\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 26, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/26\hfil Linear uncertain systems]
{Linear uncertain non-autonomous time-delay systems:
Stability and stabilizability  via Riccati equations}

\author[P. Niamsup, K Mukdasai, V. N. Phat \hfil EJDE-2008/26\hfilneg]
{Piyapong Niamsup, Kanit Mukdasai, Vu N. Phat}  


\address{Piyapong Niamsup \newline
Department of Mathematics, 
Chiangmai  University,
Chiangmai 50200, Thailand}
\email{scipnmsp@chiangmai.ac.th}

\address{Kanit Mukdasai \newline
 Department  of  Mathematics,
 Chiangmai   University,  
 Chiangmai  50200, Thailand}
\email{kanitmukdasai@hotmail.com}
 
\address{Vu Ngoc Phat \newline
 Institute of Mathematics \\
18 Hoang Quoc Viet Road, Hanoi, Vietnam}
\email{vnphat@math.ac.vn}

\thanks{Submitted September 28, 2007. Published February 22, 2008.}
\subjclass[2000]{93D15, 34D20, 49K20}
\keywords{Non-autonomous systems;  time-delay systems; uncertainties;
\hfill\break\indent
 exponential stability; stabilization; Lyapunov function; Riccati equation}

\begin{abstract}
 This paper addresses the problem of exponential stability for
 a class of uncertain linear non-autonomous time-delay
 systems. Here, the parameter uncertainties are
 time-varying and unknown but norm-bounded and the delays are
 time-varying.  Based on combination of the Riccati equation approach and
 the use of suitable Lyapunov-Krasovskii functional, new sufficient
 conditions for the robust stability   are obtained in terms
 of the solution of Riccati-type equations. The approach allows  to
 compute simultaneously the two bounds that characterize the
 exponential stability rate of the solution. As an application,
 sufficient conditions for the robust stabilization are  derived.
 Numerical examples illustrated the results are given.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}

\section {Introduction}

Since  time-delay systems are frequently encountered in various areas,
including physical and chemical processes,  biology, economics,
engineering etc.,  the  stability
problem of linear time-delay systems has attracted a lot of
attention in the past decades, e.g. see \cite{c1,g2,n1,p3}
and the  references therein. The main technique  using in
stability  investigation relies on the use of the Lyapunov
functional method \cite{y1}. The results concerning Lyapunov's direct
method for time-invariant  systems provide stability sufficient
conditions in terms of linear matrix inequalities (LMIs)
\cite{b1,k2}.
More recently, simple and systematic procedure for finding
exponential stability conditions using the Lyapunov-Krasovskii
functionals has been proposed in \cite{k1,m1} for autonomous systems
and in \cite{d1,p4} for non-autonomous systems. Among the usual
approaches to studying stabilization problem of autonomous
systems, the effective approach is to design linear feedback
control via  solving algebraic Riccati equations \cite{a1,g1}. However,
for the non-autonomous systems, the solution of Riccati
differential equation (RDE) is in general not bounded from above
and below such that it can not served as the candidate of the
Lyapunov function. Moreover, the approach used in the mentioned
above papers  can not be readily applied  to the non-autonomous
systems. Some sufficient conditions for global stabilizability for
non-autonomous periodical systems via controllability are given in
\cite{p2}. Among the results on stability analysis of uncertain linear
autonomous  systems, it is worth to mention the papers \cite{p1,y2},
where the uncertainties must verify some matching/bounded
conditions or must have a particular structure and intensive
computation is needed to test the robust stability. Applications
to linear time-delay control systems without uncertainties are
given in \cite{n1,p4,t1,w1}.

Motivated by the result on  exponential stability of
linear non-autonomous delay systems in \cite{p4}, we develop sufficient
conditions for the exponential stability of a class of uncertain
linear non-autonomous time-delay systems.  The parameter
uncertainties are time-varying and unknown but norm-bounded and
the delays are time-varying. Stability and stabilization
conditions are formulated in terms of the solution of Riccati-like
equations, which  allow to compute the decay rate as well as the
constant stability factor. These conditions generalize and improve
the LMI conditions obtained earlier for autonomous delay systems.


The paper is organized as follows. Section 2 presents notations,
definitions and some auxiliary propositions needed in the proof of
main results. In Section 3, based on the Lyapunov-Krasovskii
functional method,  sufficient conditions for the exponential
stability and stabilization are presented. Numerical examples
illustrated the obtained results  are  given in Section 4. The
paper ends with conclusions and cited references.


  \section{Preliminaries}

The following notation will be used in this paper:
 $\mathbb{R}^+$ denotes the set of all real non-negative numbers;
 $\mathbb{R}^n$ denotes the $n$-dimensional space with the
scalar product $\langle .,.\rangle$ and the vector norm $\|\cdot\|$;
 $M^{n\times r}$ denotes the space of all matrices
of $(n\times r)$-dimensions.

$A^T$ denotes the transpose of the vector/matrix $A$;
$A$ is symmetric if $A = A^T$;
 $I$ denotes the identity matrix;
 $\lambda(A)$ denotes the set of all eigenvalues of $A$;
 $\lambda_{\max}(A) = \max\{\mathop{\rm Re}\lambda:
 \lambda \in \lambda(A)\}$.

 $x_t := \{x(t+s): s\in [-h,0]\}, \|x_t\| = \sup_{s\in [-h,0]}\|x(t+s)\|$.
 $C([0, t], \mathbb{R}^n)$ denotes the set of all
 $\mathbb{R}^n$-valued continuous functions on $[0, t]$;
 $L_2([0,t],\mathbb{R}^m)$ denotes the set of all the $\mathbb{R}^m$-valued
 square integrable functions on $[0,t]$;

 Matrix $A$ is called semi-positive definite ($A \geq 0$) if
$\langle Ax,x\rangle \geq 0$, for all $x \in \mathbb{R}^n; A$
is positive definite ($ A > 0$) if $\langle Ax,x\rangle > 0$ for all
$x\neq 0$; $A > B $ means $A- B >0$.
$BM^+(0,\infty)$ denotes the set of all symmetric semi-positive definite
matrix functions  bounded on $[0,\infty)$;

In the sequel, sometimes for the sake of brevity, we will omit the
arguments of matrix-valued  functions, if it does not cause any
confusion.

Consider the  uncertain linear non-autonomous system with
time-varying  delay
\begin{equation} \label{e2.1}
\begin{gathered}
\begin{aligned}
\dot x(t) &=  [A_0(t) + \Delta A_0(t)]x(t) + [A_1(t)+\Delta A_1(t)]x(t-h(t)) \\
 &\quad + [B(t) + \Delta B(t)]u(t),
\end{aligned} \\
x(t) =  \phi (t), \quad  t \in [-h, 0],
\end{gathered}
\end{equation}
where $x(t)\in \mathbb{R}^n$, $u(t)\in \mathbb{R}^m, A_i(t)$,
$i=0, 1, B(t)$ are given matrix
functions continuous on $[0,\infty), 0\leq h(t) \leq h, h > 0$.
 Consider   the initial function $\phi(t)\in C([-h,0],\mathbb{R}^n)$
 with the norm $\|\phi\| =\sup_{t\in [-h,0]}\|\phi(t)\|$,
and the admissible control
$u(\cdot)\in L_2([0,t],\mathbb{R}^m)$, for all $t\in \mathbb{R}^+$.
The delay $h(t)$ is a continuously differentiable  function satisfying
$$
0\leq h(t) \leq h,\quad \dot h(t) \leq \delta < 1.
$$
 The uncertainties $\Delta A_0, \Delta A_1, \Delta B$ are time-varying and
 satisfy the condition:
\begin{gather*}
\Delta A_i(t) = G_i(t)F(t)H_i(t), \quad i =0,1,\\
\Delta B(t) = G_2(t)F(t)H_2(t), \\
 \|F(t)\| \leq 1, \quad \forall t\in \mathbb{R}^+,
\end{gather*}
where $G_i(t),H_i(t)$, $i=0,1,2$ are given
matrix functions of appropriate dimensions.
 \smallskip

\noindent \textbf{Definition} % \label{def2.1}
 The system \eqref{e2.1}, where $u(t) = 0$,  is robustly exponentially stable,
if there exist  numbers $\alpha >0$, $N>0$ such that every solution
$x(t,\phi)$ of the system satisfies
the inequality
$$
\|x(t,\phi)\| \leq N \|\phi\|e^{-\alpha (t-t_0)},\quad \forall t\geq t_0\geq 0,
$$
for all uncertainties $\Delta A_0, \Delta A_1$.

The system \eqref{e2.1} is robustly
stabilizable if there is a control $u(t) = K(t)x(t)$ such
that the closed-loop system
$$
\dot x(t) = [A_0(t) + (B(t) + \Delta B(t))K(t) + \Delta A_0(t)]x(t)
+ [A_1(t)+\Delta
A_1(t)]x(t-h(t))
$$
is robustly exponentially stable. The function $u(t) = K(t)x(t)$ is called
a feedback stabilizing control of the system.


\begin{proposition}[Completing the square] \label{prop2.1}
Assume that $S\in M^{n\times n}$ is a
symmetric positive definite matrix. Then for every $ Q\in M^{n\times n}$:
$$
2\langle Qy,x\rangle - \langle Sy,y\rangle
\leq \langle QS^{-1}Q^Tx,x\rangle, \quad \forall  x, y\in \mathbb{R}^n.
$$
\end{proposition}

\begin{proposition}[\cite{w1}] \label{prop2.2}
 Let $G, H, F$ be real
matrices of appropriate dimensions with $\|F\| < 1$. Then
\begin{itemize}
\item[(i)] For any $\epsilon > 0: GFH + H^TF^TG^T \leq
\frac{1}{\epsilon}GG^T + \epsilon H^TH$.

\item[(ii)] For any $\epsilon >0$ such that $\epsilon I - HH^T
> 0$,
$$
(A +GFH)(A+GFH)^T \leq  AA^T + AH^T(\epsilon I-  HH^T)^{-1}HA^T
+\epsilon GG^T.
$$
\end{itemize}
\end{proposition}

\begin{proposition}[Schur complement lemma \cite{b1}]  \label{prop2.3}
 Given constant symmetric matrices $X, Y, Z$ where $Y > 0$.
Then $X + Z^TY^{-1}Z < 0$ if and only if
$$
\begin{pmatrix} X& Z^T\\ Z& - Y\end{pmatrix} < 0\quad
\text{or}\quad \begin{pmatrix} -Y& Z\\ Z^T& X\end{pmatrix} < 0 .
$$
\end{proposition}

\begin{proposition}[\cite{g2}] \label{prop2.4}
 Consider  the time-delay system
 $$
\dot x(t) = f(t,x_t), x(t) =\phi(t), \quad t\in [-h,0].
$$
 If there exist
a Lyapunov function $V(t,x_t)$ and  $\lambda_1, \lambda _2>0$  such that
for every solution $x(t)$ of the system, the following conditions hold
\begin{itemize}
\item[(i)] $ \lambda_1\|x(t)\|^2 \leq V(t,x_t)
\leq \lambda_2\|x_t\|^2$,

\item[(ii)] $\dot V(t,x_t) \leq 0$,

\end{itemize}
then the solution of the system is bounded; i.e.,
there exists $N> 0$ such that
$\|x(t,\phi)\| \leq N\|\phi\|,  \forall t\geq 0$.
\end{proposition}

 \section{Main results}

Given  numbers $\epsilon >0$, $\epsilon_0 >0$, $\epsilon_1 >0$, $\alpha >0$,
$h >0$,  we set
\begin{gather*}
P_\epsilon(t) = P(t) + \epsilon I,\quad
A_{0,\alpha}(t) = A_0(t) +\alpha I, \\
Q(t) = \epsilon_{0}H_0^T(t)H_0(t) +   I,\quad
S(t) = \epsilon_1 I-H_1(t)H_1^T(t),\\
\begin{aligned}
R(t) &= \frac{e^{2\alpha h}}{1-\delta}[A_1(t)A_1^T(t)
+ \epsilon_1G_1(t)G_1^T(t) \\
&\quad + A_1(t)H_1^T(t)S^{-1}(t)H_1(t)A_1^T(t)]
 +\epsilon_0^{-1}G_0(t)G^T_0(t).
\end{aligned}
\end{gather*}
Consider the  Riccati differential equation
 \begin{equation} \label{e3.1}
 \dot P_\epsilon(t) + P_\epsilon(t)
A_{0,\alpha}(t) + A^T_{0,\alpha}(t)P_\epsilon(t) + P_\epsilon(t)
R(t)P_\epsilon(t) + Q(t) = 0.
\end{equation}

  \begin{theorem} \label{thm3.1}
The uncertain linear non-autonomous  system \eqref{e2.1}, where $u(t) =0$,
is robustly exponentially stable if there exist positive numbers
  $\alpha, \epsilon, \epsilon_0, \epsilon_1$, and a  matrix function
$P(t)\in BM^+(0,\infty)$  such that
$\epsilon_{1} I-H_1(t)H_1^T(t) >0$
and the RDE \eqref{e3.1} holds. Moreover, the solution $x(t,\phi)$
satisfies the inequality
$$
\|x(t,\phi)\| \leq N \|\phi\|e^{-\alpha t},\quad t\in \mathbb{R}^+,
$$
where
$$
N = \sqrt{\frac{\lambda_{\max}(P(0))}{\epsilon}
+ \frac{1}{2\alpha\epsilon}(1-e^{-2\alpha h})+1 }.
$$
\end{theorem}

\begin{proof}
Let $P_{\epsilon}(t) \in BM^+(0,\infty), t\in \mathbb{R}^+$, be a solution
of the RDE \eqref{e3.1}. We take
the change of the state variable
\begin{equation} \label{e*}
y(t) = e^{\alpha t}x(t),\quad t\in \mathbb{R}^+,
\end{equation}
then the linear delay  system \eqref{e2.1}, where $u(t) =0$,  is
transformed to the  delay system
\begin{equation} \label{e3.2}
\begin{gathered}
\dot y(t) =  [A_{0,\alpha}(t) +\Delta A_0(t)]y(t) + e^{\alpha
h(t)}[A_1(t) + \Delta A_1(t)]y(t-h(t)), \\
 y(t) =  e^{\alpha t}\phi(t),\quad t\in [-h,0],
 \end{gathered}
\end{equation}
Consider the following time-varying Lyapunov function for the
system \eqref{e3.2},
$$
V(t,y_t) = \langle P(t)y(t),y(t)\rangle + \epsilon\|y(t)\|^2+
\int_{t-h(t)}^t\|y(s)\|^2ds.
$$
It is easy to see that
\begin{equation} \label{e**}
\epsilon\|y(t)\|^2 \leq V(t,y_t) \leq (p + \epsilon +h)\|y_t\|^2,
\end{equation}
where $p=\max_{t\geq 0}|P(t)|$  which is a finite number because $P(t)\in BM^+(0,\infty)$  and hence $P(t)$ is a bounded function. 
Taking  the derivative of $V(\cdot)$ in $t$ along the solution of
 $y(t)$ of system \eqref{e3.2}, we have
\begin{align*}
\dot V(t,y_t)
&=  \langle \dot P(t)y(t),y(t)\rangle + 2 \langle
P_\epsilon(t)\dot y(t), y(t)\rangle + \|y(t)\|^2 - (1 -\dot h(t))\|y(t-h(t))\|^2\\
&=  \langle \dot P(t)y(t),y(t)\rangle + 2\langle
P_\epsilon(t)A_{0,\alpha}(t)y(t),y(t)\rangle + 2\langle
P_\epsilon(t)G_0F(t)H_0 y(t),y(t)\rangle \\
& \quad + \|y(t)\|^2 - (1 -\dot h(t))\|y(t-h(t))\|^2\\
& \quad + 2e^{\alpha h(t)}\langle P_\epsilon(t)[A_1(t) + G_1F(t)H_1]y(t-h),
y(t)\rangle \\
&\leq \langle \dot P(t)y(t),y(t)\rangle + 2\langle
P_\epsilon(t)A_{0,\alpha}(t)y(t),y(t)\rangle \\
& \quad + 2\langle P_\epsilon(t)G_0F(t)H_0 y(t),y(t)\rangle + \|y(t)\|^2 - (1 -\delta
)\|y(t-h(t))\|^2\\
&\quad + 2\langle e^{\alpha h(t)}P_\epsilon(t)[A_1(t)
 + G_1F(t)H_1]y(t-h(t)), y(t)\rangle .
\end{align*}
 From Proposition \ref{prop2.1} it follows that
\begin{align*}
&2\langle e^{\alpha h(t)}P_\epsilon(t)[A_1(t) + G_1(t)F(t)H_1(t)]y(t-h(t)),
y(t)\rangle - (1-\delta) \|y(t-h(t))\|^2 \\
&\leq \frac{e^{2\alpha h(t)}}{1-\delta}\langle
P_\epsilon(t)[A_1(t) + G_1(t)F(t)H_1(t)] [A_1(t) +
G_1(t)F(t)H_1(t)]^T P_\epsilon (t) y(t),y(t)\rangle.
\end{align*}
Using  Proposition \ref{prop2.2}, for  numbers $\epsilon_0, \epsilon_1$  such that
\begin{gather*}
2 \langle P_\epsilon G_0FH_0 y,y\rangle \leq
\frac{1}{\epsilon_0}\langle P_\epsilon G_0G_0^TP_\epsilon y,y\rangle +
\epsilon_0\langle H_0^TH_0y,y\rangle,\\
[A_1 + G_1FH_1][A_1 + G_1FH_1]^T \leq A_1A^T_1 + A_1H_1^T(\epsilon_1 I-
H_1H_1^T)^{-1}H_1A_1^T
 + \epsilon_1 G_1G_1^T,
\end{gather*}
 provided $\epsilon_{1} I- H_1H_1^T >0$.
 Furthermore, note that
 $e^{2\alpha h(t)} \leq e^{2\alpha h}, \forall t\in \mathbb{R}^+$, we obtain
\begin{align*}
&\dot V(t,y(t)) \\
&\leq  \langle \dot P(t)y(t),y(t)\rangle + 2\langle
P_\epsilon(t)A_{0,\alpha}(t)y(t),y(t)\rangle
\\
&\quad + \frac{1}{\epsilon_0}\langle P_\epsilon G_0G_0^TP_\epsilon y,y\rangle
+\epsilon_0\langle
H_0^TH_0y(t),y(t)\rangle + \|y(t)\|^2 \\
&\quad + \frac{e^{2\alpha h}}{1-\delta}\langle \{P_\epsilon [A_1A^T_1 +
A_1H_1^T(\epsilon_{1} I- H_1H_1^T)^{-1}H_1A_1^T
 + \epsilon_1G_1^TG_1]P_\epsilon\}y(t),y(t)\rangle.
\end{align*}
 Therefore,
 \begin{align*}
  \dot V(t,y(t))
 &\leq \langle \{\dot P_\epsilon + P_\epsilon
A_{0,\alpha} + A^T_{0,\alpha}P_\epsilon +
\epsilon_0^{-1}P_\epsilon G_0G^T_0P_\epsilon +\epsilon_1H_0^TH_0
+I\\
&\quad + \frac{e^{2\alpha h}}{1-\delta}[P_\epsilon
A_1A_1^TP_\epsilon + P_\epsilon A_ 1H_1^T(\epsilon_1 I-
H_1H_1^T)^{-1}H_1A_1^TP_\epsilon \\
&\quad + \epsilon_1P_\epsilon G_1^TG_1P_\epsilon]\}y(t),y(t)\rangle .
\end{align*}
Since $P(t)$ is the solution of  \eqref{e3.1}, we obtain
\begin{equation} \label{e3.3}
\dot V(t,y(t)) \leq 0, \quad \forall t\in \mathbb{R}^+.
\end{equation}
Thus,  from  (3.4), (3.5) and Proposition \ref{prop2.4} it follows
 the boundedness of the solution $y(t,\phi)$ for the system \eqref{e3.2};
  i.e., there exists $ N >0$ such that
$$
\|y(t,\phi)\| \leq N\|\phi\|,\quad \forall t\geq 0.
$$
Returning to the solution $x(t,\phi)$ of the system \eqref{e2.1}
by the transformation \eqref{e*}, we obtain
$$
\|x(t,\phi)\| \leq N\|\phi\|e^{-\alpha t},\quad \forall t\geq 0,
$$
which  gives the exponential stability of  \eqref{e2.1}.
To determine the stability factor $N$,  we integrate both sides
 of \eqref{e3.3} from $0$ to $t$ we find
$$
V(t,y(t)) - V(0,y(0)) \leq 0,\quad \forall t\in \mathbb{R}^+,
$$
and hence
\begin{align*}
&\langle P(t)y(t),y(t)\rangle  + \epsilon \|y(t)\|^2
+ \int_{t-h(t)}^t\|y(s)\|^2ds\\
&\leq \langle P_0y(0),y(0)\rangle + \epsilon \|y(0)\|^2 + \int_{
-h(0)}^0\|y(s)\|^2ds.
\end{align*}
Since
$$
\langle P(t)y,y\rangle \geq 0,\quad
\int_{t-h(t)}^t\|y(s)\|^2ds \geq  0,
$$
and
$$
 \int_{-h(0)}^0\|y(s)\|^2ds \leq \|\phi\|^2\int_{-h(0)}^0e^{2\alpha s}ds =
\frac{1}{\alpha}(1-e^{-2\alpha h(0)})\|\phi\|^2 \leq
\frac{1}{2\alpha}(1-e^{-2\alpha h})\|\phi\|^2,
$$
we have
$$
\epsilon \|y(t)\|^2 \leq \lambda_{\max}(P(0))\|y(0)\|^2 + \epsilon \|y(0)\|^2
+ \frac{1}{2\alpha}(1-e^{-2\alpha h})\|\phi\|^2.
$$
Returning to the solution $x(t,\phi)$ of system \eqref{e2.1} and noting
that
$$
\|y(0)\| = \|x(0)\| = \phi(0) \leq \|\phi\|,
$$
we have
$$
\|x(t,\phi)\| \leq  N\|\phi\|e^{-\alpha t},\quad \forall t\in
\mathbb{R}^+,
$$
where
$$
N =  \sqrt{\frac{\lambda_{\max}(P(0))}{\epsilon} +
\frac{1}{2\alpha\epsilon}(1-e^{-2\alpha h})+1 }.
$$
The proof of the theorem is complete.
\end{proof}

\begin{remark} \label{rmk3.1} \rm
When the system is time-invariant, using the Schur
complement lemma (Proposition \ref{prop2.3}),
the  RDE \eqref{e3.1} can be rewritten
in terms of the  LMI:
$$
\begin{pmatrix}  X(P) & P_\epsilon A_{1,\alpha}&
P_\epsilon G_1 & P_\epsilon G_0& P_\epsilon A_{1,\alpha}H_1^T
\\A^T_{1,\alpha}P_\epsilon & -\eta e^{-2\alpha h}I&0&0&0\\G_1^TP_\epsilon &0&-\eta\epsilon_1^{-1}e^{-2\alpha h}I&0&0
\\G_0^TP_\epsilon  &&0& -\epsilon_0I&0\\
H_1A^T_{1,\alpha}P_\epsilon &0&0&0&-\eta e^{-2\alpha h}S
\end{pmatrix} < 0,
$$
where $\eta = 1 -\delta $ and
$$
X(P) := P_\epsilon A_{0,\alpha} + A^T_{0,\alpha}P_\epsilon +
\epsilon_{0}H_0^TH_0.
$$
\end{remark}

We now apply the stability result to  global stabilization of the
uncertain linear  time-delay control  system \eqref{e2.1}.
For this, given  numbers
$\epsilon >0$, $\epsilon_0 >0$, $\epsilon_1 >0$,  $\alpha >0$, $h >0$,
we set
\begin{gather*}
P_\epsilon(t) = P(t) + \epsilon I,\quad
A_{0,\alpha}(t) = A_0(t) +\alpha I, \\
Q(t) = \epsilon_{0}H_0^T(t)H_0(t) + I,\quad
S(t) = \epsilon_1 I- H_1(t)H_1^T(t),\\
\begin{aligned}
R_1(t)  &= \frac{e^{2\alpha h}}{1-\delta}[A_1(t)A_1^T(t)
  + \epsilon_1G_1(t)G_1^T(t) +
A_1(t)H_1^T(t)S^{-1}(t)H_1(t)A_1^T(t)] \\
&\quad -B(t)B^T(t)+ \frac{1}{4}\epsilon_0B(t)H_2^T(t)H_2(t)B^T(t)\\
&\quad + \epsilon_0^{-1}[G_0(t)G_0^T(t)+G_2(t)G_2^T(t)].
\end{aligned}
\end{gather*}
Consider the  Riccati differential equation
 \begin{equation} \label{e3.4}
 \dot P_\epsilon(t) + P_\epsilon(t)
A_{0,\alpha}(t) + A^T_{0,\alpha}P_\epsilon(t) + P_\epsilon(t)
R_1(t)P_\epsilon(t) + Q(t) =0.
\end{equation}

 \begin{theorem} \label{thm3.2}
Uncertain linear non-autonomous  control delay system \eqref{e2.1}
is robustly stabilizable if there exist positive numbers $\alpha,
\epsilon, \epsilon_0, \epsilon_1$, and a matrix function $P(t)\in
BM^+(0,\infty)$  such that $\epsilon_{1} I-H_1(t)H_1^T(t) >0$ and
the RDE \eqref{e3.4} holds. Moreover,  the feedback stabilizing
control is given  by
  \begin{equation}
u(t) = -\frac{1}{2}B^T(t)P(t)x(t).
\end{equation}
\end{theorem}

\begin{proof}
Let us define
\begin{gather*}
\bar A_0(t) = A(t) + B(t)K(t),\quad
\bar G_0(t) = [G_0(t)\,\, G_2(t)],\quad
\bar H_0(t) = \begin{bmatrix} H_0(t)\\
H_2(t)K(t)\end{bmatrix},\\
\bar F(t) = \begin{pmatrix} F(t) &0\\0&F(t) \end{pmatrix},\quad
\Delta\bar A_0(t) = \bar G_0(t)\bar F(t)\bar H_0(t),
\end{gather*}
where $K(t):= - \frac{1}{2}B^T(t)P(t)$.  With the feedback 
control (3.7),  the closed-loop system of
the system \eqref{e2.1} is
\begin{equation} \label{e3.6}
\dot x(t) = [\bar A_0(t) + \Delta \bar A_0(t)]x(t) +
[A_1(t)+\Delta A_1(t)]x(t-h(t)).
\end{equation}
Therefore, the proof of the theorem is completed by using Theorem
\ref{thm3.1} for the uncertain unforced system \eqref{e3.6} with the
following transformations
\begin{gather*}
\bar G_0(t)\bar G_0^T(t) = G_0(t)G^T_0(t) + G_2(t)G^T_2(t),\\
\bar H_0^T(t)\bar H_0(t) =  H_0^T(t)H_0(t) +
\frac{1}{4}P(t)B(t)H_2^T(t)H_2(t)B^T(t)P(t),\\
\bar A_{0,\alpha}(t)= A_{0,\alpha}(t) -\frac{1}{2}B(t)B^T(t)P(t),\\
P_\epsilon(t) \bar A_{0,\alpha}(t) + \bar A^T_{0,\alpha}(t)P_\epsilon(t)
= P_\epsilon(t) A_{0,\alpha}(t) + A^T_{0,\alpha}(t)P_\epsilon(t)
- P(t)B(t)B^T(t)P(t).
\end{gather*}
\end{proof}

\begin{remark} \label{rmk3.3} \rm
As in Remark \ref{rmk3.1}, for the time-invariant systems the
Riccati equation \eqref{e3.4} can be replaced  by the  LMI:
$$
\begin{pmatrix}  X(P) & P_\epsilon A_{1,\alpha}&
P_\epsilon G_1 & P_\epsilon G_0& P_\epsilon A_{1,\alpha}H_1^T &
P_\epsilon B &P_\epsilon G_2 &P_\epsilon BH_2^T
\\A^T_{1,\alpha}P_\epsilon & -\eta e^{-2\alpha h}I&0&0&0&0&0&0\\G_1^TP_\epsilon &0&-\eta \epsilon_1^{-1}e^{-2\alpha
h}I&0&0&0&0&0
\\G_0^TP_\epsilon  &&0& -\epsilon_0I&0&0&0&0\\
H_1A^T_{1,\alpha}P_\epsilon &0&0&0&-\eta e^{-2\alpha
h}S&0&0&0\\B^TP_\epsilon &0&0&0&&I&0\\G_2^TP_\epsilon
&0&0&0&0&0&-\epsilon_0I&0\\H_2B^TP_\epsilon
&0&0&0&0&0&0&-4\epsilon_0^{-1}
\end{pmatrix} < 0,
$$
where $\eta = 1-\delta$, and
$$
X(P) := P_\epsilon A_{0,\alpha} + A^T_{0,\alpha}P_\epsilon +
\epsilon_{0}H_0^TH_0.
$$
\end{remark}

The stability conditions are given in terms of the solution
of some RDEs. Although the problem of solving RDEs
is in general still not easy, various effective approaches for
finding the solutions of RDEs  can be found in \cite{d2,l1,w2}.

\section{Examples}

\begin{example} \label{exa4.1}\rm
 Consider the uncertain linear non-autonomous
unforced system  with  time-varying  delay  \eqref{e2.1},
where $u(t) =0$,   with any
initial function $\phi(t)$ and time-delay function $h(t)=3 \sin^2(2/3)t$
and
\begin{gather*}
A_{0}(t)=\begin{bmatrix}
  -\frac{1}{2} & -1 \\
  0 & e^{-2t}-\frac{3}{4} \end{bmatrix},\quad
A_{1}(t)=\begin{bmatrix}
  -\frac{\sqrt{2-e^{-2t}}}{e^3 (e^{-2t}+1)} & 0 \\
  0 & -\frac{\sqrt{2-e^{-2t}}}{2e^3} \end{bmatrix}, \\
G_{0}(t)=\begin{bmatrix}
  \frac{e^{t}}{e^{-2t}+1} & 0 \\
  0 & \frac{e^{-t}}{\sqrt{2}} \end{bmatrix},\quad
H_{0}(t)=\begin{bmatrix}
  e^{-t} &  e^{-t} \\
   e^{t} &  e^{-t} \end{bmatrix},\\
G_{1}(t)=\begin{bmatrix}
  \frac{e^{t}}{e^{3}(e^{-2t}+1)} & 0 \\
  0 & \frac{e^{-t}}{e^3} \end{bmatrix},\quad
H_{1}(t)=\begin{bmatrix}
  e^{-t} &  0 \\
   0 &  e^{-t} \end{bmatrix}.
\end{gather*}
We see that  $h=3$, and $\dot h(t) = 2\sin(4/3t)$ and then $\delta=2$.
Taking  $\alpha=\epsilon=\epsilon_{0} = 1$ and $\epsilon_{1}=2$,  we
  have
$$
\epsilon_{1}I- H_{1}(t)H^{T}_{1}(t)=\begin{bmatrix}
    2-e^{-2t} & 0 \\
    0 & 2-e^{-2t}   \end{bmatrix}>0.
$$
We can verify that the   matrix
$P(t)=\begin{bmatrix}
    e^{-2t} & 0 \\
    0 & 1 \end{bmatrix}$
is a solution  of  \eqref{e3.1}. Therefore, by
Theorem \ref{thm3.1} the system is robustly exponentially stable and the solution
satisfies
$$
\|x(t,\phi)\|\leq (3-e^{-3})\|\phi\|e^{-t},\quad t\in \mathbb{R}^{+}.
$$
\end{example}

\begin{example} \label{exa4.2}\rm
 Consider the uncertain linear non-autonomous
control system  with  time-varying  delay  \eqref{e2.1} with any
initial  function $\phi(t)$ and time-delay function $h(t)=2 \sin^2t$ and
\begin{gather*}
A_{0}(t)=\begin{bmatrix}
  -e^{-2t}+1 & 0 \\
  -1 & -e^{-2t}+1 \end{bmatrix},\quad
A_{1}(t)=\begin{bmatrix}
  -\frac{\sqrt{2-e^{-2t}}}{e^2 (e^{-2t}+1)} & 0 \\
  0 & -\frac{\sqrt{2-e^{-2t}}}{e^2 (e^{-2t}+1)} \end{bmatrix}, \\
B(t)=\begin{bmatrix}
  -\frac{1}{e^{-2t}+1} & 0 \\
  0 & -\frac{1}{e^{-2t}+1} \end{bmatrix},\quad
G_{0}(t)=\begin{bmatrix}
  \frac{e^{t}}{e^{-2t}+1} & 0 \\
  0 & \frac{e^{-t}}{\sqrt{2}} \end{bmatrix},\quad
H_{0}(t)=\begin{bmatrix}
  e^{-t} &  e^{t} \\
   e^{-t} &  e^{-t} \end{bmatrix},\\
G_{1}(t)=\begin{bmatrix}
  \frac{e^{-2t}}{e^{2}(e^{-2t}+1)} & 0 \\
  0 & \frac{e^{t}}{\sqrt{2}e^2(e^{-2t}+1)} \end{bmatrix},\quad
H_{1}(t)=\begin{bmatrix}
  e^{-t} &  0 \\
   0 &  e^{-t} \end{bmatrix},\\
G_{2}(t)=\begin{bmatrix}
  0 & 0 \\
  0 & \frac{e^{-2t}}{(e^{-2t}+1)} \end{bmatrix},\quad
H_{2}(t)=\begin{bmatrix}
  4e^{-2t} &  0 \\
   0 &  2e^{-t} \end{bmatrix}.
\end{gather*}
  We see that  $h=2,\delta=2$. Taking
  $\alpha=\epsilon=\epsilon_{0}=1$ and $\epsilon_{1}=2$,   we
  have
$$\epsilon_{1}I- H_{1}(t)H^{T}_{1}(t)=\begin{bmatrix}
    2-e^{-2t} & 0 \\
    0 & 2-e^{-2t}
  \end{bmatrix}>0.
$$
We can verify that the   matrix $P(t)=\begin{bmatrix}
    e^{-2t} & 0 \\
    0 & e^{-2t} \end{bmatrix}$ is a solution  of the RDE \eqref{e3.4}.
Therefore, by   Theorem \ref{thm3.2} the system is robustly stabilizable
and the feedback stabilizing    control  is given by
$$
u(t)=\begin{bmatrix}
     \frac{e^{-2t}}{2(e^{-2t}+1)} & 0\\
     0 & \frac{e^{-2t}}{2(e^{-2t}+1)} \\
   \end{bmatrix}x(t),\quad t\geq 0.
$$
\end{example}

\subsection*{Conclusions}
Based on combination of the Riccati equation approach and
the use of suitable Lyapunov-Krasovskii functional, sufficient
conditions for the exponential stability and stabilizability
of linear non-autonomous delay systems with time-varying and
norm-bounded uncertainties have been
established.  The conditions are formulated in terms of the solution of
curtain Riccati differential equations, which  allow to compute the
decay rate as well as the constant stability factor.

\subsection*{Acknowledgements}
This research  was supported by the Thai Research Fund
and the Basic Program in Natural Science, Vietnam.
The authors would like to thank anonymous referee for valuable
comments and suggestions to improve this paper.

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\end{document}