TITLE: Are vectors truly independent of coordinate systems?
QUESTION [5 upvotes]: I have been told to think of vectors as existing independent of a coordinate system. This means that the magnitude of a vector should be independent of any coordinate system we choose. Galilean transformations of the form
$$ x' = x - vt $$
do not preserve the magnitude of the velocity vectors however.  How is it possible to have a vector that has a different magnitude in a different coordinate system?

REPLY [2 votes]: One thing that may cause some confusion is that there are some differences between the way physicists tend to think about vectors and scalars and the way mathematicians tend to think about them. Physicists tend to think about them in this way:

A 3-vector is a vector that transforms under a change of basis in the
same way as a spatial displacement.
A 4-vector is a vector that transforms under a change of basis in the
same way as a spacetime displacement.
A scalar is a quantity that does not change at all under a change of
basis.

Based on these ideas, I think it's helpful to introduce the idea of what I call an "incomplete object," or IO. An IO is a mathematical object that doesn't contain enough information to allow you to transform it. Suppose that I go and visit Gettysburg and stand in front of the brass plaque marking the site of the battle. I could say that I have a displacement vector $\Delta\textbf{x}=0$ between my present position in space and the position where the fighting happened. But of course this is all under the assumption that the earth is at rest. There is certainly a Galilean frame of reference in which $\Delta\textbf{x}=0$, but there are other frames in which $\Delta\textbf{x}\ne0$. This $\Delta\textbf{x}$ is an IO in the context of Galilean transformations. Suppose I tell you that $\Delta\textbf{x}=0$ in a certain frame, and then ask you to find the value of $\Delta\textbf{x}$ in some other frame, say, a frame moving toward Sirius at $10^5$ m/s. This is not enough information. In order to carry out the calculation, you would also need to know the time between the battle of Gettysburg and the present day. To make the displacement not be an IO, we would need to change it into a 4-vector.
The issue with velocity 3-vectors is basically the same as the issue with displacement 3-vectors. A velocity 3-vector is just a displacement divided by a time, so it's an IO under Galilean transformations. Relativistically, we make use of velocity 4-vectors (which have an arbitrary normalization), and these 4-vectors do transform appropriately under a Lorentz transformation (although they do not combine according to vector addition in relative motion).
Another nice example of an IO has to do with the usual method for calibrating the magnetic compass built into some handheld GPS units. The device's instructions tell you to hold it in a horizontal plane and slowly rotate 360 degrees. Because the device believes in rotational invariance, it knows how $B_x$ and $B_y$ should transform, and if it finds that $\sqrt{B_x^2+B_y^2}$ doesn't stay constant, it can recalibrate itself to eliminate the discrepancy. But if you later tilt the device so it isn't in a horizontal plane, it will get sad and confused. This tells you that $(B_x,B_y)$ is an IO, and you need to extend it to $(B_x,B_y,B_z)$.
Another example of an IO is charge density $\rho$. To make it a complete object, you need to extend it to the current 4-vector $(\rho,\textbf{j})$.
One thing to watch out for in Galilean relativity is that there is no metric. There is no unified system of measurement that measures both time and space. Therefore although you can make displacement and velocity vectors into Galilean 4-vectors, which are god-fearing complete objects rather than IOs, you can't talk about the magnitude of a Galilean 4-vector.


TITLE: I still can't figure out sig figs... how to deal with volume?
QUESTION [0 upvotes]: "Jupiter is approximately a sphere of radius $6.99 \times 10^7 \text{ m}$. (c) What is its volume in cubic kilometers? "
I have $V = \frac{4}{3}\pi r^3$. Then, $\frac{4}{3}\pi (6.99 \times 10^7\text{ m})^3$. Then, $2.93 \times 10^{22}\text{ m}^3$. Then, $10^{13}\text{ km}^3$.
But the software says the answer is "1.43e+15." I don't understand.

REPLY [0 votes]: This time I will try to break down the problem a little bit more clearly than with the area example, to hopefully help you understand this. I would highly recommend checking out some simple manipulation examples, for starters, Khan Academy on DA, Khan Academy on Sig. Figs, and Khan Academy on Scientific Notation. I think you need to work on your baseline understanding before attempting these results.

\begin{align}
\frac{4}{3} \pi r^3 &= \frac{4}{3} \cdot \pi \cdot (6.99 \times 10^7 m )^3 \\
&= \frac{4}{3}\pi \cdot (6.99)^3 \times (10^7)^3 \times (m)^3 \\
&= \frac{4}{3}\pi \cdot (6.99 \cdot 6.99 \cdot 6.99) \times (10^{7 \cdot 3} )\times (m \cdot m \cdot m )\\
&= 1430 \times (10^{21}) \times (m\cdot m\cdot m) \\
&= (1.43 \times 10^3) \times 10^{21} \times (m\cdot m\cdot m) \\
&= 1.43 \times 10^{3 + 21}\times (m \cdot m\cdot m)\\
&= 1.43 \times 10^{24} \times m\cdot m \cdot m\\
&= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1000m}{1000m}) \\
&= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1km}{1000m}) \\
&= 1.43 \times 10^{24} \times m\cdot m \cdot m \times (\frac{1km}{1000m} \cdot \frac{1km}{1000m} \cdot \frac{1km}{1000m}) \\
&=  1.43 \times 10^{24} \times (\frac{1km \cdot m}{1000m} \cdot \frac{1km\cdot m}{1000m} \cdot \frac{1km\cdot m}{1000m}) \\
&= 1.43 \times 10^{24} \times (\frac{1km}{1000} \cdot \frac{1km}{1000} \cdot \frac{1km}{1000}) \\
&= 1.43 \times 10^{24} \times (\frac{1km}{10^3} \cdot \frac{1km}{10^3} \cdot \frac{1km}{10^3}) \\
&= 1.43 \times 10^{24} \times \frac{km^3}{(10^3)^3} \\
&= 1.43 \times \frac{10^{24}}{(10^3)^3} \times km^3 \\
&= 1.43 \times \frac{10^{24}}{10^9} \times km^3 \\
&= 1.43 \times 10^{24 - 9} \times km^3 \\
&= 1.43 \times 10^{15} km^3
\end{align}


TITLE: A type of Hölder function
QUESTION [0 upvotes]: I was doing the following exercise: if $ f: \mathbb{R}^n \setminus \{ 0 \} $ defined by $ f (x) = | x |^\alpha x_i $ for a fixed i, then $ f \in C^{0, \beta} (B_1 (0)) $, for $ 0<\alpha\leq 1$.

I tried to use the same idea as in the case where $ f (x) = |x|^\alpha $ (as for example in + Proving that $f(x) = \vert x \vert^{\alpha}$ is Holder continuous, inequality help). But I cannot simplify and adapt the expression properly. A tip would be welcome!

REPLY [2 votes]: Here this is much easier since the function is Lipshitz. Since $∇|x| = \frac{x}{|x|}$, we can compute
$$
∇f(x) = \alpha \,|x|^{\alpha-2}\, x_i\, x + |x|^{\alpha} 1_i
$$
where I used the notation $1_i = (\delta_{i,j})_{j=1..n}$ to denote the vector with coordinates $0$, except the $i$-th coordinate, which is $1$.
Therefore,
$$
|∇f(x)| \leq \alpha\,|x|^{\alpha-1} |x_i| + |x|^{\alpha} ≤ (α +1) \, |x|^\alpha,
$$
from which we deduce that $\|∇f\|_{L^\infty(B_1(0))} ≤ α+1$, so that $f$ is Lipshitz continuous $f∈C^{0,1}$ (and so Hölder continuous for any $\alpha\leq 1$ since $C^{0,1}⊂ C^{0,\alpha}$).


TITLE: How to get this result for this integral computation?
QUESTION [0 upvotes]: I do not understand how to get the following:
$$
\sum_{\lvert m\rvert <N}(1-\frac{\lvert m\rvert}{N})f(m)\int_{-\pi}^{\pi}e^{i(h-m)v}\, dv=\begin{cases}(1-\frac{\lvert h\rvert}{N})f(h), & \lvert h\rvert <N\\0, & \text{otherwise}\end{cases}
$$
Do you see that and can please explain that to me?
I have no idea, unfortunately.

REPLY [0 votes]: $$ \int_{-\pi}^{\pi} e^{i(h-m)v} \, dv = \begin{cases} 2\pi & h=m \\ 0 & h \neq m \end{cases} $$
It follows that, if $\lvert h \rvert \geqslant N$, there is no term in the sum for which the integral is nonzero. On the other hand, if $\lvert h \rvert <N$, then the above identity gives $2\pi$ of the term with $m=h$, which is (almost) what you've got. (The discrepancy's not my error; did you copy something down incompletely?)


text:
\begin{document}

We illustrate our method for the computation of time-marginal distributions in numerical experiments using the following setting based on the model of Mutual Hazard Networks.
\subsection{Setting}
\label{subsection:settings}

We consider $d$ automata and sparse parameters $\PM \in \R^{d \times d}$ with a particular block-diagonal form.
Each block of size $b \times b$ characterizes a subset of $b$ automata which directly affect one another.
There are no direct effects between automata from different blocks.
In each block, we would like the effects between neighboring automata to be stronger than those between automata that are farther apart.
Therefore, we draw the entries $B[i, j]$ of the blocks from a normal distribution with mean $1$ and standard deviation $\sigma = 2^{- 1 - \vert i - j\vert}$ (and restrict them to $\R^{>0}$).
Two possible examples for $d = 4$ automata and block size $b \in \{2, 4\}$ are given below:
\begin{align*}
		\begin{pmatrix}
		{\color{blue}1.2688 }& {\color{red}1.4585 } & {\color{lightgray}1} & {\color{lightgray}1}  \\
		{\color{red}0.43529  }& {\color{blue}1.4311  }& {\color{lightgray}1}  & {\color{lightgray}1}  \\
		{\color{lightgray}1}  & {\color{lightgray}1}  &{\color{blue}1.1594}  & {\color{red} 0.67308}\\
		{\color{lightgray}1}  & {\color{lightgray}1}  & {\color{red} 0.8916} & {\color{blue}1.1713 }
		\end{pmatrix}
	\quad
		\begin{pmatrix}
			{\color{blue}1.2688 }& {\color{red}1.4585 } & {\color{gray}0.71764} & {\color{lightgray}1.0539}  \\
			{\color{red}0.43529  }& {\color{blue}1.4311  }& {\color{red}0.8916}  & {\color{gray}1.0428}  \\
			{\color{gray}1.4473 }  & {\color{red}1.6924 }  &{\color{blue}1.1594}  & {\color{red} 0.67308}\\
			{\color{lightgray}1.0453}  & {\color{gray}0.99212}  & {\color{red} 0.8916} & {\color{blue}1.1713 }
		\end{pmatrix}
\end{align*}
Parameters of the same color follow the same distribution.

For the application of~\cref{alg:uniformtization} we always choose the bound $\gamma$ as in~\cref{eq:upper_bound}, which for Mutual Hazard Networks can be reduced to
\begin{align}
\label{eq:MHN_upper_bound}
	\gamma =  \sum\limits_{\nu = 1}^d ~ \prod\limits_{\mu = 1}^d \max\{1, \PM[\nu, \mu]\}.
\end{align}
We use a canonical balanced tree tensor network for the application of the hierarchical Tucker format, i.e., the automata are assigned to the leaves following their ordering, see~\cref{figure:sv_d=8_b=4}. We perform all experiments for $100$ randomly generated sample parameters for each combination of block size $b$ and number $d$ of automata.
Unless stated otherwise, we compute low-rank approximations of the marginal distribution $\pPM$ using~\cref{alg:uniformtization} with a maximum relative truncation error of $\eps= 10^{-8}$. The algorithm stops when the norm of the relative residual  is smaller than a tolerance value of $\tol = 10^{-4}$.
The mean values we compute are arithmetic means.

\end{document}

config: arxiv


TITLE: Cesaro means and Banach limits
QUESTION [10 upvotes]: Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) assigns the same limit value.
 Does a sequence belong to this class if its Cesaro means have a limit?
Also, is the converse true?

REPLY [4 votes]: If $(x_n) \in \ell^\infty$. According to Lorenz the Banch limit is unique (also known as almost convergent) iff
$$\lim_{p\mapsto\infty} \frac{ x_n + x_{n+1} + \cdots + x_{n+p}}{p} = L  \quad (*) $$
uniformly in $n$. Setting $n=0$ yields Cesaro summability.
As Aaron says, the converse is false. If each $x_n$ is chosen uniformly at random from $\{0,1\}$ then this sequence almost never has property $(*)$ (see Connor's appropriately named article Almost none of the sequences of 0's and 1's are almost convergent)
However the Cesaro limit of this random sequence $(x_n)$ is almost always $1/2$ by the law of large numbers.


TITLE: Polytopes with six vertices
QUESTION [2 upvotes]: Let P(d,v) denote the number of convex polytopes (up to combinatorial equivalence) in $\,$d dimensions with v vertices.  The octahedron and triangular prism are two familiar examples with d=3 and v=6 .  There are five others bringing the total to P(3,6) = 7.
In four dimensions, any polytope with 6 vertices has just one more than the bare minimum needed to form a simplex. $\,$ Apparently there are P(4,6) = 4$\,$ such.
For completeness, there is only the hexagon if d=2 and only the simplex when d=5.$\,$  Therefore, the full count of all polytopes with v=6 and $2\le d$$\,$$\le 5$ $\,$is given by $\,$(1,7,4,1).
Questions:  (1) Is there a reference which shows illustrations for the seven polytopes in the$\,$ d=3, v=6 case?
(2)$\,$The set of 3 x 3 doubly stochastic matrices forms a four dimensional polytope inside$\,$ $\Bbb R^9$$\,$having six vertices (corresponding to the six permutation matrices).  Is there a nice way to embed this in$\,$$\,$$\Bbb R^4$$\,$ or to understand it in some more direct geometric way?
(3)$\,$Suppose we fix$\,$ v $\,$the number of vertices and consider the sequence P(d,v) for $\,$$2 \le d$$\,$$ \le v-1$. $\,$ Is this sequence known or conjectured to be unimodal?
Thanks very much

REPLY [1 votes]: Answer to (1): There is such a reference (by Professor Steven Dutch) showing Schlegel diagrams for nine types of polyhedra with six vertices (not seven); five under 'Polyhedra with 4 - 7 Faces' and four under 'Octahedra with 6 - 8 Vertices'.
Answer to (2): This is called the Birkhoff polytope $\Omega_3$. I believe this combinatorial type to be the same as the cyclic polytope C(6,4); also the direct sum of two triangles. It's straightforward to check that the two triangles with vertices at the even and odd permutations, respectively, aren't faces of any of the nine facets of $\Omega_3$. However, $\Omega_3$ is neighborly.


text:
\begin{document}

\title{Stability analysis of random nonlinear systems with time-varying delay
and its application}

\author{Liqiang~Yao, Weihai~Zhang${}^*$
\thanks{${}^*$  Corresponding author.}
\thanks{This work was supported by National Natural Science Foundation
of China (Nos: 61573227, 61633014), SDUST Research Fund (No.
2015TDJH105), Research Fund for the Taishan Scholar Project of
Shandong Province of China. }
\thanks{Liqiang~Yao is with the College of Electrical Engineering and Automation, Shandong
University of Science and Technology, Qingdao,  Shandong Province,
266590, P. R. China and the School of Mathematics and Information
Science, Yantai University, Yantai, Shandong Province, 264005, P. R. China.}
\thanks{Weihai~Zhang is with the College of Electrical Engineering and Automation, Shandong
University of Science and Technology, Qingdao,  Shandong Province, 266590,
P. R. China. (Corresponding author: w\_hzhang@163.com).}
\thanks{Manuscript received April 19, 2005; revised August 26, 2015.}}

\markboth{Journal of \LaTeX\ Class Files

}
{Shell \MakeLowercase{\textit{et al.}}: Bare Demo of IEEEtran.cls for IEEE Journals}

\maketitle

\begin{abstract}
This paper studies a class of random nonlinear systems with
time-varying delay,  in which the $r$-order moment ($r\geq1$)
of the random disturbance is finite. Firstly, some general conditions
are proposed to guarantee the existence and uniqueness of the global
solution to random nonlinear time-delay systems.
Secondly, some definitions and criteria on noise-to-state stability in
the moment sense and in probability sense are given by Lyapunov method
respectively. Finally, two regulation controllers are constructed
respectively for two corresponding random nonlinear time-delay
systems and the effectiveness of two proposed  recursive procedures are
demonstrated by two simulation examples.
\end{abstract}

\begin{IEEEkeywords}
Random nonlinear systems, time-varying delay, regulation controller,
noise-to-state stability.
\end{IEEEkeywords}

\IEEEpeerreviewmaketitle

\section{Introduction}\label{S1}

Time-delay is widespread in many actual systems
such as network transmission systems, hydraulic systems and power systems.
The time-delay phenomenon is often caused by  the
inherent characteristics of physical systems such as communication system
and mechanical drive system. In addition, the devices of practical
systems (for instance, controllers and actuators) need time to complete
operations during the running process of systems, which  inevitably
makes the time-delay phenomenon. The existence of time-delay phenomenon can
affect system performance seriously and brings difficulties for system
analysis and synthesis. Nevertheless, time-delay phenomenon
sometimes can be used to improve the control performance of some systems such as
repetitive control systems. Therefore, it is of great significance to study
how to eliminate and utilize time-delay phenomena. Over the past few decades,
many scholars  have committed themselves to studying time-delay
systems in theory and engineering applications ([\ref{Malek-Zavarei1987}-\ref{Lakshmanan2010}]).

On the other hand, many practical systems in engineering, which are
often subject to random disturbance from external environment, are
modeled as stochastic nonlinear systems or random nonlinear
systems ([\ref{Soong1973}-\ref{Wu2015}]).
The difference lies in the random  disturbance which appeared in
the former is a white noise process (i.e., the derivative
of Wiener process), while the
random disturbance in the latter is a stationary stochastic
process. Since the white noise has infinite bandwidth,
some actual systems are more suitable to be modeled as
random nonlinear systems from the energy view.

Along with  the work of Khas'minskii, Krsti\'{c} and Deng, Mao
and other scholars in stochastic control field, the remarkable
development has been achieved (please refer to
[\ref{Khasminskii1980}], [\ref{Krstic1998}-\ref{Mao2007}] and the
references therein). This  promotes the study of stochastic
nonlinear time-delay systems. [\ref{Fu2005}] designed an
output-feedback controller by Lyapunov-based recursive method such
that a class of stochastic lower triangular systems with constant
delay only appeared in drift term is exponentially stable. For
general stochastic nonlinear time-delay systems, [\ref{Liu2008}]
proposed some conditions to guarantee that systems solution exists
and is unique, and gave the global stability criteria in probability
sense. Besides, the adaptive output-feedback stabilization controller
for a class of stochastic lower triangular  systems with time-delay is
constructed in [\ref{Liu2008}]. By the stability results in
[\ref{Liu2008}], [\ref{Liu2011}] constructed an output-feedback
controller to achieve globally asymptotically stable in probability sense
for stochastic high-order nonlinear time-delay systems satisfying
some assumptions  and [\ref{Liu2013}] studied stabilization control
for a class of stochastic upper-triangular nonlinear systems
with time-delay by state feedback approach. However, for random nonlinear
systems, most findings in existing literature (such as
[\ref{Wu2015}, \ref{ZhangD2016}-\ref{Yao2018}]) didn't investigate
the influence of time-delay phenomenon on systems dynamics behavior.
The corresponding results about random nonlinear time-delay systems
also have not been reported in existing literature. This motivates
us to focus on random nonlinear time-delay systems.

Inspired by [\ref{Wu2015}] and [\ref{Liu2008}], this paper considers the
stability of random time-delay systems and regulation control
problems for random  nonlinear feedback time-delay systems.
The main contributions of this paper include three aspects:\\
(1) For random  time-varying delay systems with some general conditions,
this paper analyzes the existence and uniqueness of global solution. \\
(2) Some definitions and criteria on noise-to-state stability
(including in the moment sense   and in probability sense )
are proposed based on Lyapunov function approach. \\
(3) Applying the obtained stability results,
two classes of regulation control problems are studied
by constructing different Lyapunov-Krasovskii functionals, respectively.

This paper is organized as follows. Section II investigates the
existence and uniqueness of global solution for random systems with
time-delay. Some definitions and noise-to-state
stability criteria  are proposed in Section III. As
applications, two kinds of regulation problems are discussed in Section IV and
Section V, respectively. In Section VI, two corresponding simulation
examples illustrate the feasibility and effectiveness of proposed
two controller design procedures. Section VII gives the conclusions
of this paper.

{\bfseries Notions:}

$|x|$ and $x^{T}$ represent the usual Euclidean norm
and the transpose of vector $x$, respectively.
$||A||$ denotes the 2-norm of matrix A and  $||A||_F$ stands for the
Frobenius norm of matrix A.
$\mathbb{R}_+$ and $\mathbb{R}^n$ represent the set of nonnegative real numbers
and the real $n$-dimensional space, respectively.
$C([t-\tau, t]; \mathbb{R}^n)$ represents the space of continuous functions
$q$ from $[t-\tau, t]$ to $\mathbb{R}^n$ with the norm
$|q_t|=\sup_{-\tau \leq \theta \leq 0} |q(t+\theta)|$ for $\tau>0$ and $t\geq0$.
The set of functions with continuous $i$-th partial derivative is denoted as $C^i$,
and the function $W(t, x(t))\in
C^{1,1}([t_0-\tau, \infty) \times \mathbb{R}^n ; \mathbb{R}_+)$ means that
$W(t, x(t))$ are $C^1$ in $t$ and  $C^1$ in $x$.
$C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$ denotes the family of all
$\mathcal{F}_{t_0}$-measurable bounded $C([t_0-\tau, t_0]; \mathbb{R}^n)$-valued
random variable $\varphi=\{\varphi(\theta):t_0-\tau \leq \theta \leq t_0\}$
with $ t_0 \geq 0$. $\gamma(t)\in\mathcal{K}$ means that the function $\gamma(t)$
defined on $\mathbb{R}_+$ is strictly increasing, continuous and vanish at origin;
$\bar{\gamma}(t)\in\mathcal{K}_\infty$ implies that
$\bar{\gamma}(t)\in\mathcal{K}$  and $\bar{\gamma}(t)$ is
unbounded; $\beta(s,t) \in \mathcal{KL}$ denotes that
$\beta(s,t) \in \mathcal{K}$ for each fixed $t$, and
$\lim_{t \rightarrow \infty}\beta(s,t)=0$ holds for each fixed $s$.
Function $h(s)$ is convex on $D$, if  for
any $s_1, s_2 \in D$, $h(s)$ satisfies $h((s_1+s_2)/2)\leq (h(s_1)+ h(s_2))/2$.

\section{Preliminaries}\label{S2}
In this section, one discusses the existence and uniqueness of solution to
random differential delay equations (RDDEs) that are a special class of
random functional differential equations (RFDEs). So, one firstly discusses the
the existence and uniqueness of solution to RFDEs and then analyze
the existence and uniqueness of solution to RDDEs.
\subsection{The existence and uniqueness of solution to RFDEs}
Consider the following random functional differential equation
\begin{equation}\label{eq21}
\dot{x}(t)=f(t, x_t)+g(t, x_t)\xi(t), ~~t\geq t_0,
\end{equation}
with the initial data
$x_{t_0}=\varphi=\{\varphi(t_0+\theta): -\tau \leq \theta \leq 0\}
\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$,
where $x_t=\{x(t+\theta): \theta \in [-\tau, 0]\}$
is a $C([t-\tau, t]; \mathbb{R}^n)$ -valued stochastic process.
The state is $x(t)\in \mathbb{R}^{n}$,
$\xi(t)\in \mathbb{R}^{m} $ represents a  piecewise continuous and $\mathcal{F}_t$-adapted
stochastic process defined on the complete filtered probability space
$(\Omega, \mathcal{F}, \{\mathcal {F}_t\}_{t\geq t_0}, P)$,
and it has finite $r$-order moment (i.e., $\sup_{t\geq t_0} E|\xi(t)|^r<K$
with $r\geq 1$ and $K$ being positive constants). For $t_0\leq t \leq T <\infty$, functions
$f: [t_0, T] \times C([t-\tau, t]; \mathbb{R}^n)\rightarrow \mathbb{R}^n $ and
$g: [t_0, T] \times C([t-\tau, t]; \mathbb{R}^n)\rightarrow \mathbb{R}^{n\times m}$
are Borel measurable, piecewise continuous in $t$ and locally Lipschitz continuous
in $x_t\in \mathbb{R}^n $, moreover, $f(t, 0)$ and $g(t, 0)$ are bounded.
\begin{definition}\label{def21}
A solution $x(t)$ to equation (\ref{eq21}) on $[t_0-\tau, T]$ with  initial value
$x_{t_0}=\varphi$ is an  $\mathbb{R}^n$-valued stochastic process
and satisfies that \\
(i) $x(t)$ and $\{x_t\}_{t_0 \leq t \leq T}$ are continuous and $\mathcal{F}_t$-adapted.\\
(ii) $x(t)=x_{t_0} + \int_{t_0}^t f(s, x_s)ds + \int_{t_0}^t g(s, x_s)\xi(s)ds$
holds almost surely for $t_0\leq t\leq T$.
\end{definition}

The uniqueness of solution to equation (\ref{eq21}) is in almost sure sense, that is
to say, a solution $x(t)$ is called to be unique if
$P\{x(t)=\bar{x}(t), t_0-\tau \leq t\leq T\}=1$, where
$\bar{x}(t)$ is another solution to equation (\ref{eq21}).
For any $ T>t_0-\tau$, if there exists a unique solution
to equation (\ref{eq21}) on $[t_0-\tau, T]$ , equation (\ref{eq21})
has a unique solution on $[t_0-\tau, \infty)$.

In order to establish the sufficient conditions, which ensure that
the global solution to equation (\ref{eq21}) exists and is unique,
one introduces the first exit time  from a region
$U_k=\{x: |x|<k\}$ and its limit.
Let $\rho_k=\inf\{t\geq t_0 : |x(t)| \geq k\}$
and $\rho_\infty=\lim_{k\rightarrow\infty}\rho_k$ almost surely,
where $\rho_k$ are stopping times,
$ \inf\phi=\infty $, and $k \geq 1$ is any integer.

\begin{lemma}\label{lem2a1}
For equation (\ref{eq21}), if there is a
function $V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$
and constants $c_0$ and $d_0>0$ such that for any integer $k \geq 1$ and  $t\geq t_0$,
\begin{gather}
\label{lem2a1a}\lim_{k\rightarrow\infty}\inf_{|x|\geq k}V(t,x)=\infty,\\
\label{lem2a1b} EV(\rho_k \wedge t, x(\rho_k \wedge t))\leq d_0 e^{c_0t},
\end{gather}
then there exists a unique solution  on $[t_0-\tau, \infty)$ to equation (\ref{eq21}) .
\end{lemma}

\begin{proof}
For every integer $k \geq 1$, one defines the following truncation functions
\begin{align*}
f_k(t, x_t)=
\begin{cases}
f(t, x_t) &\textmd{if}~~|x_t|\leq k \\
f(t, kx_t/|x_t|) &\textmd{if}~~|x_t|>k \\
\end{cases}, \cr
g_k(t, x_t)=
\begin{cases}
g(t, x_t) &\textmd{if}~~|x_t|\leq k \\
g(t, kx_t/|x_t|) &\textmd{if}~~|x_t|>k \\
\end{cases}.
\end{align*}
Then $f_k(t, x_t)$ and $g_k(t, x_t)$ satisfy the Lipschitz condition.
Adopting a similar way as Lemma 3 in [\ref{Wu2015}], one can
prove that there exists a unique $\mathcal {F}_t$-adapted solution $x_k(t)$ satisfying
\begin{equation}\label{eq22}
\dot{x}_k(t)=f_k(t, x_{t,k})+g_k(t, x_{t,k})\xi(t), ~~t\geq t_0,
\end{equation}
with the initial value $x_{t_0, k}=\varphi \in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$,
the details of proof is omitted here.
It is clear that
\begin{equation}\label{eq23}
x_k(t)=x_{k+1}(t), ~~~\forall t\in [t_0, \rho_k).
\end{equation}
Define $x(t)=x_k(t), t\in [t_0, \rho_k), k\geq 1$,
then it follows from (\ref{eq22}) and (\ref{eq23}) that for any
$t\geq t_0$ and $k\geq 1$,
\begin{align*}
x(t \wedge \rho_k)
=&\varphi(t_0) + \int_{t_0}^{t \wedge \rho_k}[ f(s, x_{s,k})+g(s, x_{s,k})\xi(s)]ds.
\end{align*}

First, one considers the case of $\rho_\infty<T<\infty$, then
\begin{equation*}
\limsup_{t\rightarrow \rho_\infty}|x(t)|
\geq\limsup_{k\rightarrow \infty}|x(\rho_k)|
=\limsup_{k\rightarrow \infty}|x_k(\rho_k)|=\infty,
\end{equation*}
which means that equation (\ref{eq21}) has a maximal solution $x(t), t\in[t_0, \rho_\infty)$,
where $\rho_\infty$ is often called the explosion time.
The uniqueness of $x_k(t)$ on $[t_0, \rho_\infty)$ can lead to the uniqueness of
the solution $x(t)$ on $[t_0, \rho_\infty)$. By now, there exists a unique solution $x(t)$ to
equation (\ref{eq21}) on $[t_0-\tau, \rho_\infty)$ and
the  solution $x(t)$ is $\mathcal{F}_t$-adapted, which can be inferred from
the fact that $x_{k}(t)$ is $\mathcal{F}_t$-adapted.

Next, one begins to show $\rho_\infty=\infty $  almost surely.
In fact, if $\rho_\infty < \infty$ almost surely, then there exist
constants $T>0$ and $\varepsilon>0$ such that $P\{\rho_\infty\leq T\}>\varepsilon$.
Because $\lim_{k\rightarrow\infty}\rho_k=\rho_\infty$ almost surely,
there is a integer $k_0$ such that
\begin{equation}\label{eq24}
P\{\rho_k\leq T\}>\varepsilon,~~\forall k\geq k_0.
\end{equation}
For any fixed $k(k\geq k_0)$, let $t=T$ in (\ref{lem2a1b}), then one
has $EV(T\wedge\rho_k, x(T\wedge\rho_k))\leq d_0 e^{c_0T}$,
from which one also can obtain
\begin{equation}\label{eq25}
E[V(\rho_k, x(\rho_k))I_{\{\rho_k\leq T\}}]\leq d_0 e^{c_0T}.
\end{equation}

On the other hand, one defines
$\varrho_{k,T}=\inf\{V(t,x(t)):|x(t)|\geq k, t\in [t_0, T]\}$,
then $\lim_{k\rightarrow\infty}\varrho_{k,T}=\infty$  by (\ref{lem2a1a}).
It follow from (\ref{eq24}) and (\ref{eq25}) that
$d_0 e^{c_0T}\geq \varrho_{k,T} P\{\rho_k\leq T\}>\varepsilon \varrho_{k,T}$.
Since letting $k\rightarrow \infty$ yields a contradiction, one has
$\rho_\infty=\infty$ almost surely.
So, there is a unique global solution $x(t)$
on $[t_0-\tau, \infty)$ for  equation (\ref{eq21}).
\end{proof}

\begin{lemma}\label{lem2a2}
For  equation (\ref{eq21}), aussume that there are a function
$V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$
and a constant $K_1>0$ such that
\begin{gather}
\label{lem2a2a}\lim_{|x|\rightarrow\infty}\inf_{t\geq t_0}V(t,x)=\infty,\\

\label{lem2a2b}E\dot{V}(t,x)\leq K_1(1+E V(t,x)+E V(t+\theta,x(t+\theta))),
\end{gather}
where $-\tau \leq \theta \leq 0$ and $t\geq t_0$, then equation (\ref{eq21})
has a unique solution on $[t_0-\tau, \infty)$.
\end{lemma}

\begin{proof}
From the proof of Lemma \ref{lem2a1}, one knows that there is a unique
maximal local solution $x(t)$ on $[t_0-\tau, \rho_\infty)$ for equation (\ref{eq21}).
Therefore, one need to testify that  $\rho_\infty=\infty $ almost surely.
For any $k \geq 1$,
let $\varrho_k=\rho_\infty \wedge \inf\{t_0 \leq t < \rho_\infty: |x(t)| \geq k\}$.
It is certain that $\varrho_\infty \leq \rho_\infty$ and
$\varrho_\infty=\lim_{k\rightarrow\infty}\varrho_k$ almost surely.

By Fubini's theorem [Theorem 2.39, \ref{Klebaner2005}] and (\ref{lem2a2b}), one has
\begin{align}\label{lem2eq1}
&EV(t \wedge \varrho_k, x(t \wedge \varrho_k))\cr
\leq & \bar{H}(t)+K_1\int_{t_0}^{t} \sup_{t_0\leq v \leq s}
[EV(v \wedge \varrho_k, x(v \wedge \varrho_k))]ds\cr
\triangleq &  H^*(t).
\end{align}
where $t \geq t_0$ , $k \geq 1$, and
$\bar{H}(t)=EV(t_0, x(t_0))+K_1(t-t_0)+K_1\int_{t_0}^{t}EV(s+\theta, x(s+\theta))ds>0$.
Because $H^*(t)$ is a increasing function of $t$,
one can obtain the following inequality
from (\ref{lem2eq1})
\begin{align*}
&\sup_{t_0\leq v \leq t} [EV(v \wedge \varrho_k, x(v \wedge \varrho_k))]\cr
\leq & \bar{H}(t)+K_1\int_{t_0}^{t} \sup_{t_0\leq v \leq s}
[EV(v \wedge \varrho_k, x(v \wedge \varrho_k))]ds.
\end{align*}

By Gronwall's inequality [Theorem 1.20, \ref{Klebaner2005}], one gets
\begin{align*}
\sup_{t_0\leq v \leq t} [EV(v \wedge \varrho_k, x(v \wedge \varrho_k))]
\leq\bar{H}(t)e^{K_1(t-t_0)} .
\end{align*}
It is certain that
\begin{align}\label{lem2eq2}
EV(t \wedge \varrho_k, x(t \wedge \varrho_k))
\leq \bar{H}(t)e^{K_1t}, ~~\forall t \geq t_0.
\end{align}
On the other hand, for any $ R \geq 0$, one defines that
$\delta(R)=\inf_{R\leq|x|, t_0\leq t}V(t,x(t))$,
then  it is easy to show that $\delta(|x(t)|)\leq V(t,x(t))$ and
$\lim_{R\rightarrow\infty}\delta(R)=\infty$ which comes from (\ref{lem2a2a}).
It follows from (\ref{lem2eq2}) that
\begin{align*}
P\{\varrho_k\leq t\}\leq \frac{E\delta(|x(t \wedge \varrho_k)|)}{\delta(k)}
\leq \frac{\bar{H}(t)e^{K_1t}}{\delta(k)}.
\end{align*}
Letting first $k \rightarrow \infty$ and then $t \rightarrow \infty$,
$P\{\varrho_\infty < \infty \}=0$ can be obtained, that is,
$\varrho_\infty = \infty$ almost surely.
This means that $\rho_\infty = \infty$ almost surely.
Thus, equation (\ref{eq21}) has a unique solution
on $[t_0-\tau, \infty)$.
\end{proof}
\subsection{The existence and uniqueness of solution to RDDEs}
For the following random differential delay equations
\begin{align}\label{eq26}
\dot{x}=f(t,  x(t-\tau(t)), x)+g(t, x(t-\tau(t)), x)\xi(t),
\end{align}
with the initial value $x_{t_0}=\varphi=\{\varphi(t):t_0-\tau \leq t
\leq t_0\}\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0];
\mathbb{R}^n)$. System state is $x(t)\in \mathbb{R}^{n}$, and the
random disturbance  $\xi(t)\in \mathbb{R}^{m} $ is in accord
with that of system (\ref{eq21}). Borel measurable function
$\tau(t)$ is defined on $[t_0, \infty)$ and takes values on $[0, \tau]$.
Borel measurable functions $f: [t_0, \infty) \times
\mathbb{R}^n\times \mathbb{R}^n\rightarrow \mathbb{R}^n $ and $g:
[t_0, \infty) \times \mathbb{R}^n\times \mathbb{R}^n\rightarrow
\mathbb{R}^{n\times m} $  satisfy
locally Lipschitz condition with respect to $x(t-\tau(t))$ and $x(t) $,
respectively. In addition, $f(t, 0, 0)$ and $g(t,
0, 0)$ are bounded.

One introduces
$F(t, \bar{x}_t)=f(t, \bar{x}_t(-\tau(t)), \bar{x}_t(0))=f(t, x(t-\tau(t)), x(t))$ and
$G(t, \bar{x}_t)=g(t, \bar{x}_t(-\tau(t)), \bar{x}_t(0))=g(t, x(t-\tau(t)), x(t))$,
then equation (\ref{eq26}) can be rewritten as
\begin{equation}\label{eq27}
\dot{x}=F(t, \bar{x}_t)+G(t, \bar{x}_t)\xi(t), ~~t\geq t_0.
\end{equation}
This implies that RDDEs are in fact a special class of RFDEs.
Therefore, one can obtain the following sufficient conditions such that system (\ref{eq26})
has a unique global solution.

\begin{lemma}\label{lem2b1}
For  system (\ref{eq26}), if there are a function
$V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$
and constants $c_0$ , $d_0>0$ such that (\ref{lem2a1a}) and (\ref{lem2a1b})
hold for any integer $k \geq 1$ and $t\geq t_0$, then there is a unique solution
to system (\ref{eq26}) with initial value
$x_{t_0}=\varphi\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$.
\end{lemma}

\begin{proof}
For system (\ref{eq26}),  since both $f(t, x(t-\tau(t)), x)$ and
$g(t,  x(t-\tau(t)), x)$ are Borel measurable functions and
satisfy locally Lipschitz condition with respect to $x(t-\tau(t))$ and
$x$, respectively. In other words, for any $R>0$, there is a positive constant $L_R$ such that
for $\forall x_1, x_2\in U_R, x_1\neq x_2$ and
$\forall x_1(t-\tau(t)), x_2(t-\tau(t))\in U_R, x_1(t-\tau(t))\neq x_2(t-\tau(t))$,
\begin{align}\label{eq28}
&|f(t, x_1(t-\tau(t)), x_1)-f(t, x_2(t-\tau(t)), x_2)|\cr
\leq &L_R(|x_1-x_2|+|x_1(t-\tau(t))-x_2(t-\tau(t))|),\cr
&||g(t, x_1(t-\tau(t)), x_1)-g(t, x_2(t-\tau(t)), x_2)||\cr
\leq & L_R(|x_1-x_2|+|x_1(t-\tau(t))-x_2(t-\tau(t))|).
\end{align}
Note that $|q_t|=\sup_{0 \leq \tau(t) \leq \tau} |q(t-\tau(t))|$,
then one has
\begin{align}\label{lem29}
&|x_1-x_2|\vee |x_1(t-\tau(t))-x_2(t-\tau(t))|\cr
\leq &|\bar{x}_{1,t}-\bar{x}_{2,t}|.
\end{align}
It follows from (\ref{eq28}) and (\ref{lem29}) that
\begin{align}\label{eq30}
&|F(t, \bar{x}_{1,t})-F(t, \bar{x}_{2,t})| \vee ||G(t, \bar{x}_{1,t})-G(t, \bar{x}_{2,t})||\cr
\leq & 2L_R|\bar{x}_{1,t}-\bar{x}_{2,t}|.
\end{align}
This means that equation (\ref{eq27}) satisfies the locally Lipschitz condition.
From Lemma \ref{lem2a1}, there is a unique solution on $[t_0-\tau, \infty)$ to
system (\ref{eq27}) if there are a positive function
$V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$,
constant $c_0$ and positive constant $d_0$ satisfying (\ref{lem2a1a})
and (\ref{lem2a1b}).
Since equation (\ref{eq26}) is equivalent to equation
(\ref{eq27}), there exists a unique solution to system (\ref{eq26}).
\end{proof}

\begin{definition}\label{def22} [Definition 1, \ref{Wu2015}]
For any $\varepsilon>0$, if there is a positive constant $\epsilon$ such that
$P\{\sup_{t\geq t_0}|\phi(t)|>\epsilon\}\leq \varepsilon$ holds,
then the stochastic process $\phi(t)$ is  bounded in probability.
\end{definition}

\begin{lemma}\label{lem2b2}
For  system (\ref{eq26}), if there are a function
$V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$
and a constant $K_2>0$ such that for any $t\geq t_0$,
$\lim_{|x|\rightarrow\infty}\inf_{t\geq t_0}V(t,x)=\infty$ and
$E\dot{V}(t,x)\leq K_2(1+EV(t,x)+EV(t-\tau(t), x(t-\tau(t)))$ hold,
then system (\ref{eq26}) has a unique solution.
\end{lemma}

\begin{remark}\label{rem21}
Since the proof of Lemma \ref{lem2b2} is similar to that of Lemma \ref{lem2a2},
the detailed proof of Lemma \ref{lem2b2} is omitted  here.
\end{remark}

\begin{lemma}\label{lem2b3}
For  system (\ref{eq26}), if there exist a function
$V(t,x(t))\in C^{1,1}([t_0-\tau, \infty)\times \mathbb{R}^n ; \mathbb{R}_+)$
and a positive constant $d_0$ such that for any integer $k \geq 1$, (\ref{lem2a1a}) and
\begin{align}\label{lem2b2eq}
EV(t\wedge\rho_k, x(t\wedge\rho_k))\leq d_0
\end{align}
hold, then system (\ref{eq26}) has a unique solution and this unique solution is bounded in probability.
\end{lemma}

\begin{proof}
According to Lemma \ref{lem2b1}, system (\ref{eq26}) has a unique global solution.
From (\ref{lem2b2eq}), one has
\begin{align*}
EV(t, x(t))=\lim_{k\rightarrow \infty}EV(\rho_k\wedge t, x(\rho_k\wedge t))\leq d_0.
\end{align*}
On the other hand, by Lemma 1.4 in [\ref{Khasminskii1980}], one gets
\begin{align*}
EV(t, x(t))&\geq \int_{\{\sup_{t\geq t_0}|x(t)|> r\}}V(t, x(t))dP\cr
&\geq P\{\sup_{t\geq t_0}|x(t)|> r\}\inf_{t\geq t_0,~|x(t)|> r}V(t, x(t)).
\end{align*}
Thus, there is a positive constant $r$  such that for any $\varepsilon>0$,
\begin{align*}
P\{\sup_{t\geq t_0}|x(t)|> r\}\leq \frac{d_0}{\inf_{t\geq t_0, ~|x(t)|> r}V(t, x(t))}<\varepsilon.
\end{align*}
This implies that the unique global solution of system (\ref{eq26}) is bounded in probability.
\end{proof}

\begin{lemma}\label{lem2b4} [\ref{Flett1980}]
$y(t)$, $k(t)$ and $w(t)$ are continuous functions on $[t_0, \infty)$ such that
$$\limsup_{\vartriangle t\rightarrow 0^+}
\frac{y(t+\vartriangle t)-y(t)}{\vartriangle t}
=D^{+}y(t)\leq k(t)y(t)+w(t)$$
holds for almost all $t\in[t_0, \infty)$.
Then,
$$y(t)\leq y(t_0)e^{\int_{t_0}^t k(s)ds}+\int_{t_0}^t e^{\int_{s}^t k(v)dv} w(s)ds, ~~\forall t\geq t_0.$$
\end{lemma}
\begin{lemma}\label{lem2b5} [\ref{Hardy1989}]
For functions $\chi_1, \chi_2, \cdots, \chi_n$ and $l\geq 1$, the following inequality holds
\begin{align*}
|\chi_1+\chi_2+\cdots+\chi_n|^l\leq n^{l-1}(|\chi_1|^l+|\chi_2|^l+\cdots+|\chi_n|^l).
\end{align*}
\end{lemma}
\section{Main results}\label{S3}
In this section, one discusses the noise-to-state stability of random
time-delay systems (\ref{eq26}) in  the moment sense and in probability
sense, respectively. First, for random nonlinear time-delay systems (\ref{eq26}),
one gives some definitions about the ultimate boundedness of system state and
noise-to-state stability.

\begin{definition}\label{def3a1}
If there exist functions $\gamma \in \mathcal{K}$ and $\beta\in \mathcal{KL}$ such that
$E|x(t)|^m \leq \beta(|\varphi|, t-t_0)+\gamma(\sup_{t_0 \leq l \leq t} E|\xi(l)|^r)$
holds for system (\ref{eq26}) with any given initial value
$\varphi\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$,
where $m$ is a positive constant and
$|\varphi|=\sup_{t_0-\tau \leq s \leq t_0} |\varphi(s)|$,
then system (\ref{eq26}) is called to be noise-to-state stable
in the $\textbf{m}$-th moment sense(NSS-\textbf{m}-M).
\end{definition}

\begin{definition}\label{def3a2}
For any $\varepsilon>0$, if there exist functions
$\gamma \in \mathcal{K}$ and $\beta\in \mathcal{KL}$ such that
$P\{|x(t)|\leq \beta(|\varphi|, t-t_0)
+\gamma(\sup_{ t_0\leq l \leq t} E|\xi(l)|^r)\}\geq 1-\varepsilon$
holds for system (\ref{eq26}) with any given initial value
$\varphi\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$,
where $|\varphi|=\sup_{t_0-\tau \leq s \leq t_0} |\varphi(s)|$,
then system (\ref{eq26}) is called to be noise-to-state stable
in probability sense (NSS-P).
\end{definition}

\begin{definition}\label{def3a3}
Let $m>0$ be  a constant. If there is a function $\gamma \in \mathcal{K}$ such that
$\lim_{t\to\infty} E|x(t)|^m \leq \gamma(K)$ holds,
then the state $x(t)$ of system (\ref{eq26}) is  ultimately bounded in the
$\textbf{m}$-th moment sense (UB-\textbf{m}-M).
\end{definition}

\begin{definition}\label{def3a4}
For any $\varepsilon>0$, if there is a function $\gamma \in \mathcal{K}$
such that $P\{\lim_{t\to\infty}|x(t)|> \gamma(K)\}< \varepsilon$ holds,
then the state $x(t)$ of system (\ref{eq26}) is  ultimately bounded
in probability sense (UB-P).
\end{definition}

Next, one addresses the criteria on noise-to-state stability of random
time-delay systems in the moment sense and in probability sense, respectively.

\begin{theorem}\label{the31}
For system (\ref{eq26}), if there are a function
$V(t,x)\in C^{1,1}([t_0-\tau, \infty) \times \mathbb{R}^n ; \mathbb{R}_+)$,
positive constants $m, c_1, c_2, c_3, c$ and non-negative constant $d_c$
such that
\begin{gather}
\label{eq31} c_1|x(t)|^m\leq V(t, x)\leq c_2(\sup_{-\tau\leq s\leq 0}|x(t+s)|)^m, \\

\label{eq32} \dot{V}(t,x)\leq -cV(t,x)+c_3|\xi(t)|^r+d_c,
\end{gather}
then (i) system (\ref{eq26}) has a unique solution  on $[t_0-\tau, \infty)$;
(ii) system (\ref{eq26}) is NSS-\textbf{m}-M;
(iii) the state of system (\ref{eq26}) is UB-\textbf{m}-M.
\end{theorem}

\begin{proof}
From (\ref{eq31}), it yields that
\begin{align}\label{eq33}

\lim_{|x|\rightarrow\infty}\inf_{t\geq t_0}V(t,x)=\infty.
\end{align}
Taking expectations on both sides of (\ref{eq32}), then
\begin{align}\label{eq34}
E\dot{V}(t,x)\leq -cEV(t, x) +c_3E|\xi(t)|^r +d_c.
\end{align}
According to Lemma \ref{lem2b2}, it follows from (\ref{eq33}) and (\ref{eq34}) that
system (\ref{eq26}) has a unique global solution.

Letting $\nu(t)=EV(t,x)$ and using Fubini's theorem
[Theorem 2.39, \ref{Klebaner2005}], then (\ref{eq34}) can be turned into
$$\nu(t+\varepsilon)\leq \nu(t)+ \int_t^{t+\varepsilon} (-c\nu(s) +c_3K+d_c)ds,
\forall\varepsilon>0, t\geq t_0.$$
So, $$D^{+}\nu(t)\leq -c\nu(t) +c_3K+d_c.$$
By Lemma \ref{lem2b4}, one arrives at
\begin{align*}
\nu(t)&\leq \nu(t_0)e^{-c(t-t_0)}+\int_{t_0}^{t}e^{-c(t-s)}(c_3K+d_c)ds\cr
&\leq \nu(t_0)e^{-c(t-t_0)}+\frac{c_3K+d_c}{c}-\frac{c_3K+d_c}{c}e^{c(t_0-t)}\cr
&\leq \nu(t_0)e^{-c(t-t_0)}+\frac{c_3K+d_c}{c},
\end{align*}
together with (\ref{eq31}),  one can deduce that
\begin{align}\label{eq35}
E|x(t)|^m \leq \frac{c_2}{c_1}|\varphi|^m e^{-c(t-t_0)} + \frac{c_3K+d_c}{cc_1}.
\end{align}
Thus, the system (\ref{eq26}) is NSS-\textbf{m}-M.

In addition, letting $t\rightarrow \infty$ in (\ref{eq35}), it yields that
\begin{align*}
\lim_{t\rightarrow \infty}E|x(t)|^m \leq \frac{c_3K+d_c}{cc_1}.
\end{align*}
This means that the state of system (\ref{eq26}) is UB-\textbf{m}-M.
\end{proof}

\begin{theorem}\label{the32}
For system (\ref{eq26}), if there are a function $V(x)\in
C(\mathbb{R}^n ; \mathbb{R}_+)$, functions $\gamma \in \mathcal{K}$,
$\gamma_1 \in \mathcal{KL}$, $\gamma_2 \in \mathcal{KL}$ and
positive constant $a_0$ such that
\begin{gather}
\label{eq36} \gamma_1(|x(t)|)\leq V(x)\leq  \gamma_2(\sup_{-\tau\leq s\leq 0}|x(t+s)|), \\

\label{eq37} \dot{V}(x)\leq -\gamma (\sup_{-\tau\leq s\leq 0}|x(t+s)|)+a_0|\xi(t)|^r,
\end{gather}
then system (\ref{eq26}) has a unique solution.
If $\gamma \circ \gamma_2^{-1}(\cdot)$ is a convex function, then  system (\ref{eq26})
is NSS-P and the state of system (\ref{eq26}) is UB-P.
\end{theorem}

\begin{proof}
From (\ref{eq36}) and (\ref{eq37}), one  obtains  (\ref{eq33}) and
\begin{align}\label{eq38}
E\dot{V}(x)&\leq -E[\gamma (\sup_{-\tau\leq s\leq 0}|x(t+s)|)]+a_0E|\xi(t)|^r\cr
&\leq a_0K(1+EV(x)+EV( x(t-\tau(t))),
\end{align}
respectively.
So system (\ref{eq26}) has a unique solution in the light of Lemma \ref{lem2b2}.

If $\gamma \circ \gamma_2^{-1}(\cdot)$ is a convex function, then by
(\ref{eq38}) and Jensen's inequality [\ref{Hardy1989}], and  let
$\bar{\nu}(t)=EV(x)$, one gets
\begin{align*}
\dot{\bar{\nu}}(t)&\leq -E[\gamma (\sup_{-\tau\leq s\leq 0}|x(t+s)|)]+a_0E|\xi(t)|^r\cr
&\leq -\gamma \circ \gamma_2^{-1}(\bar{\nu}(t))+a_0\sup_{ t_0\leq l \leq t} E|\xi(l)|^r.
\end{align*}
Following the thread of Theorem 2 of [\ref{Wu2015}], there exist functions
$\tilde{\beta} \in \mathcal{KL}$ and $\tilde{\gamma}  \in \mathcal{K}$ such that
$P\{|x(t)|>\tilde{\beta}(|\varphi|, t-t_0)+\tilde{\gamma}(a_0\sup_{ t_0\leq l \leq t}
E|\xi(l)|^r)\}< \varepsilon $ holds, and letting $t\rightarrow\infty$, then one has
\begin{align*}
P\{\lim_{t\to\infty}|x(t)|> \tilde{\gamma}(a_0K)\}< \varepsilon.
\end{align*}
Thus, system (\ref{eq26}) is NSS-P and the system state is UB-P.
\end{proof}
\begin{remark}\label{rem31}
(i) In Theorem \ref{the31}, if (\ref{eq32}) is replaced by
$\dot{V}(t,x)\leq -c(\sup_{-\tau\leq s\leq 0}|x(t+s)|)^m+c_3|\xi(t)|^r+d_c$, the result
remains valid. In fact, by (\ref{eq31}), one has
\begin{align*}
\dot{V}(t,x)&\leq -c(\sup_{-\tau\leq s\leq 0}|x(t+s)|)^m+c_3|\xi(t)|^r+d_c\cr
&\leq -\frac{c}{c_2}V(t,x)+c_3|\xi(t)|^r+d_c.
\end{align*}
(ii) From (\ref{eq38}), it is certain that if (\ref{eq37}) is replaced by
$\dot{V}(x)\leq -\gamma \circ \gamma_2^{-1}(V(x))+a_0|\xi(t)|^r$,
Theorem \ref{the32} still holds for system (\ref{eq26}).
\end{remark}

\begin{remark}\label{rem32}
The random disturbance $\xi(t)$ in RDDE (\ref{eq26}) satisfies the assumption
$\sup_{t\geq t_0} E|\xi(t)|^r<K$, under which one gives the criteria
on noise-to-state stability of RDDE (\ref{eq26}).  If
the random disturbance $\xi(t)$ in  RDDE (\ref{eq26}) satisfies that
\begin{align}\label{eq311}
E|\xi(t)|^r\leq \bar{d}_0 e^{\bar{c}_0t}, ~~t\geq t_0,
\end{align}
where $\bar{c}_0$ is a constant and $\bar{d}_0$ is a positive constant, then by
adopting the similar way used in section III of [\ref{Wu2015}],
one knows that the result (i) and (ii)
in Theorem \ref{the31} still hold, but the result (iii) does not necessarily hold.
Similarly, If (\ref{eq36}) and (\ref{eq37}) hold in Theorem \ref{the32} for
RDDE (\ref{eq26}) satisfying (\ref{eq311}), then system (\ref{eq26}) has
a unique solution  on $[t_0-\tau,\infty)$. Furthermore,
system (\ref{eq26}) is also NSS-P if $\gamma \circ \gamma_2^{-1}(\cdot)$ is a convex function,
but the  state of system (\ref{eq26}) is  not necessarily UB-P.
\end{remark}
\section{Application 1: state feedback regulation  control}\label{S5}

Consider  the following random nonlinear strict-feedback system with
time-varying delay
\begin{equation}\label{equ41}
\begin{cases}
\dot{x}_1(t)=x_2(t)+f_1+g_1\xi_1(t),\\
\dot{x}_2(t)=x_3(t)+f_2+g_2\xi_2(t),\\
~~~~~~~~\vdots\\
\dot{x}_{n}(t)=u(t)+f_{n}+g_{n}\xi_{n}(t),\\
y(t)=x_1(t).
\end{cases}
\end{equation}
where $f_i=f_i(t, \bar{x}_i(t-\tau(t)), \bar{x}_i(t) )$,
$g_i=g_i(t, \bar{x}_i(t-\tau(t)),  \bar{x}_i(t))$,
$\bar{x}_i(t-\tau(t))=[x_1(t-\tau(t)),\cdots,x_i(t-\tau(t))]^T$
and $\bar{x}_i(t)=[x_1(t),\cdots,x_i(t)]^T$.
System state is $x(t)=[x_1(t), x_2(t), \cdots, x_n(t)]^T\in \mathbb{R}^{n}$
with the initial value
$x_{t_0}=\varphi\in C_{\mathcal{F}_{t_0}}^b([t_0-\tau, t_0]; \mathbb{R}^n)$, and
system output $y(t)\in \mathbb{R}$ and system input $u(t)\in \mathbb{R}$.
Borel measurable function $\tau(t):[t_0,\infty)\rightarrow [0,\tau]$
denotes time-varying delay and satisfies $\dot{\tau}(t)\leq
\tau^*<1$, where $\tau$ and  $\tau^*$ are known constants.
Stochastic process $\xi(t)=[\xi_1(t), \xi_2(t), \cdots, \xi_n(t)]^T \in \mathbb{R}^{n}$
defined on the complete filtered probability space $(\Omega, \mathcal{F},
\{\mathcal {F}_t\}_{t\geq t_0}, P)$ is $\mathcal{F}_t$-adapted and
piecewise continuous. $f_i, g_i, i=1, 2, \cdots, n$ are
locally Lipschitz continuous functions in  $\bar{x}_i(t-\tau(t))$ and $\bar{x}_i(t)$, respectively.
$f(t, 0, 0)$ and $g(t, 0, 0)$ are bounded.

For system (\ref{equ41}), one proposes the following assumptions:
\textbf{H0}: The random disturbance $\xi(t)$ satisfies
$$\sup_{t\geq t_0} E|\xi(t)|^4<K. $$
\textbf{H1}: The system state $x(t)=[x_1(t), \cdots, x_n(t)]^T$
is available.\\
\textbf{H2}: There exists a positive constant $\bar{\theta}$  such that
for every  $i\in\{1, 2, \cdots, n\}$,
\begin{align*}
|f_i(t,\bar{x}_i(t-\tau(t)), \bar{x}_i(t))|&\leq \bar{\theta}\sum_{j=1}^i(|x_j(t-\tau(t))|+|x_j(t)|),\cr
|g_i(t,\bar{x}_i(t-\tau(t)), \bar{x}_i(t))|^2&\leq  \bar{\theta}\sum_{j=1}^i(|x_j(t-\tau(t))|+|x_j(t)|).
\end{align*}

The aim  in this section is to design a  controller by state feedback method such that
the output $y(t)$ can be asymptotically regulated  to a neighborhood of
the zero that is  arbitrarily small and the system state is ultimately
bounded in the mean-square sense.

First, introducing the transformations
\begin{align}\label{equ51}
z_1(t)=y(t), z_{i+1}(t)=x_{i+1}(t)-\alpha_{i}(y(t),\bar{x}_{i-1}(t)),

\end{align}
where $i=1, 2, \cdots, n$ and $x_{n+1}(t)=u(t)$,
$z_{n+1}(t)=0$, $\bar{x}_{0}(t)=0$, and
stabilizing functions $\alpha_{i}(i=1, \cdots, n)$ are to be
constructed.

Next, one constructs the Lyapunov-Krasovskii functional step by
step.

\textbf{Step 1} Choosing
\begin{equation*}
V_1=V_{\tau 1}+\frac{1}{2}z_1^2, ~~
V_{\tau 1}=n\int_{t-\tau(t)}^te^{s-t}z_1^2(s)ds,
\end{equation*}
and noting that  $\dot{\tau}(t)\leq \tau^*<1$ and $0\leq \tau(t)\leq \tau$,
the derivative of $V_1$ satisfies
\begin{align}\label{equ52}
\dot{V}_1\leq &z_1(\alpha_{1}+z_2+f_1+g_1\xi_1)+nz_1^2\cr
&-ne^{-\tau}(1-\tau^*)z_1^2(t-\tau(t))-V_{\tau 1}.
\end{align}
Applying Young's inequality and (\ref{equ51}), one gets
\begin{align}
\label{equ531} z_1z_2 &\leq \frac{1}{2}z_1^2+\frac{1}{2}z_2^2, \\
\label{equ532}z_1f_1 &\leq \Omega_1z_1^2+\frac{(1-\tau^*)e^{-\tau}}{2}z_1^2(t-\tau(t)),\\
\label{equ533}
z_1g_1\xi_1&\leq \Lambda_{1}z_1^2+\frac{(1-\tau^*)e^{-\tau}}{2}z_1^2(t-\tau(t))+\pi_1|\xi_1|^4,
\end{align}
where
$\Omega_1=\bar{\theta}+\frac{1}{2(1-\tau^*)}\bar{\theta}^2e^{\tau}$,
$\Lambda_{1}=\frac{1}{2}\bar{\theta}+\frac{1}{8(1-\tau^*)}\bar{\theta}^2e^{\tau}
+\frac{1}{16\pi_1}$, and $\pi_1$ is a positive design parameter.
Substituting (\ref{equ531})-(\ref{equ533}) into (\ref{equ52}) yields
\begin{align*}
\dot{V}_1\leq&
z_1(nz_1+\Omega_1z_1+\Lambda_{1}z_1+\frac{1}{2}z_1+\alpha_{1})
+\frac{1}{2}z_2^2+\pi_1|\xi_1|^4\cr
&-(n-1)e^{-\tau}(1-\tau^*)z_1^2(t-\tau(t))-V_{\tau
1}.
\end{align*}
Selecting the stabilizing function
$\alpha_{1}=-2nz_1-\Omega_1z_1-\Lambda_{1}z_1-\frac{1}{2}z_1
\triangleq -\beta_1z_1$, then one  has
\begin{align*}
\dot{V}_1\leq & \frac{1}{2}z_2^2+\pi_1|\xi_1|^4-nz_1^2-V_{\tau 1}\cr
&-(n-1)(1-\tau^*)e^{-\tau}z_1^2(t-\tau(t)).
\end{align*}

\textbf{Step i} $(i=2, 3, \cdots, n)$ Suppose that
$V_{i-1}=V_{i-2}+\frac{1}{2}z_{i-1}^2+(n-i+2)\int_{t-\tau(t)}^te^{s-t}z_{i-1}^2(s)ds$
and a series of stabilizing functions $\alpha_j(j=1, 2, \cdots,i-1)$
have been constructed  such that
\begin{align}\label{equ54}
\dot{V}_{i-1}\leq& \frac{1}{2}z_i^2+\sum_{j=1}^{i-1}\pi_j|\xi_j|^4
+\sum_{j=1}^{i-2}\sum_{l=j+1}^{i-1}\pi_{lj}|\xi_j|^4\cr
&-(n-i+1)e^{-\tau}(1-\tau^*)\sum_{j=1}^{i-1}z_j^2(t-\tau(t))\cr
&-(n-i+2)\sum_{j=1}^{i-1}z_j^2 -\sum_{j=1}^{i-1}V_{\tau j},
\end{align}
where $V_{\tau j}=(n-j+1)\int_{t-\tau(t)}^te^{s-t}z_{j}^2(s)ds, j=1,\cdots, i-1$.
In what follows, one will prove that (\ref{equ54}) also holds
for $V_i$.

Let
$V_{i}=V_{i-1}+\frac{1}{2}z_{i}^2+(n-i+1)\int_{t-\tau(t)}^te^{s-t}z_{i}^2(s)ds$,
then
\begin{align}\label{equ55}
\dot{V}_{i}\leq&
\dot{V}_{i-1}+z_{i}(z_{i+1}+\alpha_{i}+f_i+g_i\xi_i)+(n-i+1)z_i^2\cr
&-z_{i}\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}
(x_{j+1}+f_j+g_j\xi_j)-V_{\tau i}\cr
&-(n-i+1)e^{-\tau}(1-\tau^*)z_i^2(t-\tau(t)),
\end{align}
where
\begin{align}\label{equ56}
-\frac{\partial\alpha_{i-1}}{\partial x_j}
=\prod_{k=j}^{i-1}\beta_k\triangleq \Xi_{j(i-1)}, ~~j=1, 2, \cdots
i-1.
\end{align}
By \textbf{H2}, (\ref{equ56}) and Young's inequality, one obtains
\begin{align}
\label{equ570}
&z_iz_{i+1}
\leq \frac{1}{2}z_i^2+\frac{1}{2}z_{i+1}^2,\\
\label{equ571}
&z_if_i-z_i\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}(x_{j+1}+f_j)\cr
\leq &\sum_{l=1}^{i-1}\frac{1}{2}z_l^2+\Omega_iz_i^2
+\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i}z_l^2(t-\tau(t)),\\
\label{equ572}
&z_ig_i\xi_i-z_i\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}g_j\xi_j\cr
\leq &\sum_{l=1}^{i-1}\frac{1}{2}z_l^2+\Lambda_iz_i^2
+\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i}z_l^2(t-\tau(t))\cr
&+\pi_i|\xi_i|^4+\sum_{j=1}^{i-1}\pi_{ij}|\xi_j|^4,
\end{align}
where $\pi_i$ and $\pi_{ij}$ are positive design parameters. The
detail proofs of (\ref{equ571}) and (\ref{equ572}) are given in
\textbf{Appendix A} and \textbf{Appendix B}, respectively.
Substituting (\ref{equ54}), (\ref{equ570})-(\ref{equ572}) to
(\ref{equ55}) leads
\begin{align}\label{equ58}
\dot{V}_{i}\leq &z_{i}(z_{i}+\alpha_{i}+2(n-i+1)z_{i}
+\Omega_iz_i+\Lambda_{i}z_{i})\cr
&+\frac{1}{2}z_{i+1}^2+\sum_{j=1}^{i}\pi_j|\xi_j|^4
+\sum_{j=1}^{i-1}\sum_{l=j+1}^{i}\pi_{lj}|\xi_j|^4\cr
&-(n-i+1)\sum_{j=1}^{i}z_j^2-\sum_{j=1}^{i}V_{\tau j}\cr
&-(n-i)e^{-\tau}(1-\tau^*)\sum_{j=1}^{i}z_j^2(t-\tau(t)).
\end{align}
Choosing the stabilizing function
\begin{align*}
\alpha_{i}=-z_{i}-2(n-i+1)z_{i}-\Omega_iz_i-\Lambda_{i}z_{i}
\triangleq -\beta_iz_i,
\end{align*}
then (\ref{equ58}) can be changed into
\begin{align*}
\dot{V}_{i}\leq&  \frac{1}{2}z_{i+1}^2
+\pi_i|\xi_i|^4+\sum_{j=1}^{i-1}(\pi_j+\sum_{l=j+1}^{i}\pi_{lj})|\xi_j|^4
-\sum_{j=1}^{i}V_{\tau j}\cr
&-(n-i)e^{-\tau}(1-\tau^*)\sum_{j=1}^{i}z_j^2(t-\tau(t))\cr
&-(n-i+1)\sum_{j=1}^{i}z_j^2.
\end{align*}
Let $i=n$, together with $z_{n+1}=0$, then one gets the actual
control law
\begin{align}\label{equ59}
u=-z_{n}-2z_{n}-\Omega_nz_n-\Lambda_{n}z_{n}

= -\sum_{i=1}^{n}(\prod_{j=i}^{n}\beta_j)x_i,
\end{align}
and
\begin{align}\label{equ510}
\dot{V}_{n}
\leq &-\sum_{j=1}^{n}\frac{1}{2}z_j^2-\sum_{j=1}^{n}V_{\tau j}+\pi_n|\xi_n|^4\cr
&+\sum_{j=1}^{n-1}(\pi_j+\sum_{l=j+1}^{n}\pi_{lj})|\xi_j|^4\cr
\leq &-V_{n}+\tilde{\pi}|\xi|^4,
\end{align}
where $\tilde{\pi}=\max\{\max_{1\leq j\leq
n-1}\{\pi_j+\sum_{l=j+1}^{n}\pi_{lj}\}, \pi_n\}$.

According to the above design procedure, for system (\ref{equ41})
satisfying the assumptions \textbf{H0}, \textbf{H1} and
\textbf{H2}, one  chooses the  Lyapunov-Krasovskii
functional
\begin{align*}
V(t,z(t))
&=\sum_{i=1}^n\Big[\frac{1}{2}z_{i}^2(t)+(n-i+1)\int_{t-\tau(t)}^te^{s-t}z_{i}^2(s)ds\Big].
\end{align*}
It is clear that
\begin{equation*}
V(t,z(t))\geq \sum_{j=1}^n\frac{1}{2}z_j^2=\frac{1}{2}|z(t)|^2
\end{equation*}
and
\begin{align*}
V(t,z(t))&\leq n\sum_{j=1}^n \int_{-\tau(t)}^0z_j^2(v+t)dv
+\sum_{j=1}^n\frac{1}{2}z_j^2\cr &\leq
n^2\tau\sum_{j=1}^n(\sup_{-\tau\leq v\leq
0}|z_j(v+t)|)^2+\frac{1}{2}|z(t)|^2\cr &\leq
\frac{1+2n^2\tau}{2}(\sup_{-\tau\leq v\leq 0}|z(v+t)|)^2,
\end{align*}
that is to say,
\begin{equation}\label{equ591}
\frac{1}{2}|z(t)|^2 \leq V(t,z(t)) \leq
\frac{1+2n^2\tau}{2}(\sup_{-\tau\leq v\leq 0}|z(v+t)|)^2.
\end{equation}

From (\ref{equ510}) and (\ref{equ591}),  the Lyapunov-Krasovskii
functional $V$ meets the requirements of Theorem \ref{the31}, so
the closed-loop system (including (\ref{equ41}) and (\ref{equ59}) )
is noise-to-state stable in the mean-square sense and has a unique
solution which is UB-\textbf{2}-M. Moreover, the output regulation
error of closed-loop system satisfies
\begin{align*}
\lim_{t\rightarrow \infty}E|y(t)|^2 \leq \lim_{t\rightarrow
\infty}E|z(t)|^2 \leq 2\tilde{\pi}K.
\end{align*}
According to the definitions of $\tilde{\pi}$, the regulation error
can be decreased  by tuning the positive constants
$\pi_j+\sum_{i=j+1}^{n}\pi_{ij}(1 \leq j \leq n-1)$ and $\pi_n$
small enough (i.e., letting $\pi_j$, $\pi_{ij}(i=j+1, 1 \leq j \leq
n-1)$ and $\pi_n$ arbitrarily small), where $\pi_j(1 \leq j \leq
n-1)$, $\pi_{ij}(i=j+1, 1 \leq j \leq n-1)$ and $\pi_n$ are
independent of each other. In other words, the regulation error can
be made small enough in the mean-square sense by selecting
appropriate design parameters.

From what has been analyzed above, one draws the stability analysis
result as below at present.
\begin{theorem}
If  system (\ref{equ41}) satisfies the assumptions \textbf{H0},
\textbf{H1} and \textbf{H2}, then by selecting appropriate design
parameters, the closed-loop system (\ref{equ41}) and (\ref{equ59})
is NSS-\textbf{2}-M and has a unique solution which is UB-\textbf{2}-M.
Moreover, the regulation error
satisfies
\begin{align*}
\lim_{t\rightarrow \infty}E|y(t)|^2 \leq 2\tilde{\pi}K,
\end{align*}
and the regulation error can be made arbitrarily small in the
mean-square sense by parameter-tuning technique.
\end{theorem}

\section{Application 2: adaptive output feedback regulation control}\label{S4}

In this section, one focuses on the adaptive output feedback
regulation control for system (\ref{equ41}). To this end,
one supposes that system (\ref{equ41}) satisfies the assumption
\textbf{H0} and the following two assumptions:\\
\textbf{H1'}: Only $x_1(t)$ is measurable by system output $y(t)$,
other state variables $x_2(t), \cdots, x_n(t)$ are unavailable.\\
\textbf{H2'}: For $i=1, 2, \cdots, n$, unknown functions $f_i, g_i$
satisfy
\begin{align*}
|f_i(t, \bar{x}_i(t-\tau(t)),\bar{x}_i(t))|\leq \theta_i[\phi_{\tau i}(y(t-\tau(t)))+\phi_i(y(t))], \cr
|g_i(t,\bar{x}_i(t-\tau(t)),\bar{x}_i(t))|\leq \theta_i[\psi_{\tau i}(y(t-\tau(t)))+\psi_i(y(t))],
\end{align*}
where $\phi_i, \phi_{\tau i}, \psi_i$ and $\psi_{\tau i}$ are smooth
non-negative function with $\phi_i(0)=\phi_{\tau
i}(0)=\psi_i(0)=\psi_{\tau i}(0)=0$. $\theta_i>0$ are unknown
parameters.

It follows from the assumption \textbf{H2'} that there exist smooth
functions $\bar{\phi}_i(y), \bar{\phi}_{\tau i}(y(t-\tau(t))),
\bar{\psi}_i(y)$ and $\bar{\psi}_{\tau i}(y(t-\tau(t)))$
satisfying
\begin{align}\label{equ42}

&\phi_i(y)=y\bar{\phi}_i(y), ~~~~\psi_i(y)=y\bar{\psi}_i(y),\cr
&\phi_{\tau i}(y(t-\tau(t)))=y(t-\tau(t))\bar{\phi}_{\tau i}(y(t-\tau(t))), \cr
&\psi_{\tau i}(y(t-\tau(t)))=y(t-\tau(t))\bar{\psi}_{\tau i}(y(t-\tau(t))).

\end{align}

The task in this section is to construct a  regulation controller for system (\ref{equ41})
by adaptive output feedback method such that the system output
$y(t)$ can be regulated  to  an neighborhood  of the zero whose
scope can be controlled and the system state is bounded in the
mean-square sense.

\subsection{Observer design}
Since state variables $x_2(t), \cdots, x_n(t)$ are unavailable, one
reconfigures $x_i(t) (i=2, \cdots, n)$ by the following
reduced-order observer:
\begin{equation}\label{equ43}
\begin{cases}
\dot{\hat{x}}_i(t)=\hat{x}_{i+1}(t)+\kappa_{i+1}y(t)-\kappa_i(\hat{x}_1(t)+\kappa_1y(t)),\\
i=1, 2, \dots n-2, \\
\dot{\hat{x}}_{n-1}(t)=u(t)-\kappa_{n-1}(\hat{x}_1(t)+\kappa_1y(t)),
\end{cases}
\end{equation}
where $\kappa_1, \dots, \kappa_{n-1}$ are the observer gain to be
designed.

Let $\theta^*=\max\{1, \theta_1, \cdots, \theta_n \}$, $e=[e_2,
\cdots, e_n]^T$ with
$e_j=(x_j(t)-\hat{x}_{j-1}(t)-\kappa_{j-1}y(t))/\theta^*(j=2, 3,
\cdots, n)$, then the dynamic trajectory of observer
error can be described by
\begin{equation}\label{equ44}
\dot{e}=A_ee +\frac{1}{\theta^*}\Xi_1+\frac{1}{\theta^*}\Xi_2\xi,
\end{equation}
where $A_e=(-\kappa, \bar{I}_{n-1})$,
$\bar{I}_{n-1}=diag\{1, 1, \cdots, 1, 0\}$,
$\kappa=(\kappa_1, \cdots, \kappa_{n-1})^T$,
$\Xi_1=[f_2-\kappa_1f_1, \cdots, f_n-\kappa_{n-1}f_1]^T$,
$\Xi_2=(-g_1\kappa,\Theta)$,
$\Theta=diag\{g_2, \cdots, g_n\}$.

The observer gain $\kappa_1, \dots, \kappa_{n-1}$ should be chosen
to meet the requirement that $A_e$ is Hurwitz. This means that there
is a matrix $P_{n-1}>0$  such that
$A_e^TP_{n-1}+P_{n-1}A_e+bI_{n-1}=0$, where $b$ is a positive design constant.

\subsection{Adaptive regulation controller design}\label{S42}
First, one introduces the error transformations
\begin{equation}\label{equ45}
z_1=y(t), ~~ z_{i+1}=\hat{x}_i-\alpha_{i}(y, \bar{\hat{x}}_{i-1},
\hat{\theta}),
\end{equation}
where $\bar{\hat{x}}_{i-1}(t)=(\hat{x}_1(t), \cdots,
\hat{x}_{i-1}(t))^T$, $i=1, 2, \cdots, n$, $\hat{x}_{n}=u$, $z_{n+1}=0$,
$\bar{\hat{x}}_{0}=0$, $\alpha_{i}(i=1, \cdots, n)$ are stabilizing
functions to be determined and $\hat{\theta}$ is the estimation of
$\theta$ with $\theta=\max\{\theta^{*2},\theta^{*4}\}$. In the light
of (\ref{equ43})-(\ref{equ45}), one can deduce that
\begin{align}
\label{equ451} \dot{z}_1=&e_2\theta^*+\hat{x}_1+\kappa_1y+f_1+g_1\xi_1,\\
\label{equ452}
\dot{z}_i=&z_{i+1}+\alpha_{i}-\frac{\partial\alpha_{i-1}}{\partial\hat{\theta}}\dot{\hat{\theta}}
-\frac{\partial\alpha_{i-1}}{\partial y}(e_2\theta^*+f_1)\cr
&-\frac{\partial\alpha_{i-1}}{\partial y}g_1\xi_1+\bar{\beta}_i,
~~i=2, \cdots, n,
\end{align}
where
$\bar{\beta}_i=\kappa_iy-\sum_{j=1}^{i-2}\frac{\partial\alpha_{i-1}}{\partial
\hat{x}_j}(\hat{x}_{j+1}+\kappa_{j+1}y-\kappa_i(\hat{x}_1+\kappa_1y))
-(\kappa_{i-1}+\frac{\partial\alpha_{i-1}}{\partial
y})(\hat{x}_1+\kappa_1y)$.

Next, one begins to design adaptive regulation controller.

\textbf{Step 1} Selecting the candidate Lyapunov-Krasovskii functional
\begin{align*}
V_1&=V_0+V_\tau+\frac{1}{2}z_1^2,~~
V_0=e^TPe+\frac{1}{2\mu}\tilde{\theta}^2,\cr
V_\tau&=\int_{t-\tau(t)}^t e^{s-t}Q(y(s))ds,
\end{align*}
where $\tilde{\theta}=\theta-\hat{\theta}$, both the adaptive gain
$\mu>0$ and continuous function $Q(y(s))>0$ are  to be designed. By
(\ref{equ451}), $\dot{\tau}(t)\leq \tau^*<1$ and $0\leq \tau(t)\leq \tau$,
the derivative of $V_1$  satisfies that
\begin{align}\label{equ46}
\dot{V}_1=&\dot{V}_0
+z_1(e_2\theta^*+\hat{x}_1+\kappa_1y+f_1+g_1\xi_1)-V_\tau\cr
&+Q(y(t))-e^{-\tau(t)}(1-\dot{\tau}(t))Q(y(t-\tau(t)))\cr
\leq &-b|e|^2+\frac{2}{\theta^*}e^TP\Xi_1+\frac{2}{\theta^*}e^TP\Xi_2\xi
-\frac{1}{\mu}\tilde{\theta}\dot{\hat{\theta}} -V_\tau\cr
&-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y)+z_1e_2\theta^*\cr
&+z_1(z_2+\alpha_1+\kappa_1y)+z_1f_1+z_1g_1\xi_1.
\end{align}
It follows from assumption \textbf{H2'}, (\ref{equ42}), Lemma
\ref{lem2b5} and Young's inequality  that
\begin{align}
\label{equ471}
z_1e_2\theta^*\leq &d_{11}|e|^2+\frac{1}{4}d_{11}^{-1}z_1^2\theta,\\
\label{equ472}
z_1f_1\leq &\frac{1}{4}d_{12}^{-1}z_1^2\theta + 2d_{12}y^2\bar{\phi}_1^2
+ 2d_{12}\phi_{\tau 1}^2,\\
\label{equ473}
z_1g_1\xi_1\leq &\frac{1}{8}d_{13}^{-1}z_1^4\theta +
d_{13}^{-1}y^4\bar{\psi}_1^4\cr
&+ d_{13}^{-1}\psi_{\tau 1}^4+d_{13}|\xi|^2,\\
\label{equ474}
\frac{2}{\theta^*}e^TP\Xi_1
\leq &\sum_{i=2}^n[(\bar{\phi}_i^2+\kappa_{i-1}^2\bar{\phi}_1^2)y^2
+(\phi_{\tau i}^2+\kappa_{i-1}^2\phi_{\tau 1}^2)]\cr
&\cdot4||P||_F^2d_{01}^{-1}+d_{01}|e|^2, \\
\label{equ475} \frac{2}{\theta^*}e^TP\Xi_2\xi
&\leq \sum_{i=2}^n[(\bar{\psi}_i^4+\kappa_{i-1}^4\bar{\psi}_1^4)y^4
+(\psi_{\tau i}^4+\kappa_{i-1}^4\psi_{\tau 1}^4)]\cr
&\cdot4(n-1)d_{02}^{-1}||P||_F^4+ |e|^2+d_{02}|\xi|^4,
\end{align}
where $d_{01}, d_{02}, d_{11}, d_{12}$ and $d_{13}$ are positive design
parameters. Substituting (\ref{equ471})-(\ref{equ475}) into (\ref{equ46}), one obtains
\begin{align}\label{equ48}
\dot{V}_1\leq &-b_1|e|^2+z_1(z_2+\alpha_1+\kappa_1y+\omega
+\frac{1}{8}\varpi_1\hat{\theta})+\delta_1 \cr
&+(\frac{1}{8}\varpi_1z_1-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}
+d_{02}|\xi|^4+d_{13}|\xi|^2-V_\tau\cr
&-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y),
\end{align}
where
$\omega=4||P||_F^2d_{01}^{-1}\sum_{i=2}^n[\bar{\phi}_i^2+\kappa_{i-1}^2\bar{\phi}_1^2]y
+4(n-1)||P||_F^4d_{02}^{-1}\sum_{i=2}^n[\bar{\psi}_i^4+\kappa_{i-1}^4\bar{\psi}_1^4]y^3
+2d_{12}\bar{\phi}_1^2y+ d_{13}^{-1}\bar{\psi}_1^4y^3$,
$\delta_1=\delta_0+2d_{12}\phi_{\tau 1}^2+ d_{13}^{-1}\psi_{\tau
1}^4$, $\delta_0=4||P||_F^2d_{01}^{-1}\sum_{i=2}^n[\phi_{\tau
i}^2+\kappa_{i-1}^2\phi_{\tau 1}^2]
+4(n-1)||P||_F^4d_{02}^{-1}\sum_{i=2}^n[\psi_{\tau
i}^4+\kappa_{i-1}^4\psi_{\tau 1}^4]$,
$\varpi_1=2d_{11}^{-1}z_1+2d_{12}^{-1}z_1+d_{13}^{-1}z_1^3$,
$b_1=b_0-d_{11}>0$, $b_0=b-d_{01}-1>0$.

Selecting the following tuning function $\lambda_1$ and the
stabilizing function $\alpha_1$ respectively
\begin{align*}
\lambda_1&=\frac{1}{8}\varpi_1 z_1-\frac{1}{\mu}\hat{\theta},\cr
\alpha_1(y, \hat{\theta})&=-c_1z_1-\kappa_1y-\omega
-\frac{1}{8}\varpi_1\hat{\theta}-\sigma(y),
\end{align*}
where positive constant $c_1$ and the function $\sigma(y)$ are  to
be designed, then
\begin{align}\label{equ49}
\dot{V}_1\leq &-b_1|e|^2+z_1z_2-c_1z_1^2-\sigma(y)y
+(\lambda_1-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}\cr
&+\frac{1}{\mu}\hat{\theta}\tilde{\theta}+\delta_1
+d_{02}|\xi|^4+d_{13}|\xi|^2-V_\tau\cr
&-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y).
\end{align}

\textbf{Step k} $(k= 2, \cdots, n)$ Suppose that the appropriate
stabilizing functions $\alpha_j(j=1, 2, \cdots, k-1)$ and tuning
functions $\lambda_j(j=1, 2, \cdots, k-1)$ have been designed such
that $V_{k-1}=V_{k-2}+\frac{1}{2}z_{k-1}^2$ satisfies
\begin{align}\label{equ410}
\dot{V}_{k-1}\leq
&-b_{k-1}|e|^2-\sum_{j=1}^{k-1}c_jz_j^2+z_{k-1}z_{k}-\sigma(y)y\cr
&+d_{02}|\xi|^4+\sum_{j=1}^{k-1}d_{j3}|\xi|^2
+\frac{1}{\mu}\hat{\theta}\tilde{\theta} +\delta_{k-1}\cr
&+\sum_{j=2}^{k-1}[2d_{j2}\bar{\phi}_1^2+d_{13}^{-1}\bar{\psi}_1^4
y^2]y^2
+(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}\cr
&+\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}
(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}})-V_\tau\cr
&-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y),
\end{align}
where $\delta_{k-1}=\delta_{k-2}+2d_{(k-1)2}\phi_{\tau 1}^2+
d_{(k-1)3}^{-1}\psi_{\tau 1}^4$, $b_{k-1}=b_{k-2}-d_{(k-1)1}>0$.

In what follows, one selects $V_k=V_{k-1}+\frac{1}{2}z_{k}^2$ in
\textbf{Step k} to prove that $V_k$ also satisfies (\ref{equ410}).
From  (\ref{equ452}) and  (\ref{equ410}), one has
\begin{align}\label{equ411}
\dot{V}_{k}\leq &-b_{k-1}|e|^2-\sum_{j=1}^{k-1}c_jz_j^2+\delta_{k-1}+d_{02}|\xi|^4\cr
&+\sum_{j=1}^{k-1}d_{j3}|\xi|^2+z_{k}(z_{k-1}+z_{k+1}+\alpha_{k}+\bar{\beta}_k)\cr
&-z_{k}\frac{\partial\alpha_{k-1}}{\partial y}(e_2\theta^*+f_1+g_1\xi_1)
-z_k\frac{\partial\alpha_{k-1}}{\partial\hat{\theta}}\dot{\hat{\theta}}\cr

&+\sum_{j=2}^{k-1}[2d_{j2}\bar{\phi}_1^2+d_{j3}^{-1}\bar{\psi}_1^4y^2]y^2
+\frac{1}{\mu}\hat{\theta}\tilde{\theta}-\sigma(y)y\cr
&+(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}
+\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}(\lambda_{k-1}
-\frac{1}{\mu}\dot{\hat{\theta}}) \cr
&-V_\tau-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y).
\end{align}

By Young's inequality, assumption \textbf{H2'} and (\ref{equ42}), one
gets the following inequalities
\begin{align}
\label{equ4121} -z_k\frac{\partial\alpha_{k-1}}{\partial y}e_2\theta^*\leq & d_{k1}|e|^2
+\frac{1}{4}d_{k1}^{-1}z_k^2\Big(\frac{\partial\alpha_{k-1}}{\partial y}\Big)^2\theta,\\
\label{equ4122} -z_k\frac{\partial\alpha_{k-1}}{\partial y}f_1
\leq &\frac{1}{4}d_{k2}^{-1}z_k^2\Big(\frac{\partial\alpha_{k-1}}{\partial y}\Big)^2\theta
+ 2d_{k2}y^2\bar{\phi}_1^2\cr
&+ 2d_{k2}\phi_{\tau 1}^2,\\
\label{equ4123} -z_k\frac{\partial\alpha_{k-1}}{\partial y}g_1\xi_1
\leq &
\frac{1}{8}d_{k3}^{-1}z_k^4\Big(\frac{\partial\alpha_{k-1}}{\partial
y}\Big)^4\theta + d_{k3}^{-1}y^4\bar{\psi}_1^4\cr
&+d_{k3}^{-1}\psi_{\tau 1}^4+d_{k3}|\xi|^2.
\end{align}
Substituting (\ref{equ4121})-(\ref{equ4123}) into (\ref{equ411}),
then $\dot{V}_k$ satisfies
\begin{align}\label{equ413}
\dot{V}_{k}\leq &-b_{k}|e|^2-\sum_{j=1}^{k-1}c_jz_j^2+(\frac{1}{8}z_k \varpi_k +\lambda_{k-1}
-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}\cr
&+z_{k}(z_{k-1}+z_{k+1}+\alpha_{k}+\bar{\beta}_k
+\frac{1}{8}\varpi_k\hat{\theta}
-\mu\frac{\partial\alpha_{k-1}}{\partial\hat{\theta}}\lambda_k)\cr
&-\sigma(y)y+\delta_{k}+\mu z_k\frac{\partial\alpha_{k-1}}{\partial\hat{\theta}}
(\lambda_k-\frac{1}{\mu}\dot{\hat{\theta}})+\frac{1}{\mu}\hat{\theta}\tilde{\theta}\cr
&+\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}
(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}})+d_{02}|\xi|^4\cr
&+\sum_{j=1}^{k}d_{j3}|\xi|^2
+\sum_{j=2}^{k}[2d_{j2}\bar{\phi}_1^2+d_{j3}^{-1}\bar{\psi}_1^4y^2]y^2 \cr
&- e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y)-V_\tau,
\end{align}
where $d_{k1}, d_{k2}$ and $d_{k3}$ are positive design constants.
$b_{k}=b_{k-1}-d_{k1}>0$,
$\varpi_k=2d_{k1}^{-1}z_k\Big(\frac{\partial\alpha_{k-1}}{\partial
y}\Big)^2 +2d_{k2}^{-1}z_k\Big(\frac{\partial\alpha_{k-1}}{\partial
y}\Big)^2 +d_{k3}^{-1}z_k^3\Big(\frac{\partial\alpha_{k-1}}{\partial
y}\Big)^4$ and $\delta_{k}=\delta_{k-1}+2d_{k2}\phi_{\tau 1}^2+
d_{k3}^{-1}\psi_{\tau 1}^4$.

First, let the tuning function
$\lambda_k=\lambda_{k-1}+\frac{1}{8}z_k\varpi_k$, then
\begin{align}
\label{equ4141}
&\frac{1}{8}z_k \varpi_k \tilde{\theta}
+(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}
=(\lambda_{k}-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta},\\
\label{equ4142}
&\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}
(\lambda_{k}-\frac{1}{\mu}\dot{\hat{\theta}})
-\frac{1}{8}z_k\varpi_k\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}\cr
=&\mu\sum_{j=2}^{k-1}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}
(\lambda_{k-1}-\frac{1}{\mu}\dot{\hat{\theta}}).
\end{align}
Next, one substitutes (\ref{equ4141})-(\ref{equ4142}) into
(\ref{equ413}) and chooses the stabilizing function
\begin{align}
\alpha_k(y, \bar{\hat{x}}_{k-1}, \hat{\theta})
=&-c_kz_k-z_{k-1}-\bar{\beta}_k
+\mu\frac{\partial\alpha_{k-1}}{\partial\hat{\theta}}\lambda_k\cr
&-\frac{1}{8}\varpi_k\hat{\theta}+\frac{1}{8}\varpi_k\mu\sum_{j=2}^{k-1}
z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}},
\end{align}
then (\ref{equ413}) can be changed into
\begin{align*}
\dot{V}_{k}\leq &-b_{k}|e|^2 -\sum_{j=1}^{k}c_jz_j^2+z_{k}z_{k+1}
+(\lambda_{k}-\frac{1}{\mu}\dot{\hat{\theta}})\tilde{\theta}\cr
&+\mu\sum_{j=2}^{k}z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}}
(\lambda_{k}-\frac{1}{\mu}\dot{\hat{\theta}})
+\frac{1}{\mu}\hat{\theta}\tilde{\theta}
+d_{02}|\xi|^4+\delta_{k}\cr &+\sum_{j=1}^{k}d_{j3}|\xi|^2
-\sigma(y)y+\sum_{j=2}^{k}[2d_{j2}\bar{\phi}_1^2
+d_{j3}^{-1}\bar{\psi}_1^4 y^2]y^2 \cr
&-e^{-\tau}(1-\tau^*)Q(y(t-\tau(t)))+Q(y)-V_\tau.
\end{align*}

At \textbf{Step n}, let $Q(y)=\bar{Q}(y)y$,
$\sigma(y)=\bar{Q}(y)+\sum_{j=2}^{n}[2d_{j2}\bar{\phi}_1^2(y)
+d_{j3}^{-1}\bar{\psi}_1^4(y) y^2]y$, where
\begin{align*}
\bar{Q}(y(t))
=&\frac{4e^\tau }{1-\tau^*}||P||_F^2d_{01}^{-1}y\sum_{i=2}^n
[\bar{\phi}_{\tau i}^2(y)+\kappa_{i-1}^2\bar{\phi}_{\tau 1}^2(y)]\cr
&+\frac{e^\tau }{1-\tau^*}\sum_{j=1}^{n}[2d_{j2}\bar{\phi}_{\tau1}^2(y)
+d_{j3}^{-1}\bar{\psi}_{\tau 1}^4(y)y^2]y\cr
&+\frac{4(n-1)e^\tau }{1-\tau^*}||P||_F^4d_{02}^{-1}y^3\cr
&\cdot\sum_{i=2}^n[\bar{\psi}_{\tau i}^4(y)+\kappa_{i-1}^4\bar{\psi}_{\tau 1}^4(y)].
\end{align*}
Furthermore, selecting the update law
\begin{align}\label{equ415}
\dot{\hat{\theta}}=\mu\lambda_{n},
~~\lambda_n=\lambda_{n-1}+\frac{1}{8}z_n\varpi_n,
\end{align}
and the control law
\begin{align}\label{equ416}
u=&-c_nz_n-z_{n-1}-\bar{\beta}_n-\frac{1}{8}\varpi_n\hat{\theta}
+\mu\frac{\partial\alpha_{n-1}}{\partial\hat{\theta}}\lambda_n\cr
&+\frac{1}{8}\varpi_n\mu\sum_{j=2}^{n-1}
z_j\frac{\partial\alpha_{j-1}}{\partial\hat{\theta}},
\end{align}
then one has
\begin{align}\label{equ417}
\dot{V}_{n} \leq &-b_{n}|e|^2-\sum_{j=1}^{n}c_jz_j^2
+\frac{1}{\mu}\hat{\theta}\tilde{\theta}+d_{02}|\xi|^4\cr
&+\sum_{j=1}^{n}d_{j3}|\xi|^2-V_\tau\cr
\leq &-\frac{b_n}{\sigma_M}e^TPe-\frac{1}{2\mu}\tilde{\theta}^2-V_\tau
-\sum_{j=1}^{n}c_jz_j^2 +\frac{1}{2\mu}\theta^2\cr
&+d_{02}|\xi|^4+\sum_{j=1}^{n}d_{j3}(\frac{1}{4}+|\xi|^4)\cr \leq
&-\bar{c}V_n+\bar{d}_1|\xi|^4+\bar{d}_2,
\end{align}
where $\sigma_M=\lambda_{max}(P)$,
$\bar{c}=\min\{\frac{b_n}{\sigma_M}, 1, 2c_1, 2c_2, \cdots, 2c_n
\}$, $\bar{d}_1=d_{02}+\sum_{j=1}^{n}d_{j3}$, and
$\bar{d}_2=\frac{1}{2\mu}\theta^2+\frac{1}{4}\sum_{j=1}^{n}d_{j3}$.

\subsection{Stability analysis}
\begin{theorem}\label{the41}
Suppose that assumptions \textbf{H0}, \textbf{H1'} and \textbf{H2'}
hold for system (\ref{equ41}), then  the closed-loop system
including (\ref{equ41}), (\ref{equ415}) and (\ref{equ416})
is NSS-\textbf{2}-M and the closed-loop system solution on $[t_0-\tau, \infty)$
is UB-\textbf{2}-M. Furthermore, the regulation error in the
mean-square sense satisfies
\begin{align*}
\lim_{t\rightarrow \infty}E|y(t)|^2 \leq
\frac{2\bar{d}_1K+2\bar{d}_2}{\bar{c}},
\end{align*}
and by selecting appropriate design parameters, the upper bound of the regulation error
can be made arbitrarily small.
\end{theorem}

\begin{proof}
For system (\ref{equ41}), one chooses the Lyapunov-Krasovskii
functional
\begin{equation*}
V=e^TPe+\frac{1}{2\mu}\tilde{\theta}^2+\sum_{j=1}^{n}\frac{1}{2}z_j^2
+\int_{t-\tau(t)}^te^{s-t}Q(y(s))ds ,
\end{equation*}
from which one knows that $V$ is a radially unbounded functional and
\begin{align}\label{equ418}
\dot{V}\leq &-\bar{c}V+\bar{d}_1|\xi|^4+\bar{d}_2,
\end{align}
where $\bar{c}=\min\{\frac{b_n}{\sigma_M}, 1, 2c_1, 2c_2, \cdots,
2c_n \}$, $\bar{d}_1=d_{02}+\sum_{j=1}^{n}d_{j3}$, and
$\bar{d}_2=\frac{1}{2\mu}\theta^2+\frac{1}{4}\sum_{j=1}^{n}d_{j3}$
are all positive constants. Taking expectations on both sides of
(\ref{equ418}), one obtains
\begin{align*}
E\dot{V}&\leq -\bar{c}EV +\bar{d}_1E|\xi(t)|^4+\bar{d}_2.

\end{align*}
By Lemma \ref{lem2b2}, the closed-loop system including
(\ref{equ41}), (\ref{equ415}) and (\ref{equ416}) has a unique
solution.

Defining $\tilde{\nu}(t)=EV$ and $z(t)=[z_1(t),  \cdots, z_n(t)]^T$,
one adopts the same way  for (\ref{equ418}) as one has
done for (\ref{eq32}) in the proof of Theorem \ref{the31}, then the
inequality

$\tilde{\nu}(t)\leq
\tilde{\nu}(t_0)e^{-\bar{c}(t-t_0)}+(\bar{d}_1K+\bar{d}_2)/\bar{c}$
holds,

together with $V \geq \frac{1}{2}|z(t)|^2$, one gets
\begin{align}\label{equ419}
E|z(t)|^2 \leq 2E(V|_{t=t_0})e^{-\bar{c}(t-t_0)} +
\frac{2\bar{d}_1K+2\bar{d}_2}{\bar{c}},
\end{align}
which implies that the closed-loop system including (\ref{equ41}),
(\ref{equ415}) and (\ref{equ416}) is NSS-\textbf{2}-M. Furthermore,
letting $t\rightarrow \infty$ in (\ref{equ419}) yields
\begin{align}\label{equ420}
\lim_{t\rightarrow \infty}E|z(t)|^2 \leq
\frac{2\bar{d}_1K+2\bar{d}_2}{\bar{c}}.
\end{align}
This means that the state of the closed-loop system including
(\ref{equ41}), (\ref{equ415}) and (\ref{equ416}) is UB-\textbf{2}-M.

Additionally, it follows from (\ref{equ420}) that
\begin{align*}
\lim_{t\rightarrow \infty}E|y(t)|^2 \leq \lim_{t\rightarrow
\infty}E|z(t)|^2 \leq \frac{2\bar{d}_1K+2\bar{d}_2}{\bar{c}},
\end{align*}
which implies that the upper bound of regulation error in the
mean-square sense can be made small enough by  selecting $d_{02},
d_{j3}(1 \leq j \leq n)$ sufficiently small and $\mu$ large enough.
The parameters consisting of $\bar{c}, d_{02}, \mu, d_{j3}(1 \leq j
\leq n)$ are independent from each other. One completes the proof.
\end{proof}

\begin{remark}\label{rem51}
From the above design procedure of adaptive output feedback regulation controller in
this section, the observer is introduced to estimate the unmeasurable states,
which lead to the assumption  \textbf{H0} is proposed.
If the state of system (\ref{equ41}) is measurable
(i.e.,the assumption \textbf{H1} holds for  system (\ref{equ41})), and
system (\ref{equ41}) satisfies the assumptions \textbf{H0'}
(the random disturbance $\xi(t)$ satisfies $\sup_{t\geq t_0} E|\xi(t)|^2<K $)
and \textbf{H2'}, then one can design an adaptive state feedback controller to
achieve regulation control objective. The design procedure of state feedback controller
is as similar as the procedure given above, thus one omits the detailed process.
\end{remark}
\section{Simulation examples}\label{S6}

\textbf{Example 1:} The time-delay phenomenon widespreadly exists in
the chemical industry, such as  the chemical reactor recycle system
([\ref{Hua2009}], [\ref{Chen2011}]).
Consider a  two-stage chemical reactor system that is  modeled as
\begin{equation}\label{equ600}
\begin{cases}
\dot{x}_1=-l_1(t)x_1-\frac{1}{\mu_1}x_1+\frac{1-r_1}{v_1}x_2,\\
\dot{x}_2=\frac{\lambda_2}{v_2}u-l_2(t)x_2-\frac{1}{\mu_2}x_2
+\frac{r_1}{v_2}x_{1t}+\frac{r_2}{v_2}x_{2t},\\
y=x_1,
\end{cases}
\end{equation}
where $x_{1t}=x_1(t-\tau(t))$ and $x_{2t}=x_2(t-\tau(t))$,
$x_1$ and $x_2$ stand for produced streams from the two reactors;
the residence time of two reactors are $\mu_1$ and $\mu_2$; $v_1$ and $v_2$
represent reactor volumes;  $r_1$ and $r_2$ denote the recycle flow rates;
the feed rate is $\lambda_2$;  $\tau(t)=0.1(1-\sin t)$ represents the time delay;
$l_1(t)$ and $l_2(t)$ indicate the uncertain reaction functions.

For system (\ref{equ600}), [\ref{Chen2011}] studies the impact of
the white noise random disturbance for the  reactors by introducing It$\hat{o}$
stochastic differential equation. Here, one considers random differential equation by
introducing  random noise disturbance that is not white noise. Let the uncertain
reaction functions $l_i(t)=l_{i0}+\xi_i(t)/\sqrt{|x_i(t)|}$, $i=1,2$, where $l_{i0}$
are reaction constants and random disturbance $\xi(t)=[\xi_1(t), \xi_2(t)]^T$ satisfies
$\sup_{t\geq 0} E|\xi(t)|^4<K$, and one chooses $r_1=r_2=0.5$, $\mu_1=\mu_2=2$,
$l_{10}=l_{20}=0.5$, $\lambda_2=v_1=v_2=0.5$. System (\ref{equ600}) can be
rewritten as
\begin{equation}\label{equ601}
\begin{cases}
\dot{x}_1=x_2dt+f_1dt+g_1\xi_1,\\
\dot{x}_2=udt+f_2dt+g_2\xi_2,\\
y=x_1,
\end{cases}
\end{equation}
where $f_1=-x_1, g_1=-\sqrt{|x_1|},  f_2=-x_2+x_2(t-\tau(t))+x_1(t-\tau(t)), g_2=-\sqrt{|x_2|}$.
One supposes that the state variables $x_1$ and $x_2$ are available. The random disturbance
 $\xi_1(t)$ and $\xi_2(t)$ are produced by
\begin{align}\label{equ602}
\xi_1(t)&=a_1\cos(\ell_1t+U_1)\sin(\bar{\ell}_1t), \cr
\xi_2(t)&=a_2\cos(\ell_2t+U_2)\sin^2(\bar{\ell}_2t),
\end{align}
where $U_1$ and $U_2$ are random variables uniformly distributed on $[0,2\pi]$
and $a_1, a_2, \ell_1, \ell_2, \bar{\ell}_1, \bar{\ell}_2$ are real constants, then
$E\xi_1^4(t)\leq a_1^4$ and $E\xi_2^4(t)\leq a_2^4$.
It is clear that (\ref{equ601}) satisfies  assumptions \textbf{H0},
\textbf{H1} and \textbf{H2}.

In simulation, one chooses
$a_1=1, a_2=1, \ell_1=1.2, \ell_2=0.8,  \bar{\ell}_1=1.5, \bar{\ell}_2=1$,
$\pi_1=\pi_2=\pi_{21}=1$, and initial value $x_1(0)=0.5, x_2(0)=-0.5$.
The responses of closed-looped system (\ref{equ601}) and (\ref{equ59})
are shown in Figure 1, which illustrates that
the output regulation error of system (\ref{equ601}) can be controlled in
a small enough neighborhood of zero and the another state $x_2(t)$
is bounded in the mean-square sense.
\begin{figure}[htb]
 \centering
\rotatebox{360}{\scalebox{0.3}[0.3]{\includegraphics{fig1}}}
\center{ {\small Figure 1.  The responses of closed-loop system
 (\ref{equ601}) and (\ref{equ59}).}}
\vspace*{0.5cm}
\end{figure}

\textbf{Example 2:} For the following random nonlinear feedback time-delay systems
\begin{equation}\label{equ621}
\begin{cases}
\dot{x}_1=x_2+\theta_1 x_{1t}^2+\theta_1 x_1\xi_1,\\
\dot{x}_2=u+\theta_2 x_1\cos(x_{2t})
+\theta_2 x_{1t}\sin(x_2)\xi_2,\\
y=x_1,
\end{cases}
\end{equation}
where  $x_{1t}=x_1(t-\tau(t))$ and $x_{2t}=x_2(t-\tau(t))$,
$\tau(t)=0.5(1+\cos t)$ that satisfies $0 \leq \tau(t)\leq 1 $
and $\dot{\tau}(t)\leq 0.5$.
Both $\theta_1$ and  $\theta_2$ are unknown parameters.
Only the state variable $x_1$ is measurable
by output $y$, the state variable  $x_2$ is unavailable.
The random disturbance $\xi_1$ and $\xi_2$ are
generated by (\ref{equ602}) with
$a_1=0.8, a_2=1.2, \ell_1=1 , \ell_2=0.5, \bar{\ell}_1=0.5, \bar{\ell}_2=1$.
So system (\ref{equ621})  satisfies assumptions \textbf{H0}, \textbf{H1'} and \textbf{H2'}.

For simulation purpose, one selects
$\theta_1=1.2$ and $\theta_2=1.5$, and initial values
$x_1(0)=0.01$,  $x_2(0)=-0.3$, $\hat{x}_1(0)=0.1, \hat{\theta}(0)=-0.5$,
the design parameters $\mu=1, \kappa_1=1, b=2$,
$d_{01}=1, d_{02}=0.1, d_{11}=d_{12}=1$, $d_{13}=0.1$, $d_{21}=d_{22}=1$,
$d_{23}=0.1$, $c_1=1, c_2=1$. The responses of closed-loop system
(\ref{equ621}), (\ref{equ415}) and (\ref{equ416}) is given in Figure 2, which
shows that the state $x_2$ is bounded in the mean-square sense and the output
regulation error $e$ can be made small enough in the mean-square sense.

\begin{figure}[htb]
 \centering
\rotatebox{360}{\scalebox{0.3}[0.3]{\includegraphics{fig2}}}
\center{ {\small Figure 2.  The responses of closed-loop system
(\ref{equ621}), (\ref{equ415}) and (\ref{equ416}).}}
\vspace*{0.5cm}
\end{figure}

\section{Conclusion}\label{S7}
This paper investigates the stability of random nonlinear systems with time-delay
and its application in regulation control. The existence and
uniqueness of solution to random nonlinear time-delay systems is guaranteed
by proposed conditions. Next, some criteria on noise-to-state stability of random
nonlinear systems with time-delay are established. As applications,
two kinds of regulation  control problems are solved by state feedback method
and adaptive output feedback method.

Many related problems deserves further investigation. For instance, how
to apply the obtained results in stability analysis and synthesis of
practical systems with time-delay, such as Hamiltonian mechanical systems
and Lagrangian mechanical system; how to generalize this result to random
switched systems with time-delay.

\appendices

\section{The Proof of (\ref{equ571})}
\renewcommand{\theequation}{A.\arabic{equation}}\setcounter{equation}{0}
\begin{proof}
From \textbf{H2} and (\ref{equ51}), one can obtain the following inequality
\begin{align}\label{equA1}
&z_if_i-z_i\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}(x_{j+1}+f_j)\cr
\leq &|z_i|(\bar{\theta}\sum_{l=1}^i(|x_l|+|x_l(t-\tau(t))|)\cr
&+\sum_{j=1}^{i-1}\Xi_{j(i-1)}
(|x_{j+1}|+\bar{\theta}\sum_{l=1}^j(|x_l|+|x_l(t-\tau(t))|))\cr
\leq & |z_i|\eta_i(\sum_{l=1}^i(|x_l|+|x_l(t-\tau(t))|)
+\sum_{j=1}^{i-1}|x_{j+1}|\cr
&+\sum_{j=1}^{i-1}\sum_{l=1}^j(|x_l|+|x_l(t-\tau(t))|))\cr
=& |z_i|\eta_i(\sum_{l=2}^i(i+2-l)|x_l|\cr
&+\sum_{l=1}^{i}(i+1-l)|x_l(t-\tau(t))|+i|x_1|)\cr
\leq &|z_i|\bar{\eta}_i(|z_1|+\sum_{l=2}^i(|z_l|+\beta_{l-1}|z_{l-1}|)+|z_1(t-\tau(t))|\cr
&+\sum_{l=2}^i(|z_l(t-\tau(t))|+\beta_{l-1}|z_{l-1}(t-\tau(t))|))\cr
\leq &|z_i|\tilde{\eta}_i(\sum_{l=1}^{i-1}(|z_l|+|z_l(t-\tau(t))|)\cr
&+|z_i|\bar{\eta}_{i}(|z_i|+|z_i(t-\tau(t))|),
\end{align}
where
$\eta_i=\max_{1\leq j\leq i-1}\{\bar{\theta}, \Xi_{j(i-1)},\Xi_{j(i-1)}\bar{\theta}\}$,
$\bar{\eta}_{i}=\eta_ii$,
$\tilde{\eta}_{i}=\bar{\eta}_{i}\max_{1\leq l\leq i-1}\{1+\beta_l\}$,
$i=2, 3, \cdots, n$.

By Young's inequality, one gets
\begin{align}
\label{equA2}
|z_i|\tilde{\eta}_i\sum_{l=1}^{i-1}|z_l|\leq& \frac{(i-1)}{2}\tilde{\eta}_i^2 z_i^2
+\sum_{l=1}^{i-1}\frac{1}{2}z_l^2,\\
\label{equA3}
|z_i|\bar{\eta}_{i}|z_i(t-\tau(t))|
\leq&\frac{e^{-\tau}(1-\tau^*)}{2}z_i^2(t-\tau(t))\cr
&+\frac{e^{\tau}}{2(1-\tau^*)}\bar{\eta}_i^2 z_i^2, \\
\label{equA4}
|z_i|\tilde{\eta}_i\sum_{l=1}^{i-1}|z_l(t-\tau(t))|
\leq &\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i-1}z_l^2(t-\tau(t))\cr
&+\frac{(i-1)e^{\tau}}{2(1-\tau^*)}\tilde{\eta}_i^2 z_i^2.
\end{align}
Substituting (\ref{equA2})-(\ref{equA4}) to (\ref{equA1}) leads
\begin{align*}
&z_i(f_i-\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}(x_{j+1}+f_j))\cr
\leq &\sum_{l=1}^{i-1}z_l^2+\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i}z_l^2(t-\tau(t))
+\Omega_iz_i^2,
\end{align*}
where $\Omega_i=\frac{(i-1)}{2}\tilde{\eta}_i^2
+\frac{(i-1)}{2(1-\tau^*)}e^{\tau}\tilde{\eta}_i^2
+\bar{\eta}_i+\frac{1}{2(1-\tau^*)}e^{\tau}\bar{\eta}_i^2$
is a positive constant.
\end{proof}

\section{The Proof of (\ref{equ572})}
\renewcommand{\theequation}{B.\arabic{equation}}\setcounter{equation}{0}
\begin{proof}
It follows from \textbf{H2} and (\ref{equ51}) that
\begin{align}\label{equB1}
&z_ig_i\xi_i-z_i\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}g_j\xi_j\cr
\leq &\frac{1}{2}|z_i|(g_i^2+|\xi_i|^2)
+\frac{1}{2}|z_i|\sum_{j=1}^{i-1}\Xi_{j(i-1)}(g_j^2+|\xi_j|^2)\cr
\leq & |z_i|\zeta_i(\sum_{l=1}^i(|x_l|+|x_l(t-\tau(t))|)\cr
&+\sum_{j=1}^{i-1}\sum_{l=1}^j(|x_l|+|x_l(t-\tau(t))|))\cr
&+\frac{1}{2}|z_i|(|\xi_i|^2+\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2)\cr
=& |z_i|\zeta_i\sum_{l=1}^{i}(i-l+1)(|x_l|+|x_l(t-\tau(t))|)\cr
&+\frac{1}{2}|z_i|(|\xi_i|^2+\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2)\cr
\leq& |z_i|\bar{\zeta}_i\sum_{l=1}^{i}(|x_l|+|x_l(t-\tau(t))|)\cr
&+\frac{1}{2}|z_i|(|\xi_i|^2+\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2)\cr
\leq& |z_i|\bar{\zeta}_i(|z_1|+|z_1(t-\tau(t))|
+\sum_{l=2}^i(|z_l|+\beta_{l-1}|z_{l-1}|)\cr
&+\sum_{l=2}^i(|z_l(t-\tau(t))|+\beta_{l-1}|z_{l-1}(t-\tau(t))|))\cr
&+\frac{1}{2}|z_i|(|\xi_i|^2+\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2)\cr
\leq &|z_i|\tilde{\zeta}_i(\sum_{l=1}^{i-1}(|z_l|+|z_l(t-\tau(t))|)\cr
&+|z_i|\bar{\zeta}_{i}(|z_i|+|z_i(t-\tau(t))|)\cr
&+\frac{1}{2}|z_i|(|\xi_i|^2+\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2),
\end{align}
where
$\zeta_i=\max_{1\leq j\leq i-1}\{\frac{1}{2}\bar{\theta}, \frac{1}{2}\bar{\theta}\Xi_{j(i-1)}\}$,
$\bar{\zeta}_i=\zeta_ii$,
$\tilde{\zeta}_{i}=\bar{\zeta}_{i}\max_{1\leq l\leq i-1}\{1+\beta_l\}$,
$i=2, 3, \cdots, n$.

By Young's inequality, one obtains
\begin{align}
\label{equB2}
\frac{1}{2}|z_i||\xi_i|^2 \leq& \frac{1}{16\pi_i}z_i^2+\pi_i|\xi_i|^4,\\
\label{equB3}
|z_i|\tilde{\zeta}_i\sum_{l=1}^{i-1}|z_l|
\leq& \frac{(i-1)}{2}\tilde{\zeta}_i^2 z_i^2+\sum_{l=1}^{i-1}\frac{1}{2}z_l^2,\\
\label{equB4}
\frac{1}{2}|z_i|\sum_{j=1}^{i-1}\Xi_{j(i-1)}|\xi_j|^2
\leq& \sum_{j=1}^{i-1}\frac{1}{16\pi_{ij}}\Xi_{j(i-1)}^2z_i^2\cr
&+\sum_{j=1}^{i-1}\pi_{ij}|\xi_j|^4,\\
\label{equB5}
|z_i|\bar{\zeta}_{i}|z_i(t-\tau(t))|
\leq& \frac{e^{-\tau}(1-\tau^*)}{2}z_i^2(t-\tau(t))\cr
&+\frac{e^{\tau}}{2(1-\tau^*)}\bar{\zeta}_i^2 z_i^2, \\
\label{equB6}
|z_i|\tilde{\zeta}_i\sum_{l=1}^{i-1}|z_l(t-\tau(t))|
\leq&\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i-1}z_l^2(t-\tau(t))\cr
&+\frac{(i-1)e^{\tau}}{2(1-\tau^*)}\tilde{\zeta}_i^2 z_i^2.

\end{align}
Substituting (\ref{equB2})-(\ref{equB6}) to (\ref{equB1}) yields
\begin{align*}
&z_ig_i\xi_i-z_i\sum_{j=1}^{i-1}\frac{\partial\alpha_{i-1}}{\partial x_j}g_j\xi_j\cr
\leq &\sum_{l=1}^{i-1}z_l^2+\Lambda_iz_i^2+\pi_i|\xi_i|^4+\sum_{j=1}^{i-1}\pi_{ij}|\xi_j|^4\cr
&+\frac{e^{-\tau}(1-\tau^*)}{2}\sum_{l=1}^{i}z_l^2(t-\tau(t)),
\end{align*}
where $\Lambda_i=\frac{(i-1)}{2}\tilde{\zeta}_i^2
+\frac{(i-1)}{2(1-\tau^*)}e^{\tau}\tilde{\zeta}_i^2
+\bar{\zeta}_i+\frac{1}{2(1-\tau^*)}e^{\tau}\bar{\zeta}_i^2
+\frac{1}{16\pi_i}
+\sum_{j=1}^{i-1}\frac{1}{16\pi_{ij}}\Xi_{j(i-1)}^2$ is a positive constant.
\end{proof}

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi


TITLE: Determinant of matrix with coefficients in field of 4 elements
QUESTION [1 upvotes]: I want to calculate the determinant of the matrix
$M=\begin{pmatrix}
1 & 1 & 1\\
x & y & z\\
x^2 & y^2 & z^2
\end{pmatrix}$
where $x,y,z\in\mathbb{F}_4=\{0,1,a,b\}$ (the field of four elements). $x,y,z$ could be distinct elements of $\mathbb{F}_4$ or not.
All I know is that for any element $\mathbb{F}_4$, say $t$, then $t+t=0$, that $b=a^2$ and that $+$ and $\cdot$ are commutative in $\mathbb{F}_4$. Because $t+t=0$, then $t=-t$ for all $t\in\mathbb{F}_4$. I have found that:
$\det{M}=1\cdot(y\cdot z^2-y^2\cdot z)-1\cdot(x\cdot z^2-z\cdot x^2)+1\cdot(x\cdot y^2-y\cdot x^2)$
Using $t=-t$ and the fact that $1\cdot t=t$ for all $t\in\mathbb{F}_4$, then we have:
$\det{M}=yz^2+zy^2+xz^2+zx^2+xy^2+yx^2$
Is there a way of simplifying things further?

REPLY [2 votes]: Hint: The formula for the Vandermonde determinant gives
$$
\det(M)=(x+y)(x+z)(y+z)
$$
over any field of characteristic $2$. Now use the addition and multiplication table for $\mathbb{F}_4=\{0,1,a,b\}$ (if you google MSE, then you will find it here, or at other MSE-questions).


TITLE: Why are ring extensions only discussed in the context of $\mathbb{C}$?
QUESTION [0 upvotes]: I'm watching the great (imo) set of lectures on abstract algebra from the harvard extension school that's available on youtube.
Now this lecture is about extending a ring. The lecturer talk about this in the context of extending subrings of $\mathbb{C}$. However, I wonder (as did someone during the lecture but the prof didn't really answer the question) why this is only discussed in the context of subrings of $\mathbb{C}$. But couldn't the same constructions also be made in more general rings? Where a subring of an arbitrary ring is extended by just adjoining some elements from the large ring?
Thanks

REPLY [0 votes]: Never mind, just a little bit later he explains that it is indeed not limited to $\mathbb{C}$, I should have been a little bit more patient.


TITLE: What is the inverse of the function $f(x)=x^2-9$? $f(x)= \sqrt{x+9}$ or $f(x)= -\sqrt{x+9}$ or both?
QUESTION [1 upvotes]: What is the inverse of the function $f(x)=x^2-9$? $f(x)= \sqrt{x+9}$ or $f(x)= -\sqrt{x+9}$ or both?
I have to find the inverse of the function $f(x)=x^2-9$ for $x \in \mathbb{R}$.
I can see that the range of this function is $y \in [-9, ∞)$.
Inverse relation:

$f(x)= \sqrt{x+9}$ for  $x  \in [-9, ∞)$
$f(x)= -\sqrt{x+9}$ for  $x  \in [-9, ∞)$

So range becomes $y \in [0, ∞)$ for positive part inverse function and $y \in [0, -∞)$ for negative part inverse function. But since its a function It has to be either 1 to 1 or many to 1 mapped, hence it can't be both, so which part to keep as the answer and why?
Also I have read that range and domain of inverse functions are swapped, if so, we must include both parts of the inverse relation, but that would result in violation of the {(1 or many) to 1} mapping. I am a bit confused how to go about it. Thanks for the help.

REPLY [3 votes]: Whether you choose $f^{-1}(x)=\sqrt{x+9}$ or $f^{-1}(x)=-\sqrt{x+9}$ will depend on the restriction you put on the domain of $f(x)$. A restriction is necessary here because, as you correctly point out, $\pm\sqrt{x+9}$ is not a function. This is due to the fact that $f(x)$ is not injective ("one-to-one"), so its inverse will not pass the vertical line test (i.e. not be a function).
Let's say you restrict the domain of $f(x)$ to be $[0,\infty)$. This means the range of the function will be $[-9,\infty)$. The domain of $f(x)$ will be the range of $f^{-1}(x)$ and the range of $f(x)$ will be the domain of $f^{-1}(x)$, as you also correctly point out. Notice that the domain of $f(x)$ is nonnegative; this means the range of $f^{-1}(x)$ will be nonnegative, so $f^{-1}(x)=\sqrt{x+9}$ in that case.
In the case that you choose the domain restriction of $(-\infty,0]$, the range of the function will be $[-9,\infty)$. Notice that the domain of $f(x)$ is now nonpositive; this means the range of $f^{-1}(x)$ will be nonpositive, so $f^{-1}(x)=-\sqrt{x+9}$ in that case.


TITLE: 6 Circle in Equilateral Triangle
QUESTION [1 upvotes]: i have one doubt in this when there are 6 circles in equilateral triangle then how the angle is 30 degree

REPLY [0 votes]: Here's a more Geometric approach:
$OY$ is the radius of circle $O$ and is tangent to $\triangle ABC$.
From that, we have $\triangle YOC \cong\triangle XOC$ because $OY\cong XO$ and $CO\cong CO$. (HL congruency for right triangles)
Therefore, $\angle OCY\cong\angle OCX$. And since they add up to $60^{\circ}$, we have $$\angle OCY+\angle OCX=60\\2\angle OCY=60\\\angle OCY=\angle OCX=30^{\circ}$$


TITLE: Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals?
QUESTION [5 upvotes]: Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.
$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$$
where I assume that $x_0$ is a unique minimum of the function $f(x)$. These types of computations are omnipresent in physics, for example. I am interested if there are examples in the reverse direction, when finding the minimum of $f(x)$ directly is challenging, but computing something like
$$\frac{\int e^{-\lambda f(x)}x}{\int e^{-\lambda f(x)}}\approx x_0,\qquad \lambda\to\infty$$
is tractable and allows to locate the minimum. Perhaps generalization to many variables or functional integrals would be more appropriate.
I hope the question is meaningful despite the lack of precision and rigor. Any comments are welcome!

REPLY [0 votes]: This is a very particular instance of what's known as moment problem, or sometimes "inverse moment problem". One has an unknown measure $\mu$ in some space of measures, and one tries to reconstruct $\mu$ from knowlegde of $\mu_k:=\int f_k(x) d\mu(x)$, $k=0,1,\dots$, for a well-behaved sequence of functions $f_k$.
This readily generalises to several dimensions.


TITLE: What’s the definition indefinite integral?
QUESTION [1 upvotes]: I am very confused that sometimes an indefinite integral represents the anti derivative, and other times it represents the area under the graph I.e. $\int f(x) dx = \lim_{x\rightarrow \infty} \int_{-x}^x f(x) dx$. I wonder if the representation of anti derivative an abuse of notation?

REPLY [2 votes]: This is actually a good question and one that is awfully confusing without a proper background in analysis.
An antiderivative of $f(x)$ on a  set $X$ is a function $F(x)$ such that $F'(x)=f(x)$ for all $x\in X$. Note that antiderivatives are certainly not unique, which is why a $+C$ is added (derivative of constant is zero).
The first fundamental theorem of calculus says that if $f$ is integrable on $[a,x]$ and an antiderivative $F(x)$ exists for all $x\in[a,b]$, then $\int_a^x f=F(x)-F(a)$. We call $F(x)=F(a)+\int_a^x f(y)\,\mathrm{d}y$ an indefinite integral. Obviously, $F'(x)=f(x)$ by the second fundamental theorem if $f$ is continuous at $x$.
Where things get subtle is considering $F(x)$ in general when $f$ is integrable, but not necessarily continuous. It is a perfectly legitimate, continuous function. However, it's not necessarily an antiderivative of $f$ if $f$ is not continuous at $x$.
For example, consider $$f(x)=\begin{cases}1,&\quad x\in[0,1] \\
2,& \quad x\in(1,2]\end{cases}$$
There is no antiderivative function $F(x)$ on $[0,2]$. One might suspect that
$$
F(x)=\begin{cases}x,&\quad x\in[0,1] \\
2x,& \quad x\in(1,2]\end{cases}
$$
works. However, while each piece of $F(x)$ serves as an antiderivative on $[0,1]$ and $(1,2]$ respectively, this piecewise function is not an antiderivative on $[0,2]$, even though $F(x)=F(0)+\int_0^x f(y)\,\mathrm{d}y$ is continuous. Another way to see it is since $\int_0^2 f=1+2=3$, which is not equal to $F(2)-F(0)=4-0=4$, we would be contradicting the fundamental theorem to claim $F$ is an antiderivative.
Now consider when $f$ is continuous. Then it's always true that $F(x)=C+\int_a^x f(y)\,\mathrm{d}y$ serves the role of an antiderivative. The reason this notation is common is because this holds even if the antiderivative does not have an elementary form. For example, consider $f(x)=e^{x^2}$, which is continuous on any $[a,b]$. Then it's true that $F(x)=\int_a^x e^{y^2}\,\mathrm{d}y$ satisfies $F'(x)=f(x)$, but the notation is more instructive since an elementary form does not exist.
I hope some of this helps. The above also underscores the fact that all differentiable functions are continuous, and that a function can be differentiable without the derivative being continuous.


TITLE: Commas separating adjectives to describe mathematical objects.
QUESTION [3 upvotes]: Is there a convention which rules that in mathematics an object with some properties should be referred to as a
"property 1 property 2 property 3 ...object"
rather than a
"property 1, property 2, property 3, ...object"?
Specific example is an "oriented compact smooth manifold". By googling this question I found that it would appear the convention is without commas, but I could not find a source confirming that it actually was.
Can someone confirm this is the correct convention, and perhaps from which organization the convention comes from?
I hope this falls within the bounds of MathStack; although it is obviously not about mathematics per se, I'm guessing nearly everyone on this site has written something like what I'm asking about!

REPLY [5 votes]: This has nothing to do with mathematics or the mathematical objects in question.
Note that in this case compact, smooth and even oriented are all adjectives.
I forward you to the concepts cumulative adjective and coordinate adjective.
What's below was copied from this link.
This link from English SE should also be useful.

Coordinate Adjectives
These are adjectives that separately modify a noun. Their order can be scrambled and they can be joined by and. A comma is needed to separate each coordinate adjective. But if there is a the word and no comma is needed.
Example: While strolling in the woods, they found a strange, mysterious and frightened child.
Cumulative Adjectives
When adjectives pile up to jointly describe the noun and need to be arranged in a specific order, they are called cumulative adjectives. No commas are needed to separate the adjectives.
Example: He bought a wonderful old French car.
Cumulative adjectives generally follow a certain order of arrangement.
  $$
\begin{array}{c|c|c|c}
 1. & \text{Opinion} & \text{good, attractive, beautiful, delicious, }\ldots \\
\hline
 2. &\text{Size} & \text{large, small, enormous, }\ldots \\
 3. & \hline
\text{Age} & \text{old, new, modern, young, }\ldots \\
\hline
 4. & \text{Length or shape} & \text{long, short, square, round, }\ldots\\
\hline
 5. & \text{Color} & \text{red, blue, green, }\ldots\\
\hline
 6. & \text{Origin (nationality, religion)} & \text{American, French, Muslim, Christian, }\ldots \\
\hline
 7. &\text{Material} & \text{plastic, woolen, wooden, cotton, }\ldots \\
\hline
8. & \text{Purpose} & \text{electric (wire) , tennis (shirt), }\ldots
\end{array}$$


TITLE: Can every manifold be dominated by a parallelizable one?
QUESTION [19 upvotes]: A closed, oriented $d$-manifold $M$ is said to dominate another such manifold $N$ if there exists a map $M \to N$ of non-zero degree. (This notion should not be confused with the unrelated concept of homotopical domination that Wall's finiteness obstruction relates to.)
This relation turns $d$-manifolds into a poset (not quite; the relation is not antisymmetric even if we identify diffeomorphic manifolds, as there are examples of manifolds that dominate each other and are not homotopy equivalent; but this subtlety is rather irrelevant) with $S^d$ as its smallest element. It has been of interest to geometric topologists at least since the times of Hopf to study this poset. For a survey on this topic, see http://www.map.mpim-bonn.mpg.de/images/c/cf/Brouwer_degree.pdf.
My question is: can every manifold $N$ be dominated by another manifold $M$ that is (stably)  parallelizable, i.e., has trivial stable tangent bundle?

REPLY [10 votes]: I think this follows from the stable Hurewicz map $\pi_n^s(N) \to H_n(N;\mathbb{Z})$ being an isomorphism modulo torsion.  If we set $n = \dim(M)$ then some positive multiple of the fundamental class of $M$ is hit.  Under the Pontryagin-Thom isomorphism this gives a stably framed $M$ and a continuous $f: M \to N$ such that $f_*([M])$ is a positive multiple of $[N]$.


TITLE: continuity for TVS
QUESTION [2 upvotes]: I have some problem as following：
Let $X$ be the space of all sequences of scalars.
Define addition and scalar (= Real numbers) multiplication in the usual way. If $x_n$, $y_n\in X$, define
$\displaystyle d(x, y) = \sum_{n=1}^{\infty} 2^{-n}\frac{|x_n - y_n|}{1+|x_n - y_n|}$.
It's already known $d$ is a metric, but I have some trobule to show the continuity of addition and continuity of scalar multiplication (i.e. to show that $X$ is topological vector space (TVS)).
I'm trying to estimate by inequality (e.g. Triangle inequality), but there is no way. How can I do this gap？ Thanks.

REPLY [0 votes]: Showing that $d$ is a metric is not entirely trivial. But if, as you say, you already know that then proving addition is continuous is simple, from the triangle  inequality. Note first that it's clear that $d$ is translation invariant, meaning  that $d(x+z,y+z)=d(x,y)$. So $$\begin{align}
d(x+y,a+b)&=d((x-a)+(y-b),0)
\\&\le d((x-a)+(y-b),(y-b))+d(y-b,0)
\\&=d(x-a,0)+d(y-b,0)
\\&=d(x,a)+d(y,b).\end{align}$$So if $d(x-a)<\epsilon/2$ and $d(y-b)<\epsilon/2$ then $d(x+y,a+b)<\epsilon$.