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\markboth{\hfil Exponential stability \hfil EJDE--2001/34}
{EJDE--2001/34\hfil N. M. Linh \& V. N. Phat \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 34, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
 Exponential stability of nonlinear time-varying differential equations
and applications
 %
\thanks{ {\em Mathematics Subject Classifications:} 34K20, 34D10, 49M37.
\hfil\break\indent
{\em Key words:} Exponential stability, time-varying equations,
    Lyapunov function, \hfil\break\indent
    Lipschitz condition,  control systems.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted September 20, 2000. Published May 14, 2001.} }
\date{}
%
\author{N. M. Linh \& V. N. Phat}
\maketitle

\begin{abstract}
 In this paper, we give sufficient conditions for the exponential stability
 of a class of nonlinear time-varying differential equations.
 We use the Lyapunov method with functions that are not necessarily
 differentiable; hence we extend previous results.
 We also provide an application to exponential stability
 for nonlinear time-varying control systems.
\end{abstract}


\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}{Proposition}[section]

\section{Introduction}

The investigation of stability analysis of nonlinear systems using
the second Lyapunov function method has produced a vast body of
important results and been widely studied \cite{b2,b3,k1,l1,z1}.
This is due to the theoretical interests in powerful
tools for system analysis and control design.  It is
recognized that the Lyapunov function method serves as a main
technique to reduce a given complicated system into a relatively
simpler system and provides useful applications to control theory
\cite{c1,l2,n1,p1,z1}. There have been a number of interesting
developments in searching the stability criteria for nonlinear
differential systems, but most have been restricted to finding the
asymptotic stability conditions  \cite{a1,b1,h1,p2}. Unlike the linear
systems, where the asymptotic stability implies the exponential
stability, the exponential stability for nonlinear differential
systems, in general, may not be easily verified. Only a few
investigations have dealt with  exponential stability
conditions for nonlinear time-varying systems \cite{s2,s3}. Moreover,
the problem of Lyapunov characterization of exponential stability
of nonlinear time-varying differential equations with non-smooth
Lyapunov functions has remained open.


The purpose of this paper is to establish sufficient conditions for
the exponential stability of a class of nonlinear time-varying systems.
In the spirit of a result of \cite{s3}, we develop the exponential stability
with more general assumptions on the Lyapunov function $V(t,x)$
in two aspects:
\\
(i) Proposing a class of Lyapunov-like functions, we prove new sufficient
conditions for the exponential stability of nonlinear time-varying systems with
more general comparable conditions.
\\
(ii) The results are extended to the systems with non-smooth Lyapunov functions,
 which need not be differentiable in $t$ and in $x$, and then the stability
results are applied to some stabilization problems of  nonlinear
time-varying control systems.

The paper is organized as follows. In Section 2, we introduce  notation,
definitions, and other preliminaries. Section 3 gives new sufficient
conditions for the exponential stability with the extended Lyapunov-like
functions. An application to exponential stability of a class of
nonlinear time-varying control systems is given in Section 4.


\section{Preliminaries}

The following  notation will be used  this paper:
$\mathbb{R}^n$ is the $n$-dimensional Euclidean vector space;
$\mathbb{R}^+$ is the set of all non-negative real numbers;
$\|x\|$ is the Euclidean norm of a vector $x \in \mathbb{R}^n$.


Consider the nonlinear system described by the
time-varying differential equations
\begin{equation} \begin{gathered}
 \dot x(t) = f(t, x(t)), \quad t \geq 0,\\
 x(t_0) = x_0,\quad t_0 \geq 0
\end{gathered}\end{equation}
 where $x(t)\in \mathbb{R}^n$, $f(t, x) : \mathbb{R}^+\times \mathbb{R}^n \to \mathbb{R}^n$ is a given
nonlinear function satisfying $f(t, 0) = 0$ for all $t \in \mathbb{R}^+$.
We shall assume that conditions are imposed on system (1) such that the
existence of its solutions is guaranteed.

\paragraph{Definition 2.1} The zero solution of system (1) is  exponentially
stable if any solution $x(t, x_0)$ of (1) satisfies
$$\|x(t,x_0)\| \leq \beta(\|x_0\|,t_0)e^{-\delta (t-t_0)},\quad
\forall t \geq t_0,$$
where  $\beta(h, t):\mathbb{R}^+\times \mathbb{R}^+\to \mathbb{R}^+$ is a non-negative function
increasing in $h\in \mathbb{R}^+$, and  $\delta$ is a positive constant.

If the function $\beta(.)$ in the above definition does not depend on $t_0$,
the zero solution is called uniformly exponentially stable. From now on, to shorten expressions, instead of saying the zero solution
is stable, we say that the system is stable.

Associated with system (1) we consider a nonlinear time-varying control
system
\begin{equation}
\dot x(t) = f(t,x(t), u(t)), \quad t \geq 0,
\end{equation}
where $x\in \mathbb{R}^n$, $u\in \mathbb{R}^m$, $f(t, x, u): \mathbb{R}^+\times \mathbb{R}^n\times \mathbb{R}^m \to \mathbb{R}^n$.

\paragraph{Definition 2.2} Control system (2) is exponentially stabilizable
by the feedback control $u(t) = h(x(t))$, where $h(x) : \mathbb{R}^n \to \mathbb{R}^m$, if the
closed-loop system
$$\dot x(t) = f(t, x(t), h(x(t)))$$
is exponentially stable.

Let $D\subset \mathbb{R}^n$ be an open set containing the origin,  and let
$V(t, x): \mathbb{R}^+\times D \to R$ be a given function.
Then we define $W = \mathbb{R}^+\times D$ and
$$D^+_fV(t,x) = \limsup_{h\to 0^+}\frac{V(t+h, x+hf) - V(t,x)}{h},$$
where $f(.)$ is the right-hand side function of (1). $D^+_fV$ is called
the upper Dini derivative of $V(.)$ along the trajectory of (1). Let $x(t)$ be a solution of (1) and denote by $d^{+}V(t,x)$ the upper right-hand derivative of $V(t,x(t))$, i.e.
$$d^{+}V(t, x(t)) = \limsup_{h\to 0^+}\frac{V(t+h, x(t+h)) - V(t,x(t))}{h}.$$

\paragraph{Definition 2.3} A function $V(t,x): \mathbb{R}^+\times \mathbb{R}^n \to R$ is
Lipschitzian in $x$ (uniformly in $t\in \mathbb{R}^+$) if there is a number $L > 0$
such that for all $t \in \mathbb{R}^+$,
$$|V(t,x_1) - V(t, x_2)| \leq L\|x_1 - x_2\|,\quad \forall (x_1,x_2)
\in \mathbb{R}^n\times \mathbb{R}^n.$$

In the sequel we assume that $V(t,x)$ is continuous in $t$ and Lipschitzian in
$x$ (uniformly in $t$) with the Lipschitz constant $L > 0$.
In which case, $d^+V$ and $D^+_fV$ related as follows:
\begin{eqnarray*}
\lefteqn{V(t+h,x(t+h)) - V(t, x(t)) }\\
&=& V(t+h,x(t+h)) - V(t+h,x+hf(t,x)) \\
&&+ V(t+h,x+hf(t,x)) - V(t, x(t)).
\end{eqnarray*}
Also
\begin{eqnarray*}
\lefteqn{\limsup_{h\to 0^+}\frac{V(t+h, x+hf(t,x))) - V(t,x(t))}{h} }\\
&\leq& \limsup_{h\to 0^+}\frac{V(t+h, x(t+h)) - V(t,x(t))}{h} \\
&&+ L \{\lim_{h\to 0^+}\frac{\|x(t+h) - x(t)\|}{h} - f(t, x(t))\},
\end{eqnarray*}
which gives
\begin{equation}
d^+V(t,x) \leq \limsup_{h\to 0^+}\frac{V(t+h, x+hf)) - V(t,x)}{h}
= D^+_fV(t,x).
\end{equation}
As shown in \cite{m1},  if $D^+_fV(t,x) \leq 0$ and consequently, by (3),
$d^+V(t,x) \leq 0$, the function $V(t, x(t))$ is a non-increasing function of
$t$, which means that $V(t,x)$ is non-increasing along a solution
of (1). To study exponential stability of (2) we need the following
 comparison theorem presented in \cite{l1,y1}.
 Consider a scalar differential equation
\begin{equation}
\dot u(t) = g(t,u), \quad t \geq 0,
\end{equation}
where $g(t,u)$ is continuous in $(t,u).$

\begin{proposition} % Proposition 2.1
 Let $u(t)$ be the maximal solution of (4) with $u(t_0) = u_0$.
If a continuous function $v(t)$ with $v(t_0) = u_0$ satisfies
$$d^+v(t) \leq g(t, u(t)), \quad \forall t \geq t_0,$$
then
$$v(t) - v(t_0) \leq \int_{t_0}^tg(s,u(s))ds, \quad \forall t\geq t_0.$$
\end{proposition}

Let us set
$$D_fV(t,x) = \frac{dV(t,x)}{dt} + \frac{dV(t,x)}{dx}f(t,x).$$

\paragraph{Definition 2.4}
A function $V(t,x): W\to R$ is called a  Lyapunov-like function
for (1) if $V(t, x)$ is continuously differentiable in $t\in \mathbb{R}^+$ and
in $x\in D$, and there exist positive  numbers
$\lambda_1, \lambda_2, \lambda_3, K, p, q, r, \delta$ such that
\begin{gather}
\lambda_1\|x\|^p \leq V(t, x) \leq \lambda_2\|x\|^q,
\quad \forall (t, x) \in W\,,\\
D_fV(t, x) \leq - \lambda_3\|x\|^r + Ke^{-\delta t},
\quad \forall t\geq 0, \, x\in D\setminus \{0\}
\end{gather}

\paragraph{Definition 2.5} A function $V(t,x):W \to R$ is called a
generalized Lyapunov-like function for (1) if $V(t, x)$ is continuous
in $t\in \mathbb{R}^+$ and Lipschitzian in $x\in D$ (uniformly in $t$) and there exist
positive functions $\lambda_1(t), \lambda_2(t), \lambda_3(t)$, where $\lambda_1(t)$
is non-decreasing, and  there exist positive numbers $K, p, q, r, \delta$
such that
\begin{gather}
\lambda_1(t)\|x\|^p \leq V(t, x) \leq \lambda_2(t)\|x\|^q,
\quad \forall (t, x) \in W\,,\\
D^+_fV(t, x) \leq - \lambda_3(t)\|x\|^r + Ke^{-\delta t},
\quad \forall t\geq 0, x)\in D\setminus \{0\}
\end{gather}

\section{ Main results }

We start this section  by giving a result from \cite{s3} on the exponential
stability of (1), with the  existence of a uniform Lyapunov function.

\begin{theorem}[\cite{s2}] % Theorem 3.1.
 Assume that (1) admits a Lyapunov-like function, where $p = q = r$.
 The system (1) is uniformly exponentially stable if
$$\delta > \frac{\lambda_3}{\lambda_2}.$$
\end{theorem}

In the two theorems below, we give sufficient conditions for the exponential stability of (1) with a more general Lyapunov-like function.

\begin{theorem} % Theorem 3.2.
The system (1) is uniformly exponentially stable if it admits a Lyapunov-like
function and the following two conditions hold for all $(t,x) \in W$:
\begin{gather}
\delta > \frac{\lambda_3}{[\lambda_2]^{r/q}}\,,\\
\exists \gamma > 0\text{ such that }
 V(t,x) - V(t,x)^{r/q} \leq \gamma e^{-\delta t}.
\end{gather}
\end{theorem}

\paragraph{Proof.} Consider any initial time $t_0 \geq 0$, and let $x(t)$
be any solution of (1) with $x(t_0) = x_0$. Let us set
$$Q(t,x) = V(t,x) e^{M(t-t_0)},\quad M = \frac{\lambda_3}{[\lambda_2]^{r/q}}.
$$
Then
$$\dot Q(t, x(t)) = D_f V(t,x)e^{M(t-t_0)} + MV(t,x)e^{M(t-t_0)}.
$$
Taking (6) into account, for all $t \geq t_0, x \in D$, we have
$$\dot Q(t, x) \leq  (-\lambda_3)\|x\|^r + Ke^{-\delta t})e^{M(t-t_0)} + MV(t, x)e^{M(t-t_0)}.$$
By the condition (5) we have
$\|x\|^q \geq  \frac{V(t, x)}{\lambda_2}$,
equivalently
$$- \|x\|^r \leq - [\frac{V(t, x)}{\lambda_2}]^{r/q}.$$
Therefore, we have
$$\dot Q(t, x) \leq \{-V(t,x)^{r/q}\frac{\lambda_3}{[\lambda_2]^{r/q}} + K e^{-\delta t}\}e^{M(t-t_0)} + MV(t,x)e^{M(t-t_0)}.$$
Since
$$\frac{\lambda_3}{[\lambda_2]^{r/q}} =  M, \quad \forall t\geq 0,$$
we have
$$\dot Q(t, x)\leq M\{V(t,x) - V(t,x) ^{r/q}\}e^{M(t-t_0)} + Ke^{(M -\delta)(t-t_0)}.$$
Using (10), we obtain
$$\dot Q(t, x)\leq (K + M\gamma )e^{(M -\delta)(t-t_0)}.$$
Integrating both sides of the above inequality from $t_0$ to $t$, we obtain
\begin{eqnarray*}
Q(t, x(t)) - Q(t_0, x_0)  &\leq&
 \int_{t_0}^t (K + M\gamma )e^{()M-\delta)(s-t_0)}ds,\\
&=& (K + M\gamma )\frac{1}{M - \delta}\{e^{(M-\delta)(t-t_0)} - 1 \}.
\end{eqnarray*}
Setting  $\delta_1= -(M - \delta)$, by (9) we have $\delta_1 > 0$ and
\begin{eqnarray*}
Q(t,x(t)) &\leq & Q(t_0,x_0) + \frac{K + M\gamma}{\delta_1}
- \frac{K+M\gamma}{\delta_1}e^{(M-\delta)(t-t_0)}\\
&\leq& Q(t_0,x_0) + \frac{K + M\gamma}{\delta_1}\,.
\end{eqnarray*}
Since
$Q(t_0,x_0) = V(t_0,x_0) \leq \lambda_2\|x_0\|^q$,
we have
$$Q(t,x(t)) \leq   \lambda_2\|x_0\|^q +
\frac{K + M\gamma}{\delta_1}.$$
Setting
$$\lambda_2\|x_0\|^q +
 \frac{K+M\gamma}{\delta_1} = \beta(\|x_0\|)  > 0,$$
we have
\begin{equation}
Q(t, x(t)) \leq \beta(\|x_0\|), \quad \forall t\geq t_0.
\end{equation}
On the other hand, from (5) it follows that
$$\displaylines{
\lambda_1\|x(t)\|^p \leq V(t, x(t)),\cr
\|x(t)\| \leq \{\frac{V(t,x(t))}{\lambda_1}\}^{1/p}.
}$$
Substituting
$V(t,x) = Q(t,x) / e^{M(t-t_0)}$ into the last inequality, we obtain
\begin{equation}
\|x(t)\| \leq \{\frac{Q(t,x(t))}{e^{M(t-t_0)}\lambda_1)}\}^{1/p}.
\end{equation}
Combining (11) and (12) gives
\begin{equation}
\|x(t)\| \leq \{\frac{\beta(\|x_0\|}{e^{M(t-t_0)}\lambda_1}\}^{1/p}
=
\{\frac{\beta(\|x_0\|}{\lambda_1}\}^{1/p}e^{-\frac{M}{p}(t-t_0)},
\quad \forall t \geq t_0,
\end{equation}
This inequality shows that (1) is uniformly exponentially stable. Therefore,
the present proof is  complete.  \hfill$\diamondsuit$\medskip

Note that Theorem 3.1 is a special case of Theorem 3.2: $p = q = r$.

\paragraph{Example 3.1} Consider a nonlinear differential equation
\begin{equation}
\dot x = -\frac{1}{4}x^{\frac{3}{5}} + xe^{-2t}, \quad t \geq 0.
\end{equation}
Let us take a Lyapunov function
$V(t,x):\mathbb{R}^+\times D\to \mathbb{R}^+$, $V(t,x) = x^6$,
where $D = \{x: |x| \leq 1\}$. Note that
$$|x|^7 \leq V(t,x) \leq |x|^6, \quad \forall x\in D.$$
Then condition (5) holds with
$\lambda_1 = \lambda_2 = 1$, $p = 7$, $q = 6$.
On the other hand, we have
$$\dot V(t,x) = 6x^5\dot x = 6x^5(-\frac{1}{4}x^{\frac{3}{5}} + xe^{-2t})
= -\frac{3}{2} x^{\frac{28}{5}} + 6x^6e^{-2t}.$$
Therefore,
$$\dot V(t,x) \leq - \frac{3}{2}x^{\frac{28}{5}} + 6e^{-2t},
\quad \forall x \in D.$$
Conditions (9), (10) of Theorem 3.2 are also satisfied with
$\lambda_3 = 3/2$, $K = 6$, $\delta = 2$, $r = 28/5$.
Moreover, we also have
$$V(t,x) - V(t,x)^{r/q} = x^6 - x^{\frac{28}{5}}
= x^{\frac{28}{5}} ( x^{\frac{2}{5}} - 1) \leq 0 \leq e^{-2t},
\quad \forall x \in D\,.$$
Therefore, (14) is exponentially stable. \medskip

We now give a sufficient condition for the exponential stability of (1)
admitting a generalized Lyapunov-like function.

\begin{theorem} % Theorem 3.3.
System (1) is exponentially stable if it admits a generalized \break
 Lyapunov-like
function and  the following two conditions hold for all $(t, x)\in W$:
\begin{gather}
\delta > \inf_{t\in \mathbb{R}^+}\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}} > 0\,.\\
\exists \gamma > 0 \text{ such that }
 V(t,x) - [V(t,x)]^{r/q} \leq \gamma e^{-\delta t}.
\end{gather}
\end{theorem}

\paragraph{Proof.} We consider the function
$Q(t, x(t)) = V(t,x(t))e^{M(t-t_0)}$,
where
$$M=\inf_{t\in \mathbb{R}^+}\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}}$$
We see that $ M < \delta$ and
$$D^+_fQ(t,x) = D^+_fV(t,x)e^{M(t-t_0)} + MV(t,x(t))e^{M(t-t_0)}.$$
By the same arguments used in the proof of Theorem 3.2, we arrived at the fact that
$$D^+_f Q(t, x) \leq  (-\lambda_3(t)\|x\|^r + Ke^{-\delta t})e^{M(t-t_0)} + MV(t, x)e^{M(t-t_0)}.$$
Taking condition (7) into account and since, by the assumption, $\lambda_2(t) > 0$ for all $t\in \mathbb{R}^+$, we have
$$ \|x\|^q \geq  \frac{V(t, x)}{\lambda_2(t)},$$
equivalently
$$- \|x\|^r \leq - [\frac{V(t, x)}{\lambda_2(t)}]^{r/q}.$$
Therefore, we have
$$D^+_f Q(t, x) \leq \{-V(t,x)^{r/q}\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}} + K e^{-\delta t}\}e^{M(t-t_0)} + MV(t,x)e^{M(t-t_0)}.$$
Since
$$\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}} \geq M, \quad \forall t\geq 0,$$
and by the condition  (16) we obtain
\begin{eqnarray*}
D^+_f Q(t, x)&\leq& M\{V(t,x) - V(t,x) ^{r/q}\}e^{M(t-t_0)}
+ Ke^{(M -\delta)(t-t_0)}\\
&\leq& M\gamma e^{-\delta t}e^{M(t-t_0)} + Ke^{-\delta t}e^{M(t-t_0)} \\
&=& (K+M\gamma)e^{-\delta t}e^{M(t-t_0)} \\
&\leq& (K+M\gamma)e^{-\delta(t-t_0)}e^{M(t-t_0)}.
\end{eqnarray*}
Therefore,
$D^+_f Q(t, x)\leq (K + M\gamma)e^{(M -\delta)(t-t_0)}$.
Thus, applying Proposition 2.1 to the case
$$v(t) = Q(t, x(t)), \quad g(t,u(t)) = (K+M\gamma)e^{(M-\delta)(t-t_0)},$$
we obtain
\begin{eqnarray*}
Q(t, x(t)) - Q(t_0, x_0)
&\leq& \int_{t_0}^t (K + M\gamma)e^{(M-\delta)(s-t_0)}ds \\
&=& (K + M\gamma)\frac{1}{M - \delta}\{(e^{(M-\delta)(t-t_0)} - 1 \}.
\end{eqnarray*}
Setting $\delta_1= -(M - \delta) $, by condition (15) we have $\delta_1 > 0$
and
\begin{eqnarray*}
Q(t,x(t)) &\leq& Q(t_0,x_0) + \frac{K + M\gamma}{\delta_1} -
\frac{K+M\gamma}{\delta_1}e^{(M-\delta)(t-t_0)}\\
&\leq& Q(t_0,x_0) + \frac{K + M\gamma}{\delta_1}.
\end{eqnarray*}
Since
$Q(t_0,x_0) = V(t_0,x_0) \leq \lambda_2(t_0)\|x_0\|^q$,
we get
$$Q(t,x(t)) \leq   \lambda_2(t_0)\|x_0\|^q +
\frac{K + M\gamma}{\delta_1}.$$
Letting
$$\lambda_2(t_0)\|x_0\|^q +
 \frac{K+M\gamma}{\delta_1} = \beta(\|x_0\|, t_0)  > 0,$$
we have
\begin{equation}
Q(t, x(t)) \leq \beta(\|x_0\|,t_0), \quad \forall t\geq t_0.
\end{equation}
Furthermore, from condition (7), it follows that
$$\displaylines{
\lambda_1(t)\|x(t)\|^p \leq V(t, x(t)),\cr
\|x(t)\| \leq \{\frac{V(t,x(t))}{\lambda_1(t)}\}^{1/p}.
}$$
Since $\lambda_1(t)$ is non-decreasing, $\lambda_1(t)\geq \lambda_1(t_0)$, we have
$$\|x(t)\| \leq \{\frac{V(t,x(t))}{\lambda_1(t_0)}\}^{1/p}.$$
Substituting
$$V(t,x) = \frac{Q(t,x)}{e^{M(t-t_0)}},$$
into the last inequality, we obtain
\begin{equation}
\|x(t)\| \leq \{\frac{Q(t,x(t))}{e^{M(t-t_0)}\lambda_1(t_0)}\}^{1/p}.
\end{equation}
Combining (17) and (18),
\begin{equation}
\|x(t)\| \leq \{\frac{\beta(\|x_0\|,t_0)}{e^{M(t-t_0)}\lambda_1(t_0)}\}^{1/p}
 =
\{\frac{\beta(\|x_0\|,t_0)}{\lambda_1(t_0)}\}^\frac{1}{p}e^{-\frac{M}{p}(t-t_0)},
\quad \forall t \geq t_0,
\end{equation}
The relation (19) shows that system (1) is exponentially stable.
Theorem is proved. \hfill$\diamondsuit$

\paragraph{Remark 3.1.} Note that in Theorem 3.3 we assume that the
function $\lambda_1(t)$ is non-decreasing. In the case if the function
$\lambda_1(t)$ satisfies the condition
\begin{equation}
\exists a > 0 :  a < M,\quad \lambda_1(t) \geq e^{-at}, \quad \forall t \geq 0,
\end{equation}
then we can replace the non-decreasing assumption by the above condition (20), where
$$ M=\inf_{t\in \mathbb{R}^+}\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}}.$$

The examples below illustrate our results in the case the Lyapunov function
satisfies either more general comparable conditions or non-differentiability
conditions, which include, as a special case, the results of \cite{s2,s3}.

\paragraph{Example 3.2} Consider the system
\begin{equation}
\dot x = -\frac{1}{6}e^tx^\frac{3}{5} + \frac{x}{12} + e^{-\frac{3t}{2}}\sin x, \quad t\geq 0.
\end{equation}
We take the Lyapunov function $V(t,x):\mathbb{R}^+\times D\to \mathbb{R}^+$, where $D$ is defined as in Example 3.1, given by
$$V(t,x) = e^{-t/2}x^6.$$
In this case, we have
$p = q = 6$, $\lambda_1(t) = e^{-t/2}$, $\lambda_2(t) = 1$,
and
\begin{eqnarray*}
\dot V(t,x) &=& -\frac{1}{2}e^{-t/2}x^6 + 6e^{-t/2}
x^5(-\frac{1}{6}e^tx^{\frac{3}{5}} + \frac{x}{12} + e^{-\frac{3t}{2}}\sin x)\\
&=& -e^{\frac{t}{2}}x^{\frac{28}{5}} + 6x^5e^{-2t}\sin x,\\
&&\dot V(t,x)\leq  -e^{\frac{t}{2}}x^{\frac{28}{5}} + 6e^{-2t}, \quad
\forall x \in D.
\end{eqnarray*}
Therefore, we see that
$r = 28/5$, $\delta = 2$, $K = 6$, $\lambda_3(t) =  e^{t/2}$,
and
$$M = \inf_{t\in \mathbb{R}^+}\frac{\lambda_3(t)}{[\lambda_2(t)]^{r/q}} = 1 < \delta = 2,$$
Moreover, we see that (20) holds: $a = \frac{1}{2} < M = 1$,
and we can check condition (16) of Theorem 3.3 as follows:
\begin{eqnarray*}
V(t,x) - [V(t,x)]^{r/q} &=&  e^{-t/2}x^6 - \{e^{-t/2}x^6\}^{28/30} \\
&=& e^{-t/2}x^6 - e^{-14t/30}x^{28/5} \leq 0\leq e^{-2t},
\quad \forall x\in D,
\end{eqnarray*}
because of $x\in D = \{x: \|x\|\leq 1 \}$, and
$$e^{-t/2}x^6 \leq e^{-t/2}x^{\frac{28}{5}} \leq e^{-\frac{14t}{30}}x^{\frac{28}{5}}.$$
System (21) is exponentially stable.

\paragraph{Example 3.3} Consider the system
\begin{equation}
\dot x = -\frac{1}{5} x^{1/3} + xe^{-2t}
\end{equation}
with the  Lyapunov function $V(t,x) = |x|^5$, where $D$ is defined the same
as in Example 3.1. We have
$$V(t,x) = \begin{cases} x^5,&\text{if } x \geq 0\\
 - x^5 & \text{if } x < 0 \,.\end{cases}$$
Then we calculate
$$
D^+_fV(t,x) = \begin{cases}
5x^4(-\frac{1}{5}x^{1/3} + xe^{-2t}) = -x^{-13/3} + 5x^5e^{-2t}
&\text{if } x \geq 0\\
 -5x^4(-\frac{1}{5}x^{1/3} + xe^{-2t})  = x^{14/3}- 5x^5e^{-2t}
& \text{if } x < 0\end{cases}
 $$
Therefore,
$$D^+_fV(t,x) = -|x|^{13/3} + 5|x|^5e^{-2t}
\leq -|x|^{13/3} - 5e^{-2t}, \quad \forall x\in D\,.$$
It follows that the conditions (8) and  (15) hold for
$\lambda_1 = \lambda_2 = \lambda_3 = 1$, $p = q = K = 5$,
$ \delta = 2$, $r = 13/3$, $\delta > \lambda_3/\lambda_2^{r/q} = 1$.
Condition (16) of Theorem 3.3 is also true because of:
$$V(t,x) - [V(t,x)]^{r/q} = |x|^5 - |x|^{13/3}
= |x|^{13/3}(|x|^{2/3} - 1) \leq 0, \quad \forall x\in D.$$
Then system (22) is uniformly exponentially stable.

\paragraph{Example 3.4} For the system
$$\dot x = -x^{1/3} + x^3e^{-2t},$$
we take $V(t,x) = |x|$ with $x\in D$. We have
$\lambda_1 = \lambda_2 = p = q =1$,
and
$$D^+_fV(t,x) = \begin{cases} -x^{1/3} + x^3e^{-2t} &\text{if } x\geq 0,\\
x^{1/3} - x^3e^{-2t} &\text{if } x < 0\,.
\end{cases}$$
Then, for all $x\in D$, we have
$$D^+_fV(t,x) =  -|x|^{1/3} + |x|^3e^{-2t} \leq -|x|^{1/3} + e^{-2t}.$$
All the conditions of Theorem 3.3 hold with $\lambda_3 = 1$, $r = 1/3$,
$\delta = 2$, $K = 1$. The system is uniformly exponentially stable.


\section{Applications to control systems}

Consider the  nonlinear control system (2), assuming  that
$f(t,0,0) = 0$, for all $t \geq 0$.
We recall that system (2) is asymptotically stabilizable
by a feedback control $u(t) = h(x,t)$, where $h(x) : \mathbb{R}^n \to \mathbb{R}^m$,
$h(0) = 0$, if the zero solution of the system without control
\begin{equation}\begin{gathered}
\dot x(t) = f(t, x(t), h(x)),\quad t \geq 0, \\ x(t_0) =
x_0,\quad t_0 \geq 0
\end{gathered}\end{equation}
is asymptotically stable in the Lyapunov sense \cite{l2,y1}. If the zero
solution of (23) is exponentially stable, we say that  (2) is
exponentially stabilizable.

Stabilization problem of system (2) has attracted a lot of
attention from many researches in control theory in the last decade
\cite{l2,n1,p1,z1}. Some sufficient conditions bellow for stability using
Lyapunov functions were obtained for a class of time-invariant systems of
the form
\begin{equation}
 \dot x(t) = f(x(t), u(t)),\quad t \geq 0,
\end{equation}
using smooth Lyapunov functions.

\begin{theorem}[\cite{l2}] %Theorem 4.1
 Consider time-invariant system (24).
If there exist a function $h(x) : \mathbb{R}^n \to \mathbb{R}^m$,
$h(0) = 0$, where $h(x)$ is continuously differentiable in $x$ and
a positive definite function $V(x) : \mathbb{R}^n \to \mathbb{R}^+$, which is continuously
differentiable in $x$ such that
\\
(i) $V(x) \to \infty$ as $\|x\| \to \infty$.
\\
(ii)
$ \frac{\partial V}{\partial x_i} f^i (x, h(x)) < 0$,
$i = 1, 2, \dots, n$,  for all $x \neq 0$.
Then the system is asymptotically stabilizable by the feedback control
$u(t) = h(x(t))$.
\end{theorem}

Some other sufficient conditions for stabilization of (24) using Lyapunov
control functions can be found in \cite{a2,k2,s1}, where the Lyapunov
function $V(x)$ is assumed to be proper (i.e. the condition (i) in
Theorem 4.1 holds) and has a negative lower Dini derivative along a solution
of the system. Based on the stability results obtained in the previous
section, we can derive the following sufficient conditions for the exponential
stability of nonlinear control system (2) with non-smooth Lyapunov
functions.

\begin{theorem} % Theorem 4.2
Assume that there is a function $h(x) : \mathbb{R}^n \to \mathbb{R}^m$, $h(0) = 0$ with
$h(x)$ continuous in $x$, such that system (23) admits a Lyapunov-like function satisfying  (9) and (10). Then the nonlinear control system (2) is exponentially stabilizable by feedback  $u(t) = h(x(t))$.
\end{theorem}

\begin{theorem} % Theorem 4.3
Assume that there is a function $h(x) : \mathbb{R}^n \to \mathbb{R}^m$ with $h(0) = 0$
and $h(x)$ continuous in $x$,  such that system (23) admits a generalized Lyapunov-like function satisfying (15), (16).
Then the nonlinear control system (2) is exponentially stabilizable
by feedback control $u(t) = h(x(t))$.
\end{theorem}

\paragraph{Conclusions}
Exponential stability of a class of nonlinear time-varying systems by the
second Lyapunov method has been studied.  New sufficient conditions for the
exponential stability and applications to exponential stabilization problem of
nonlinear control systems were given.

\paragraph{Acknowledgments.}
This research was supported by the National
Basic Program in Natural Sciences, Vietnam.

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\noindent\textsc{Nguyen Manh Linh}\\
Institute of Mathematics \\
National Center for Sciences and Technology \\
P.O. Box 631 Bo Ho, 10.000 Hanoi, Vietnam \medskip

\noindent\textsc{Vu Ngoc Phat} \\
Institute of Mathematics \\
National Center for Sciences and Technology \\
P.O. Box 631 Bo Ho, 10.000 Hanoi, Vietnam\\
e-mail: vnphat@hanimath.ac.vn; phvu@syscon.ee.unsw.edu.au


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