is a mean-convex domain and $t\colon \Omega \to\mathbb R$ solves (\ref{lse}) and attains its maximum $T$ at the point $x_0\in \Omega.$ If
	\begin{align}
		t(x) = T - \frac{|x-x_0|^2}{2n} +O\left( |x-x_0|^N\right) \label{asympt}
	\end{align}
	as $x\to x_0,$ for every integer $N> 2,$ then $t(x) = T - |x - x_0|^2/(2n)$ and $\Omega = B_{(2nT)^{1/2}}(x_0)$ is the ball of radius $(2n T)^{1/2}$ centered at $x_0.$
\end{theorem}

\emph{Remarks.}

1. Let $\Omega$ be a mean-convex domain and suppose $t\colon \Omega\to\mathbb R$ satisfies (\ref{lse}). We already mentioned that $t$ is twice differentiable on its domain (this was proved in \cite{CoMi16}).\footnote{It was shown in \cite{Wh00} (Theorem 1.2) and \cite{Wh11} that any tangent flow of a smooth mean convex mean curvature flow is a generalized cylinder. From this one can figure out what the Hessian of the arrival time function must be if it exists. The remaining issue was to show that the Hessian exists, which is equivalent to the problem of uniqueness of tangent flows. This was solved in \cite{CoMi15}.} The Hessian of $t$ at a critical point $x_0$ is then
\begin{align*}
	\nabla^2 t(x_0) &= - \frac{1}{2k}P,
\end{align*}
where $k$ is an integer between $1$ and $n$ and $P$ is a projection onto a $n-k$ dimensional hyperplane. See \cite{CoMi18}. The hypothesis of Theorem \ref{unique-cont} implies that $k=n.$ In this case $t$ is actually $C^2$ in a neighborhood of the maximum, and the corresponding mean curvature flow becomes extinct at $x_0$ in such a way that the rescaled mean curvature flow converges to a round sphere. In particular, the isolated point $\{x_0\}$ is a connected component of the level set $\{x\colon t(x) = T\},$ and nearby level sets are convex. Because a mean curvature flow cannot coincide with a sphere at any time unless it is a shrinking sphere, it is therefore sufficient to prove Theorem \ref{unique-cont} in case $\Omega$ is a convex domain.

2. The hypothesis (\ref{asympt}) for a fixed $N>2$ likely implies (for a solution to (\ref{lse})) that $t$ is $C^{N-1}$ near $x_0.$ This is known in case $N=3$ (Theorem 6.1 of \cite{Hu93}) and in case $N=4$ (Corollary 5.1 of \cite{Se08}). This fact is not required for our result, however, and we do not investigate it here. Of course, Theorem \ref{unique-cont} implies that $t$ is analytic if it satisfies (\ref{asympt}) for \emph{all} $N.$

\section{Proof of Theorem \ref{unique-cont}}

To prove Theorem \ref{unique-cont}, we relate the asymptotic behavior of $t$ near its maximum to the behavior of the corresponding mean curvature flow near its singularity. We show that the hypothesis (\ref{asympt}) for all $N>2$ implies that the rescaled mean curvature flow converges to a stationary round sphere at a rate that is faster than any exponential, and then that this cannot happen unless the rescaled flow is identically equal to the stationary sphere. As mentioned in the remarks following the theorem, it is sufficient to prove Theorem \ref{unique-cont} in case $\Omega$ is a convex domain, and we restrict attention to this case from here on.

We now briefly describe the rescaled mean curvature flow for a convex surface. Let $\Omega\subset\mathbb R^{n+1}$ be a bounded convex region and let $\{M_\tau\}_{\tau\in [0,T]}$ be the mean curvature flow starting from $M_0 = \partial \Omega.$ By a theorem of Huisken, \cite{Hu84}, the flow $M_t$ shrinks down to a single point $x_0$ (that is, $M_T =\{x_0\}$) and the translated and rescaled flow $(T-\tau)^{-1/2}(M_\tau- x_0)$ converges in $C^k,$ for any $k,$ to the round sphere $\bf{S}^n$ of radius $(2n)^{1/2}$ centered at the origin.

The following rescaling procedure is standard in the study of mean curvature flow.\footnote{See the first sections of \cite{CoMi12} for an excellent introduction and overview.} We perform the substitution $s = -\log{(T-\tau)}$ and denote by $\Sigma_s$ the surface $(T-\tau)^{-1/2}(M_\tau - x_0).$ The $1$-parameter family $\{\Sigma_s\}_{s\in [-\log{T},\infty)}$ is said to be a \emph{rescaled mean curvature flow}. By rescaling the surfaces $M_\tau,$ we can arrange that the time $T$ of extinction is $1,$ and therefore that the rescaled flow is defined for $s\geq 0.$ Its position vector $y(s)$ satisfies the equation
\begin{align*}
	\frac{\mathrm d y}{\mathrm ds} \bdot N(y,s) = - H(y,s) + \frac{1}{2}y\bdot N(y,s),
\end{align*}
where $H$ and $N$ are the scalar mean curvature and outer unit normal, respectively, for $\Sigma_s.$

Let $\bf{n}(x) = x/|x| = x/(2n)^{1/2}$ be the outer unit normal to the sphere $\bf{S}^n$ at the point $x.$ Because $\Sigma_s$ converges in $C^k$ to $\bf{S}^n$ as $s\to\infty,$ the surface $\Sigma_s$ is a normal graph over $\bf{S}^n$ for sufficiently large $s$: there exists $s_0\geq0$ and a scalar function $u\colon \bf{S}^n\times [s_0,\infty)\to\mathbb R$ for which
\begin{align}
	\Sigma_s &= \{x+ u(x,s)\bf{n}(x)\colon x\in\bf{S}^n\}. \label{defn.u}
\end{align}
The function $u$ is uniquely determined as the solution of a quasilinear parabolic PDE on the sphere. The parabolic evolution has the constantly zero function for a solution (a stationary state), corresponding to the fact that the stationary sphere $\bf{S}^n$ satisfies the rescaled MCF equation. The linearization of the parabolic operator at the zero solution is $\partial_s - \Delta - 1,$ where $\Delta$ is the Laplacian on the sphere. The following lemma converts Theorem \ref{unique-cont} to a problem about functions $u$ satisfying this parabolic equation: if $t$ is a solution to the level set equation satisfying the assumption (\ref{asympt}) for all $N,$ then the corresponding solution to rescaled MCF converges to $\bf{S}^n$ faster than any exponential. The problem is then reduced to showing that this cannot happen unless the rescaled MCF is identically $\bf{S}^n.$

Note we may assume by translating everything that $x_0 = 0.$ Thus we state the lemma for the case when the mean curvature flow becomes extinct at the origin.

\medskip
\begin{lemma}
	\label{conv} Suppose $\Omega\subset\mathbb R^{n+1}$ is a convex domain and $t\colon \Omega \to\mathbb R$ solves (\ref{lse}) and attains its maximum $T$ at the origin. Suppose
	\begin{align}
		t(x) = T - \frac{|x|^2}{2n} +O\left( |x|^N\right) \label{asympt'}
	\end{align}
	as $x\to 0,$ for every integer $N> 2.$ Then for any integer $k\geq 0,$ the rescaled MCF $\{\Sigma_s\}$ that $t$ defines converges to $\bf{S}^n$ in the Sobolev space $H^k$ faster than any exponential.

\end{lemma}
\begin{proof}
We use the following fact: if $\{\Sigma_s\} = \{x+ u(x,s)\bf{n}(x)\colon x\in\bf{S}^n\}$ is a rescaled MCF converging as $s\to\infty$ to the sphere in $L^2,$ that is, for which $u$ converges to zero in $L^2(\bf{S}^n),$ then $u$ is bounded in $H^k(\bf{S}^n)$ for every $k\geq 1$ (actually $u$ converges exponentially in $H^k$). On account of the interpolation inequalities
\begin{align*}
	\|u\|_{H^k}\leq \|u\|_{L^2}^{1/2} \|u\|_{H^{2k}}^{1/2},
\end{align*}
this reduces the proof of the lemma to showing that the function $u$ converges to zero faster than any exponential in $L^2(\bf{S}^n).$

Now we show that (\ref{asympt'}) holding for all $N$ implies that $u(s)$ converges to zero faster than any exponential in $L^\infty(\bf{S}^n),$ hence in $L^2(\bf{S}^n).$

First observe: if (\ref{asympt'}) holds for some $N,$ then
\begin{align*}
	|x|^2 = (2n)(T-t(x)) + O\left((T-t(x))^{N/2}\right)
\end{align*}
as $T-t\to0,$ for the same $N.$ Next, use this to write
\begin{align*}
	(T-t(x))^{-1/2}x  &= \left( |x|^2/(2n) + O\left((T-t(x))^{N/2}\right) \right)^{-1/2}x  \\
	&= (2n)^{1/2} \frac{x}{|x|} + O\left( \frac{(T- t(x))^{N/2}}{|x|^2}\right) \frac{x}{|x|} \\
	&= (2n)^{1/2} \frac{x}{|x|} + O\left((T-t(x))^{N/2-1}\right)\frac{x}{|x|}.
\end{align*}
This says exactly that if $s = -\log{(T-\tau)}$ and
\begin{align*}
	\{y+ u(y,s) \bf{n}(y)\colon y\in \bf{S}^n\} = \Sigma_s = \left\{ (T-\tau)^{-1/2} x\colon t(x) = \tau\right\}
\end{align*}
is the rescaled MCF defined by $t$ (it will be a graph over $\bf{S}^n$ for $\tau$ sufficiently close to $T$), then
\begin{align*}
	|u(y,s)| = O\left((T-\tau)^{N/2 - 1}\right) = O\left(e^{-(N/2 - 1) s}\right)
\end{align*}
as $s \to\infty.$ Therefore if (\ref{asympt'}) holds for all $N,$ then $u$ converges to zero faster than any exponential in $L^\infty(\bf{S}^n).$ Since $\bf{S}^n$ has finite volume, $u$ converges to zero faster than any exponential in $L^2(\bf{S}^n)$ as well.
\end{proof}

Having established Lemma \ref{conv}, Theorem \ref{unique-cont} is a consequence of the following theorem.

\medskip
\begin{theorem}
	\label{rapid} Let $r>n/2+1$ be an integer. Suppose $\Sigma_s = \{y+u(y,s)\bf{n}(y)\colon y\in \bf{S}^n\},$ $s\geq 0,$ is a rescaled MCF and suppose that it converges to $\bf{S}^n$ in $H^{r}(\bf{S}^n)$ faster than any exponential in the sense that
	\begin{align*}
		\lim_{s\to\infty} e^{\sigma s}\| u(\cdot,s)\|_{H^{r}(\bf{S}^n)} = 0
	\end{align*}
	for all $\sigma>0.$ Then $u$ is identically zero and $\Sigma_s = \bf{S}^n$ for all $s.$
\end{theorem}

The equation satisfied by $u$ in order for the normal graph $\Sigma_s = \{y+u(y,s)\bf{n}(y)\colon y\in \bf{S}^n\}$ to evolve by rescaled MCF can be written in the form
\begin{align}
	\partial_s u &= \Delta u + u + N(u,\nabla u,\nabla^2u) \label{rMCF'}
\end{align}
where $\Delta$ is the Laplacian on $\bf{S}^n$ and $N$ is a nonlinear term of the form
\begin{align}
	N(u,\nabla u,\nabla^2 u) = f(u,\nabla u) + \on{trace}(B(u,\nabla u)\nabla^2u), \label{nonlin-error}
\end{align}
where $f$ and $B$ are smooth, $B(0,0) = f(0,0) = \mathrm df(0,0) = 0.$ In other words, $N$ vanishes up to quadratic error at zero.

The important feature of this equation is that the linear operator $\partial_s - \Delta -1$ gives a good approximation to the nonlinear operator in (\ref{rMCF'}): in a Sobolev space $H^r(\bf{S}^n)$ of high enough order $r,$ a function $u$ the normal graph of which evolves by rescaled MCF satisfies
\begin{align*}
	\|(\partial_s - \Delta - 1)u\|_{H^r(\bf{S}^n)} \leq C \|u\|_{H^{r+1}(\bf{S}^n)} \|u\|_{H^{r+2}(\bf{S}^n)}.
\end{align*}
This bound is what implies that if $u$ converges to zero as $s\to+\infty,$ it must do so at an exponential rate (unless it is identically zero). We state this bound as a lemma. This is a textbook result, but the proof is also written in a companion paper.

\medskip
\begin{lemma}
\label{nonlin-bd}
	Suppose $N$ is a smooth function satisfying (\ref{nonlin-error}). If $r>n/2+1$ is an integer and $\|u\|_{H^r(\bf{S}^n)}\leq 1,$ then
	\begin{align*}
		\|N(u,\nabla u,\nabla^2 u)\|_{H^r(\bf{S}^n)} \leq C \|u\|_{H^{r+1}(\bf{S}^n)} \|u\|_{H^{r+2}(\bf{S}^n)}
	\end{align*}
	for some constant $C$ depending only on $N$ and $r.$
\end{lemma}

Using Lemma \ref{nonlin-bd}, we now prove Theorem \ref{rapid}.

\begin{proof}[Proof of Theorem \ref{rapid}]
	Throughout the proof, we abbreviate the $H^r(\bf{S}^n)$ norm $\|\cdot\|_{H^r(\bf{S}^n)}$ by $\|\cdot\|_r,$ we abbreviate $u(\cdot,t)$ by $u(t),$ and we abbreviate the nonlinear error $N(u,\nabla u,\nabla^2 u)$ by $N(u).$  Let $\lambda_1<\lambda_2<\cdots$ be the eigenvalues of the operator $-\Delta - 1,$ and denote by $\Pi_k$ orthogonal projection onto the direct sum of eigenspaces corresponding to $\lambda_j$ with $j\geq k.$

	We will prove that a solution $u$ satisfying (\ref{rMCF'}) obeys, for each positive integer $k$ and each $s_0\geq 0,$ the inequality
\begin{align}
	e^{\lambda_k (s-s_0)}\|u(s)\|_r \leq \|\Pi_{k} u(s_0)\|_r + \int_{s_0}^\infty e^{\lambda_k (t-s_0)} \|N(u(t))\|_r\,\mathrm dt. \label{arrival.0th.1}
\end{align}
Lemma \ref{nonlin-bd} implies that there is a constant $C$ depending on $N$ with the property that
\begin{align*}
	\|N(u)\|_r \leq C\|u\|_{r+1}\|u\|_{r+2}\leq C\|u\|_{r+2}^2\leq C\|u\|_r\|u\|_{r+4}.
\end{align*}
The last inequality follows from Cauchy--Schwarz and integration by parts for example.

It then follows that
\begin{align*}
	e^{\lambda_k (s-s_0)}\|u(s)\|_r &\leq \|\Pi_{k} u(s_0)\|_r + C\int_{s_0}^\infty e^{\lambda_k (t-s_0)} \|u(t)\|_{r}\|u(t)\|_{r+4}\,\mathrm dt \\
	&\leq  \|\Pi_{k} u(s_0)\|_r + C\left(\sup_{t\geq s_0}e^{\lambda_k (t-s_0)} \|u(t)\|_r\right) \int_{s_0}^\infty \|u(t)\|_{r+4}\,\mathrm dt.
\end{align*}
Taking the supremum over $s\geq s_0$ on the left side then gives
\begin{align*}
	\left(\sup_{t\geq s_0}e^{\lambda_k (t-s_0)} \|u(t)\|_r\right)\left(1 - C\int_{s_0}^\infty \|u(t)\|_{r+4}\,\mathrm dt\right)\leq \|\Pi_{k} u(s_0)\|_r
\end{align*}
for all $k.$ Because convergence to the sphere is necessarily exponential in $C^k$ for every $k,$ the $H^{r+4}(\bf{S}^n)$ norm $\|u(t)\|_{r+4}$ is integrable (as are all other $H^s$-norms), and therefore we can choose $s_0$ so large that
\begin{align}
	C\int_{s_0}^\infty \|u(t)\|_{r+4}\,\mathrm dt\leq 1/2. \label{small-int}
\end{align}
Moreover, we choose $s_0$ to be the least nonnegative number for which (\ref{small-int}) holds. For this $s_0$ we obtain
\begin{align*}
	\sup_{t\geq s_0}e^{\lambda_k (t-s_0)} \|u(t)\|_r\leq 2 \|\Pi_{k} u(s_0)\|_r
\end{align*}
for all positive integers $k,$ and since the right side vanishes in the limit $k\to\infty$ and the left side is non-decreasing in $k$ it follows that $\|u(t)\|_r = 0$ for $t\geq s_0.$

Now we show that $s_0=0.$ Since $\|u(t)\|_r = 0$ for $t\geq s_0,$ we also have $\|u(t)\|_{r+4} = 0$ for $t\geq s_0.$ Consequently,
\begin{align*}
	C\int_{s_0}^\infty \|u(t)\|_{r+4}\,\mathrm dt = 0.
\end{align*}
If $s_0>0,$ then it cannot possibly be the smallest positive number for which (\ref{small-int}) holds, and we arrive at a contradiction. Thus $u$ is in fact identically zero for all $s\geq 0.$

Therefore it is enough to prove (\ref{arrival.0th.1}). Write $L = \Delta+1$ for brevity. We will briefly explain how the assumption that $u$ converges to zero faster than any exponential leads to the following representation formula:
\begin{align*}
	e^{\lambda_k (s-s_0)} u(s) & = e^{(\lambda_k + L)(s-s_0)} \Pi_{k} u(s_0)   + \int_{s_0}^s e^{(\lambda_k + L)(s-t)} e^{\lambda_k(t-s_0)} \Pi_{k} N(u(t))\,\mathrm dt \\
	&\quad  - \int_s^\infty e^{(\lambda_k + L)(s-t)}e^{\lambda_k (t-s_0)}(1-\Pi_{k}) N(u(t))\,\mathrm dt,
\end{align*}
 Notice that $\lambda_k +L$ is non-positive definite on the range of $\Pi_{k}$ and positive definite on the range of $1-\Pi_{k}.$ Thus (\ref{arrival.0th.1}) follows by simply taking the $H^r$ norm of both sides and applying the triangle inequality repeatedly to the right side.

 The representation can be derived from the variation of constants formula
\begin{align*}
	u(s_1)  &= e^{L(s_1-s)} u(s) + \int_{s}^{s_1} e^{L(s_1-t)} N(u(t))\,\mathrm dt,
\end{align*}
where $s_1\geq s,$ in the following way.\footnote{This is a standard trick in the construction of invariant manifolds for ODE.} Apply the projection $1-\Pi_k$ to both sides of the variation of constants formula. Apply $e^{-L(s_1-s)},$ which is a bounded operator on the finite-dimensional image of $1-\Pi_k,$ to both sides and rearrange to obtain
\begin{align*}
	(1-\Pi_k)u(s) &= e^{-L(s_1-s)} (1-\Pi_k)u(s_1) - \int_{s}^{s_1} e^{L(s-t)}(1-\Pi_k) N(u(t))\,\mathrm dt.
\end{align*}
Since $e^{-L(s_1 - s)}$ is bounded by $e^{\lambda_{k-1}} (s_1- s)$ on the image of $1-\Pi_k,$ we can send $s_1\to\infty$ and by our assumption that $u_1-u_2$ vanishes more rapidly than any exponential, the term $e^{-L(s_1 - s)} (1-\Pi_k) u(s_1)$ converges to zero. The integral, meanwhile, is absolutely convergent. Thus we obtain
\begin{align*}
	(1-\Pi_k)u(s) &= - \int_s^\infty e^{L(s-t)}(1-\Pi_k) N(u(t))\,\mathrm dt,
\end{align*}
for any $k.$ Substituting this equation into the variation of constants formula (with $s_1$ replaced by $s$ and $s$ replaced by $s_0$) gives the representation described above.
\end{proof}

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TITLE: irreducibility in $\mathbb Z/p\mathbb Z[X]$
QUESTION [2 upvotes]: I've read somewhere that $q\in\mathbb Z/p\mathbb Z[X]$ is irreducible iff $\forall a\in\mathbb Z/p\mathbb Z:q (a)\neq0$. Now for degree one polynomials its not true. For higher degree it is true that if $q(a)=0$ for some $a$ then $q$ is reducible. But I couldn't verify the stronger statement. ($p$ a prime)

REPLY [1 votes]: Over any field, the product of two irreducible polynomials of degree $>1$ is a reducible polynomial with no roots in that field.


TITLE: Orthonormal basis of a vector space with polynomials of the form $ax+b$
QUESTION [0 upvotes]: Find the Orthonormal basis of vector space $V$ of the linear polynomials of the form $ax+b$  such that  $\:$, $p:[0,1] \to \mathbb{R}$. with inner product
$$\langle p,q \rangle= \int_0^1 p(x)q(x) dx$$ for $q, p \in V$
The polynomial in $V$ are linear so the basis of $V$ are $\left \{1,x  \right \}$ further i am not getting how to proceedes further
please help
Thankyou.

REPLY [1 votes]: A standard approach for these types of problems is to use the Gram-Schmidt procedure:

You already found a basis of $V$, so let $\mathbf{v}_1=1$ and $\mathbf{v}_2=x$. Then an orthogonal basis $\{\mathbf{u}_1,\mathbf{u}_2\}$ is given by:
$$\mathbf{u}_1=\mathbf{v}_1$$
$$\mathbf{u}_2=\mathbf{v}_2-\frac{\langle \mathbf{v}_2,\mathbf{u}_1\rangle}{\langle \mathbf{u}_1,\mathbf{u}_1\rangle}\mathbf{u}_1$$
So in your case, all you have to do is compute two simple integrals.
An orthonormal basis $\{\mathbf{w}_1,\mathbf{w}_2\}$ can be found by normalizing the two vectors $\mathbf{u}_1$ and $\mathbf{u}_2$. Namely:
$$\mathbf{w}_1=\frac{\mathbf{u}_1}{\|\mathbf{u}_1\|}=\frac{\mathbf{u}_1}{\sqrt{\langle \mathbf{u}_1,\mathbf{u}_1\rangle}}$$
$$\mathbf{w}_2=\frac{\mathbf{u}_2}{\|\mathbf{u}_2\|}=\frac{\mathbf{u}_2}{\sqrt{\langle \mathbf{u}_2,\mathbf{u}_2\rangle}}$$
This requires the computation of one additional (easy) integral.


TITLE: Convergence of non-negative series
QUESTION [2 upvotes]: Let $x_n$ be a sequence of non-negative numbers. I want to show
$\sum_{n=1}^\infty x_n<\infty \Longleftrightarrow \sum_{n=1}^\infty \frac{x_n}{1+x_n}<\infty$
One direction is easy:
$x_n\geq\frac{x_n}{1+x_n}\;\;\forall n$, therefore $\sum_{n=1}^\infty x_n<\infty \Longrightarrow \sum_{n=1}^\infty \frac{x_n}{1+x_n}<\infty$
Any hints for the other direction?

REPLY [2 votes]: Any convergent series of positive numbers has a maximum element.  Therefore, if $\sum{x_n\over1+x_n}$ converges, then there exists an $N$ such that
$${x_n\over1+x_n}\le{x_N\over1+x_N}\quad\text{for all }n$$
From this we see that
$${x_n\over1+x_n}=x_n\left(1-{x_n\over1+x_n}\right)\ge x_n\left(1-{x_N\over1+x_N}\right)=x_n\left(1\over1+x_N\right)$$
and therefore
$$\sum_{n=1}^\infty x_n\le(1+x_N)\sum_{n=1}^\infty{x_n\over1+x_n}$$


TITLE: roots of a linear DE $(D^2+2cD+k)y=0$ given $c<0,k>0, c^2>k$
QUESTION [4 upvotes]: Let y(x) be a non-trivial solution of the second order linear differential equation
$$\dfrac{d^2y}{dx^2}+2c\dfrac{dy}{dx}+ky=0 $$, where $c<0,k>0, c^2>k$. Then,
(a) $|y(x)|\to\infty$ as $x\to\infty$
(b) $|y(x)|\to 0$ as $x\to\infty$
(c) $\lim\limits_{x\to\pm\infty}|y(x)|$ exists and is finite.
(d) none of the above
Now, solving the ODE, I get the roots as $-c\pm\sqrt{c^2-k}$.
The two roots will be real, distinct and positive and hence, option (a) should be the answer.
Did I solve this correctly ?

REPLY [1 votes]: Yes : $y=e^{dx}$ so that $$ d=-c\pm\sqrt{c^2-k}$$


TITLE: matrix calculus (differentiation of complex matrix)
QUESTION [1 upvotes]: I know that $f(x)=||Ax-b||_2^2$ (real matrix) has gradient $\partial f/\partial x=A^T(Ax-b)$.
Now suppose $A$ is complex, then how can I prove that $\partial f/\partial x=A^*(Ax-b)$?

REPLY [4 votes]: I assume that $x$ is real in both cases. For real-valued $A$ and $b$ you have
$$f(x)=||Ax-b||_2^2=(Ax-b)^T(Ax-b)=x^TA^TAx-2x^TA^Tb+b^Tb$$
So
$$\partial f/\partial x=2A^T(Ax-b)$$
For complex $A$ and $b$ we get
$$f(x)=||Ax-b||_2^2=(Ax-b)^H(Ax-b)=x^TA^HAx-x^TA^Hb-b^HAx+b^Hb=\\
=x^TA^HAx-2x^T\text{Re}\{A^Hb\}+b^Hb$$
where $^H$ denotes hermitian conjugation.
The derivative w.r.t. $x$ is then
$$\partial f/\partial x=2(A^HAx-\text{Re}\{A^Hb\})$$