\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2008(2008), No. 48, 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-2008/48\hfil Lyapunov-Razumikhin method] {Lyapunov-Razumikhin method for asymptotic stability of sets for impulsive functional differential equations} \author[I. M. Stamova, G. T. Stamov\hfil EJDE-2008/48\hfilneg] {Ivanka M. Stamova, Gani T. Stamov} \address{Ivanka M. Stamova \newline Department of Mathematics, Bourgas Free University, 8000 Bourgas, Bulgaria} \email{istamova@abv.bg} \address{Gani T. Stamov \newline Department of Mathematics, Technical University at Sliven, 8800 Sliven, Bulgaria} \email{gstamov@abv.bg} \thanks{Submitted December 2, 2006. Published April 4, 2008.} \subjclass[2000]{34K45} \keywords{Global stability of sets; Lyapunov's direct method; \hfill\break\indent impulsive functional differential equations} \begin{abstract} In the present paper, we study the global stability of sets of sufficiently general type with respect to impulsive functional differential equations with variable impulsive perturbations. The main results are obtained by means of piecewise continuous Lyapunov functions and the use of the Razumikhin technique. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Stability of impulsive ordinary differential equations is discussed in \cite{b2,b3,l1,l2}, and recently the stability of impulsive functional differential equations is investigated in \cite{b4,b5,s1,s2,y1}. When the impulses are realized at fixed moments the results are easier to obtain by means of the corresponding results in the continuous case. In the investigation of the impulsive functional differential equations with variable impulsive perturbations there arise a number of difficulties related to the phenomena of ``beating'' of the solutions, bifurcation, loss of the property of autonomy, etc. The wider application, however, of these type of equations requires the formulation of effective criteria for stability of their solutions. In the present paper the problem of global stability of sets with respect to systems of impulsive functional differential equations with variable impulsive perturbations is considered by means of Lyapunov's direct method. We use the piecewise continuous Lyapunov's functions. Moreover, the technique of investigation essentially depends on the choice of minimal subsets of a suitable space of piecewise continuous functions, by the elements of which the derivatives of Lyapunov's functions are estimated \cite{l2,r1}. It is well known that Lyapunov-Razumikhin function method has been widely used in the treatment of the stability of functional differential equations without impulses \cite{h1,h2,k1}. \section{Statement of the problem, preliminary notes and definitions} Let $\mathbb{R}^n$ be the $n$-dimensional Euclidean space with norm $|\cdot|$, scalar product $\langle .,.\rangle$ and distance $d(\cdot,\cdot)$; $\mathbb{R}_{+}=[0,\infty)$; $\mathbb{R}=(-\infty,\infty)$. Let $t_{0}\in \mathbb{R}$, $r>0$. Consider the system of impulsive functional differential equations \begin{equation} \begin{gathered} \dot{x}(t)=f(t,x_t),\quad t\neq {\tau_k(x(t))},\\ \Delta{x}(t)=I_{k}(x(t-0)),\quad t=\tau_k(x(t)),\; k=1,2, \dots, \end{gathered} \label{e1} \end{equation} where $f:(t_{0},\infty)\times{D}\to{\mathbb{R}^n}$; $D=\{\phi:[-r,0]\to{\mathbb{R}^n},\quad \phi(t)$ is continuous everywhere except at finite number of points $\tilde{t}$ at which $\phi(\tilde{t}-0)$ and $\phi(\tilde{t}+0)$ exist and $ \phi(\tilde{t}-0)=\phi(\tilde t)\}$; $I_{k}:\mathbb{R}^n\to{\mathbb{R}^n},k=1,2,\dots$; $\tau_k:\mathbb{R}^n \to(t_0,\infty)$; $\Delta{x}(t)=x(t+0)-x(t-0)$ and for $t>t_0$, $x_{t}\in\quad D\quad $ is defined by $x_{t}=x(t+s)$, $-r\leq s\leq\quad 0$. Let $\tau_0(x)\equiv t_0$ for $x\in \mathbb{R}^n$. We shall assume that: \begin{itemize} \item[(a)] $\tau_k\in C[\mathbb{R}^n,(t_0,\infty)]$, $k=1,2,\dots$. \item[(b)] $t_0<\tau_1(x)<\tau_2(x)<\dots$, $x\in \mathbb{R}^n$. \item[(c)] $ \tau_k(x)\to\infty$ as $k\to\infty$ uniformly on $x\in \mathbb{R}^n$. \end{itemize} Assuming that (a), (b) and (c) are fulfilled, we introduce the notation: \begin{gather*} G_k=\big\{(t,x)\in[t_0,\infty)\times \mathbb{R}^n\colon \tau_{k-1}(x)0)$ is an $\varepsilon$- neighbourhood of $M(t)$;\\ $C_0=C[[-r,0],\mathbb{R}^n]$; $d_0(\varphi,M_0(t))=\max_{t\in[t_0-r,t_0]}d(\varphi(t-t_0),M_0(t)), \varphi\in{C_0}$;\\ $M_0(t,\varepsilon)=\{\varphi\in{C_0}: d_0(\varphi,M_0(t))<\varepsilon\}$;\\ $S_{\alpha}=\{x\in{\mathbb{R}^n}: |x|<{\alpha}\},\; \alpha>0$; $\overline{S_{\alpha}}=\{x\in{\mathbb{R}^n}: |x|\leq{\alpha}\}$;\\ $\overline{S_{\alpha}}(C_0)=\{\varphi\in{C_0}: ||\varphi||\leq{\alpha}\}$, where $||\varphi||=\max_{t\in[t_0-r,t_0]}|\varphi(t-t_0)|$ is the norm of the function $\varphi\in{C_0}$;\\ $K=\{a\in{C[R_+,R_+]}: a(r)\text{ is strictly increasing and } a(0)=0\}$;\\ $CK=\{a\in{C[(t_0,\infty)\times{R_+},R_+]}: a(t,.)\in{K} \text{ for any fixed }t\in{(t_0,\infty)}\}$;\\ $K^*=\{a\in{C[R_+\times{R_+},R_+]}: a(.,s)\in{K} \text{ for any fixed } s\in{R_+}\}$. We also introduce the following conditions: \begin{itemize} \item[(H1)] $M(t)\neq\emptyset$ for $t\in(t_0,\infty)$. \item[(H2)] $M_0(t)\neq\emptyset$ for $t\in[t_0-r,t_0]$. \item[(H3)] For any compact subset $F$ of $(t_0,\infty)\times{\mathbb{R}^n}$ there exists a constant $K>0$ depending on $F$ such that if $(t,x)$, $(t',x)\in{F}$, then the following inequality is valid $$ |d(x,M(t))-d(x,M(t'))|\leq{K|t-t'|}. $$ \item[(H4)] The integral curves of the \eqref{e1} meet successively each one of the hypersurfaces $\sigma_1,\sigma_2,\dots$. exactly once. \end{itemize} Condition (H4) guarantees absence of the phenomenon ``beating'' of the solutions to the \eqref{e1}; i.e., a phenomenon when a given integral curve meets more than once or infinitely many times one and the same hypersurface. Efficient sufficient conditions which guarantee the absence of ``beating'' of the solutions of such systems are given in \cite{b1}. Let $t_1, t_2 ,\dots$ $(t_0t_0$. Note that \cite{b1,b2,l1} if $f\in{C}[(t_0,\infty)\times{D},\mathbb{R}^n]$, the function $f$ is Lipschitz continuous with respect to its second argument in $(t_0,\infty)\times{D}$ uniformly on $t\in(t_0,\infty)$, $|f(t,\tilde x)|\leq L<\infty$ for$(t,\tilde x)\in{(t_0,\infty)\times{D} }$, $L>0$, for any $k=1,2,\dots$ the following inequality is valid $|I_k(x_1)-I_k(x_2)|\leq{c}|x_1-x_2|$, $x_1,x_2\in {\mathbb{R}^n},\quad c>0$, and (H4) are met, then $t_k \to\infty$ as $k\to\infty$ and $J^+(t_0,\varphi_0)=(t_0,\infty)$. \begin{definition} \label{def1} \rm The solutions of \eqref{e1} are said to be uniformly $M$-bounded if \begin{gather*} (\forall\eta>0)(\exists\beta=\beta(\eta)>0)(\forall{t_0}\in{R}) (\forall\alpha>0)\\ (\forall\varphi_0\in{\overline{S_{\alpha}}}(C_0) \cap\overline{M_0(t,\eta)})(\forall{t}>t_0): x(t;t_0,\varphi_0)\in{M}(t,\beta); \end{gather*} \end{definition} \begin{definition} \label{def2} \rm The set $M$ is said to be \begin{itemize} \item[(a)] stable with respect to \eqref{e1} if \begin{gather*} (\forall{t_0}\in{R})(\forall\alpha>0)(\forall\varepsilon>0) (\exists\delta=\delta(t_0,\alpha,\varepsilon)>0)\\ (\forall\varphi_0\in{\overline{S_{\alpha}}(C_0)}\cap{M_0}(t,\delta)) (\forall{t>t_0}): x(t;t_0,\varphi_0)\in{M}(t,\varepsilon); \end{gather*} \item[(b)] uniformly stable with respect to \eqref{e1} if the number $\delta$ from point $(a)$ depends only on $\varepsilon$; \item[(c)] uniformly globally attractive with respect to \eqref{e1} if \begin{gather*} (\forall\eta>0)(\forall{\varepsilon}>0)(\exists\sigma =\sigma(\eta,\varepsilon)>0)\\ (\forall{t_0}\in{R})(\forall\alpha>0) (\forall\varphi_0\in{\overline{S_{\alpha}}}(C_0)\cap{M_0(t,\eta)})\\ (\forall{t}\geq{t_0+\sigma}): x(t;t_0,\varphi_0)\in{M}(t,\varepsilon); \end{gather*} \item[(d)] uniformly globally asymptotically stable with respect to \eqref{e1} if $M$ is a uniformly stable and uniformly globally attractive set of \eqref{e1} and if the solutions of \eqref{e1} are uniformly $M$-bounded. \end{itemize} \end{definition} Also we introduce the notations: $I=[t_{0}-r,\infty)$; $I_{0}=[t_{0},\infty)$. In the further considerations we shall use the class $V_0$ of piecewise continuous auxiliary functions $V:I_0\times{\mathbb{R}^n}\to{R_+}$ which are analogues of Lyapunov's functions. \begin{definition} \label{def3}\rm We say that the function $V:\ [t_0,\infty)\times \mathbb{R}^n\to R_+$, belongs to the class $V_0$ if the following conditions are fulfilled: \begin{enumerate} \item The function $V$ is continuous in $ {\cup_{k=1}^{\infty} G_k}$ and locally Lipschitz continuous with respect to its second argument $x$ on each of the sets $G_k$, $k=1,2,\dots$. \item $V(t,x)=0$ for $(t,x) \in M$, $t\geq t_0$ and $V(t,x)>0$ for $(t,x) \in \{[t_0,\infty) \times \mathbb{R}^n \} \setminus M$. \item For each $k=1,2,\dots$ and $(t_0^*,x_0^*) \in \sigma_k$ there exist the finite limits \begin{gather*} V(t_0^*-0,x_0^*)=\lim_{{(t, x)\to (t_0^*,x_0^*)},\, {(t, x) \in G_k}}V(t,x), \\ V(t_0^*+0,x_0^*)=\lim_{{(t, x)\to (t_0^*,x_0^*)},\, {(t, x) \in G_{k+1}}}V(t,x) \end{gather*} \item For each $k=1,2,\dots$ the following equalities are valid $$V(t_0^*-0,x_0^*)= V(t_0^*,x_0^*).$$ \item For each $k=1,2,\dots$ the following inequalities are valid \begin{equation} V(t+0,x(t)+I_k(x(t)))\leq{V(t,x(t))},\quad t=\tau_k(x(t)), \quad k=1,2,\dots\label{e3} \end{equation} \end{enumerate} Let $J\subset{R}$ be an interval. Define the following classes of functions: \\ $PC[J,R^{n}]=\{\sigma:J\to{\mathbb{R}^n}: \sigma(t)$ is continuous everywhere except some points $t_{k}$ at which $\sigma(t_{k}-0)$ and $\sigma(t_{k}+0)$ exist and $\sigma(t_{k}-0)=\sigma(t_{k}),\; k=1,2,\dots\}$; $PC^{1}[J,R^{n}]=\{\sigma\in{PC[J,R^{n}]}: \sigma(t)$ is continuously differentiable everywhere except some points $t_{k}$ at which $\dot{\sigma}(t_{k}-0)$ and $\dot{\sigma}(t_{k}+0)$ exist and $\dot{\sigma}(t_{k}-0)=\dot{\sigma}(t_{k}),$ $k=1,2,\dots\}$;\\ $\Omega_1=\{x\in{PC[I_0,\mathbb{R}^n]}: V(s,x(s))\leq{V(t,x(t))},\; t-r\leq{s}\leq{t},\; t\in{I_0},\; V\in{V_0}\}$. Let $V\in{V_0}$. For $x\in{PC[I_0,\mathbb{R}^n]}$ and $t\in I_0$, $t\neq{t_k(x(t))}$, $k=1,2,\dots$ we define the function $$ D_{-}V(t,x(t))=\lim_{h\to{0^-}}\inf h^{-1}[V(t+h,x(t)+hf(t,x_t))-V(t,x(t))]. $$ \end{definition} \begin{definition}[\cite{b2}] \label{def4} \rm Let $\lambda:(t_0,\infty)\to{R_+}$ be measurable. Then we say that $\lambda(t)$ is integrally positive if $\int_{J}\lambda(t)dt=\infty$ whenever $J=\bigcup_{k=1}^{\infty}[\alpha_k,\beta_k]$, $\alpha_k<\beta_k<\alpha_{k+1}$ and $\beta_k-\alpha_k\geq{\theta}>0$, $k=1,2,\dots$. \end{definition} In the proof of the main results we shall use the following lemma. \begin{lemma}[\cite{s2}] \label{lem1} Let {\rm (H4)} and the following conditions hold: \begin{enumerate} \item The solution $x=x(t;t_0,\varphi_0)$ of the problem \eqref{e1}, \eqref{e2} is such that $x\in{PC[I,S_{\rho}]}\cap{PC^1[I_0,S_{\rho}]}$. \item $g\in{PC[[t_0,\infty)\times{R_+},R]}$ and $g(t,0)=0$ for $t\in[t_0,\infty)$. \item $B_k\in{C[R_+,R_+]}$, $B_k(0)=0$ and $\psi_k(u)=u+B_k(u)$ are nondecreasing with respect to $u$, $k=1,2,\dots$. \item The maximal solution $u^+(t;t_0,u_0)$ of the problem \begin{gather*} \dot{u}(t)=g(t,u(t)),\quad t>t_0,\; t\neq{t_k},\; k=1,2,\dots,\\ u(t_0+0)=u_0\geq{0},\\ \Delta{u(t_k)}=B_k(u(t_k)),\; k=1,2,\dots \end{gather*} is defined in the interval $[t_0,\infty)$. \item The function $V\in{V_0}$, $V:I_0\times{S_{\rho}}\to{R_+}$ is such that $V(t_0+0,\varphi_0(0))\leq{u_0}$ and the inequalities \begin{gather*} D_{-}V(t,x(t))\leq{g(t,V(t,x(t)))},\quad t\neq{\tau_k(x(t))},\; k=1,2,\dots \\ V(t+0,x(t)+I_k(x(t)))\leq{B_k(V(t,x(t)))},\quad t=\tau_k(x(t)),\; k=1,2,\dots \end{gather*} are valid for each $t\in{I_0}$ and $x\in{\Omega_1}$. \end{enumerate} Then $V(t,x(t;t_0,\varphi_0))\leq{u^+}(t;t_0,u_0)$, $t\in{I_0}$. \end{lemma} \begin{corollary} \label{coro1} Let the condition {\rm (H4)} be satisfies and the function $V\in{V_0}$ be such that the inequality $$ D_{-}V(t,x(t))\leq{0},\quad t\neq{\tau_k(x(t))},\quad k=1,2,\dots $$ is valid for each $t>{t_0}$ and $x\in{\Omega_1}$. Then $V(t,x(t;t_0,\varphi_0))\leq{V(t_0,\varphi_0(t_0))}$, $t\in[t_0,\infty)$. \end{corollary} \section{Main results} \begin{theorem} \label{thm1} Assume that {\rm (H1)-(H4)} and the following conditions are met: \begin{enumerate} \item The functions $V\in{V_0}$ and $a,b\in{K}$ are such that $$ a(d(x,M(t)))\leq{V(t,x)}\leq b(d(x,M(t))), $$ for $(t,x)\in[t_0,\infty)\times{\mathbb{R}^n}$ and $a(r)\to\infty$ as $r\to\infty$ . \item The inequality $$ D_{-}V(t,x(t))\leq{-p(t)c(d(x(t),M(t)))},\quad t \neq{\tau_k(x(t))},\quad k=1,2\dots $$ is valid for any $t>{t_0}$, $x\in{\Omega_1}, V\in{V_0}$, $p:[t_0,\infty)\to(0,\infty)$, $c\in{K}$. \item $\int_{0}^{\infty}p(s)c[b^{-1}(\eta)]ds=\infty$ for each sufficiently small value of $\eta>0$. \end{enumerate} Then the set $M$ is uniformly globally asymptotically stable with respect to \eqref{e1}. \end{theorem} \begin{proof} Let $\varepsilon>0$. Choose $\delta=\delta(\varepsilon)>0$, $ \delta<\varepsilon$ so that $b(\delta)< a(\epsilon)$. Let $\alpha>0$ be arbitrary, $\varphi_0\in{\overline{S_{\alpha}}}(C_0)\cap{M_0(t,\delta)}$ and $x(t)=x(t;t_0,\varphi_0)$. From conditions 1 and 2, and \eqref{e3} it follows that for $t\in{J^+(t_0,\varphi_0)}$, \begin{align*} a(d(x(t;t_0,\varphi_0),M(t))) &\leq{V(t,x(t))}\\ &\leq {V(t_0,\varphi_0(t_0))}\\ &\leq b(d(\varphi_0(t_0),M_0(t_0)))\\ &\leq b(d_0(\varphi_0,M_0(t)))\\ &< b(\delta)< a(\varepsilon). \end{align*} Since $J^+(t_0,\varphi_0)=(t_0,\infty)$, then $x(t)\in{M(t,\varepsilon)}$ for all $t>t_0$. This proves that the set $M$ is uniformly stable. Now let $\eta>0$ and $\varepsilon>0$ be given and let the number $\sigma=\sigma(\eta,\varepsilon)>0$ be chosen so that \begin{equation} \int_{t_0}^{t_0+\sigma}p(s)c[b^{-1}(\frac{a(\varepsilon)}{2})]ds> b(\eta). \label{e4} \end{equation} (This is possible in view of condition 3). Let $\alpha>0$ be arbitrary, $\varphi_0\in{\overline{S_{\alpha}}}(C_0)\cap{M_0(t,\eta)}$ and $x(t)=x(t;t_0,\varphi_0)$. Assume that for any $t\in[t_0,t_0+\sigma]$, $$ d(x(t),M(t))\geq{b^{-1}}(\frac{a(\varepsilon)}{2}). $$ Then by condition 2 and \eqref{e4}, it follows that \begin{equation} \int_{t_0}^{t_0+\sigma}D_{-}V(s,x(s))ds\leq{-\int_{t_0}^{t_0+\sigma}}p(s)c[b^{-1}(\frac{a(\varepsilon)}{2})]ds < -b(\eta).\label{e5} \end{equation} On the other hand, if $t_0+\sigma\in(\tau_r,\tau_{r+1}]$, then from \eqref{e3} we obtain \begin{align*} &\int_{t_0}^{t_0+\sigma}D_{-}V(s,x(s))ds\\ &=\sum_{k=1}^{r}\int_{\tau_{k-1}}^{\tau_k}D_{-}V(s,x(s))ds +\int_{\tau_r}^{t_0+\sigma}D_{-}V(s,x(s))ds\\ &=\sum_{k=1}^{r}[V(\tau_{k},x(\tau_{k}))-V(\tau_{k-1}+0, x(\tau_{k-1}+0))]+V(t_0+\sigma,x(t_0+\sigma))\\ &\quad -V(\tau_r+0,x(\tau_r+0))\geq{V(t_0+\sigma,x(t_0+\sigma))} -V(t_0,\varphi_0(t_0)), \end{align*} whence, in view of \eqref{e5} and condition 2, it follows that $V(t_0+\sigma,x(t_0+\sigma))<0$, which contradicts condition 1. The contradiction obtained shows that there exists $t^*\in[t_0,t_0+\sigma]$, such that $$ d(x(t^*),M(t^*))0$ and let $\beta=\beta(\eta)>0$ be such that $a(\beta)>\gamma b(\eta)$. Choose arbitrary $\alpha>0$, $\varphi_0\in{\overline{S_{\alpha}}}(C_0)\cap\overline{M_0(t,\eta)}$ and let $x(t)=x(t;t_0,\varphi_0)$. Then for $t>t_0$, \begin{align*} a(d(x(t),M(t)))&\leq{V(t,x(t))}\\ &\leq{V(t_0,\varphi_0(t_0))}\\ &\leq {\gamma} b(d(\varphi_0(t_0),M_0(t_0)))\\ &\leq{\gamma} b(d_0(\varphi_0,M_0(t)))\\ &\leq{\gamma }b(\eta)t_0$. \end{proof} \begin{theorem} \label{thm2} Assume that {\rm (H1)--(H4)} and Condition 1 of Theorem \ref{thm1} are met. Also assume that there exists an integrally positive function $\lambda(t)$ such that $$ D_{-}V(t,x(t))\leq{-\lambda(t)c(d(x(t),M(t)))},\quad t\neq{\tau_k(x(t))},\ k=1,2\dots $$ holds for any $t>{t_0}$, $x\in{\Omega_1}, V\in{V_0}$ and $c\in{K}$. Then the set $M$ is uniformly globally asymptotically stable with respect to \eqref{e1}. \end{theorem} \begin{proof} The fact that the set $M$ is uniformly stable with respect to the \eqref{e1} and the uniform $M$-boundedness of the solutions of \eqref{e1} are proved as in the proof of Theorem \ref{thm1}. Now we shall prove that the set $M$ is uniformly globally attractive with respect to the \eqref{e1}. Let again $\varepsilon>0$ and $\eta>0$ be given. Choose the number $\delta=\delta(\varepsilon)>0$ so that $b(\delta)0$ such that for any solution $x(t)=x(t;t_0,\varphi_0)$ of \eqref{e1} for which $t_0\in{R}$, $\varphi_0\in{\overline{S_{\alpha}}}(C_0){\cap{M_0(t,\eta)}}$ $(\alpha>0$ - arbitrary) and for any $t^*\in[t_0,t_0+\sigma]$, \begin{equation} d(x(t^*),M(t^*))<\delta(\varepsilon).\label{e6} \end{equation} Suppose that this is not true. Then for any $\sigma>0$ there exists a solution $x(t)=x(t;t_0,\varphi_0)$ of \eqref{e1} for which $t_0\in{R}$, $\varphi_0\in{\overline{S_{\alpha}}}(C_0){\cap{M_0(t,\eta)}}$, $\alpha>0$, such that \begin{equation} d(x(t),M(t))\geq{\delta(\varepsilon)},\label{e7} \end{equation} for $t\in[t_0,t_o+\sigma]$. From the third condition in this theorem and \eqref{e3} it follows that \begin{align*} V(t,x(t))-V(t_0,\varphi_0(t_0)) &\leq{\int_{t_0}}^{t}D_{-}V(s,x(s))ds\\ &\leq{-\int_{t_0}^t}\lambda(s)c(d(x(s),M(s)))ds,\quad t>t_0.\label{e8} \end{align*} From the properties of the function $V(t,x(t))$ in the interval $(t_0,\infty)$ it follows that there exists the finite limit \begin{equation} \lim_{t\to\infty}V(t,x(t))=v_0\geq{0}.\label{e9} \end{equation} Then from condition 1 of Theorem \ref{thm1}, \eqref{e7}-\eqref{e9} it follows that $$ \int_{t_0}^{\infty}\lambda(t)c(d(x(t),M(t)))dt\leq{b(\eta)}-v_0. $$ From the integral positivity of the function $\lambda(t)$ it follows that the number $\sigma$ can be chosen so that $$ \int_{t_0}^{t_0+\sigma}\lambda(t)dt >\frac{b(\eta)-v_0+1}{c(\delta(\varepsilon))}. $$ Then \begin{align*} b(\eta)-v_0 &\geq{\int_{t_0}}^{\infty}\lambda(t)c(d(x(t),M(t)))dt\\ &\geq{\int_{t_0}}^{t_0+\sigma}\lambda(t)c(d(x(t),M(t)))dt\\ &\geq{c(\delta(\varepsilon))}\int_{t_0}^{t_0+\sigma}\lambda(t)dt\\ &>b(\eta)-v_0+1. \end{align*} The contradiction obtained shows that there exists a positive constant $\sigma=\sigma(\epsilon,\eta)$ such that for any solution $x(t)=x(t;t_0,\varphi_0)$ of \eqref{e1} for which $t_0\in{R}$, $\varphi_0\in{\overline{S_{\alpha}}}(C_0){\cap{M_0(t,\eta)}}$, $\alpha>0$, there exists $t^*\in[t_0,t_0+\sigma]$ such that \eqref{e6} holds. Then for $t\geq{t^*}$ (hence for any $t\geq{t_0+\sigma}$ as well) the following inequalities are valid \begin{align*} a(d(x(t),M(t))) &\leq{V(t,x(t))}\\ &\leq{V(t^*+0,x(t^*+0))}\\ &\leq{b}(d(x(t^*),M(t^*)))\\ &<{b}(\delta)0,\; t\neq{\tau_k(x(t))},\\ 0,& x(t)\leq{0},\; t\neq{\tau_k(x(t))}; \end{cases}\\ \Delta{x}(t)=\begin{cases} C_k x(t), & x(t)>0,\; t=\tau_k(x(t)),\\ 0,& x(t)\leq{0},\; t={\tau_k(x(t))}, \end{cases} \end{gathered}\label{e10} \end{equation} where $t>t_0$; $x\in{PC}[(t_0, \infty),\mathbb{R}^n]$; $A(t)$ and $B(t)$ are $(n\times{n})$ matrix-valued functions, $C_k$, $k=1, 2,\dots$ are $(n\times n)$ matrices; $h \in C[(t_0, \infty), R_+]$. Such systems seem to have application, among other things, in the study of active suspension height control. In the interest of improving the overall performance of automotive vehicles, in recent years, suspension incorporating active components have been developed. The designs may cover a spectrum of of performance capabilities, but the active components alter only the vertical force reactions of the suspensions, not the kinematics. The conventional passive suspensions consist of usual components with spring and damping properties, which are time-invariant. The interest in active or semi-active suspensions derives from the potential for improvements to vehicle ride performance with no compromise or enhancement in handling. The full active suspensions incorporate actuators to generate the desired forces in the suspension. They actuators are normally hydraulic cylinders. Let $\tau =\inf_{t\geq t_0}( t-h(t))$ and $\varphi_1\in{C}[[\tau,t_0],\mathbb{R}^n]$. Denote by $x(t)=x(t;t_{0},\varphi_{1})$ the solution of system \eqref{e10} satisfying the initial condition \begin{equation} x(t;t_{0},\varphi_{1})=\varphi_{1}(t),\quad \tau\leq t\leq\quad t_0, \label{e11} \end{equation} and by $J^+(t_0,\varphi_1)$ - the maximal interval of the type $(t_0,\beta)$, at which the solution $x(t;t_0,\varphi_1)$ is defined. \begin{theorem} \label{thm3} Let {\rm (H4)} and the following conditions hold: \begin{enumerate} \item The matrix functions $A(t)$ and $B(t)$ are continuous for $t \in (t_0, \infty)$. \item $t-h(t) \to \infty$ as $t \to \infty$. \item For each $k=1,2,\dots$ the elements of the matrix $C_k$ are nonnegative. \item There exists a continuous real $(n\times{n})$ matrix $D(t)$, $t \in (t_0, \infty)$, which is symmetric, positive definite, differentiable for $t\neq{\tau_k(x(t))}$, $k=1,2,\dots$ and such that for each $k=1,2\dots$, \begin{gather} x^T [A^T (t)D(t)+D(t)A(t)+\dot {D} (t)]x \leq{- c(t)} |x|^2,\quad x\in \mathbb{R}^n,\; t\neq{\tau_k(x(t))}, \label{e12} \\ x^T [C_k^T D(t)+D(t)C_k+C_k^T D(t) C_k]x \leq{0}, \quad t = {\tau_k(x(t))}, \label{e13} \end{gather} where $c(t)>0$ is a continuous function. \item There exists an integrally positive function $\lambda(t)$ such that \begin{gather} d(t)=c(t)-\max\{\alpha (t)\lambda(t), \beta(t) \lambda(t)\}\geq{0}, \label{e14} \\ \frac {2 \beta ^{1/2}(t)} {\alpha ^{1/2}(t-h(t))} |D(t)B(t)|\leq{d(t)}, \label{e15} \end{gather} where $\alpha (t)$ and $\beta(t)$ are respectively the smallest and the greatest eigenvalues of matrix $D(t)$. \end{enumerate} Then the set $M=[\tau-t_0,\infty) \times {\{x\in{\mathbb{R}^n}: x\leq{0}\}}$ is uniformly globally asymptotically stable with respect to system \eqref{e10}. \end{theorem} \begin{proof} Consider the function $$ V(t,x)=\begin{cases} x^T D(t)x,&\text{for } x>0,\\ 0, &\text{for } x\leq{0}. \end{cases} $$ From the condition that $D(t)$ is real symmetric matrix it follows that for $x \in \mathbb{R}^n$ and $x \neq 0$ it holds \begin{equation} \alpha (t)|x|^2 \leq x^T D(t) x \leq \beta(t)|x|^2.\label{e16} \end{equation} From thins inequalities it follows that condition 1 of Theorem \ref{thm1} is satisfied. For the chosen function $V(t,x)$ the set $\Omega_1$ is $$ \Omega_1=\{x\in{PC[I_0,\mathbb{R}^n]}: x^T (s) D(s) x(s) \leq x^T (t) D(t) x(t),\; \tau \leq{s}\leq{t},\; t\in{I_0}\}. $$ For $t>t_0$ and $x\in{\Omega_1}$ the following inequalities are valid: \begin{align*} \alpha (t-h(t)) |x(t-h(t))|^2 &\leq x^T (t-h(t)) D(t-h(t)) x(t-h(t))\\ &\leq x^T (t) D(t) x(t)\leq \beta (t) |x(t)|^2, \end{align*} from which we obtain the estimate \begin{equation} |x(t-h(t))| \leq \frac { \beta ^{1/2}(t)} {\alpha ^{1/2}(t-h(t))} |x(t)|. \label{e17} \end{equation} Let $t\neq{\tau_k(x(t))}$ and $x\in{\Omega_1}$. >From \eqref{e12}, \eqref{e14}, \eqref{e15} and \eqref{e17}, we have \begin{align*} D_{-}V(t,x(t))&=\begin{cases} -c(t)|x(t)|^2 + 2|D(t)B(t)| |x(t)| |x(t-h(t))|,& x(t)>0,\\ 0,& x(t)\leq{0} \end{cases} \\ &\leq \begin{cases} -[c(t)-d(t)] |x(t)|^2, & x(t)>0,\\ 0,& x(t)\leq{0} \end{cases} \\ &\leq -\lambda(t){V}(t,x(t)). \end{align*} Let $t = {\tau_k(x(t))}$. Then from \eqref{e13} we have \begin{align*} &V(t+0,x(t)+C_k x(t))\\ &=\begin{cases} (x^T (t)+x^T (t)C_k^T) D(t)(x(t)+C_k x(t)), &x>0,\\ 0, &x(t)\leq{0} \end{cases} \\ &=\begin{cases} x^T (t) D(t) x(t) + x^T (t) [C_k^T D(t) +D(t)C_k + C_k^T D(t)C_k ] x(t), &x(t)>0,\\ 0,& x(t)\leq 0 \end{cases} \\ &\leq{V}(t,x(t)). \end{align*} Thus we have checked that all the conditions of Theorem \ref{thm2} are satisfied. Hence the set $M=[\tau-t_0,\infty) \times {\{x\in{\mathbb{R}^n}: x\leq{0}\}}$ is uniformly globally asymptotically stable with respect to system \eqref{e10}. \end{proof} \begin{thebibliography}{00} \bibitem{b1} {D.\ D. Bainov, A. B. Dishliev}; \emph{The Phenomenon ``Beating'' of the Solutions of Impulsive Functional Differential Equations}, Communications in Applied Analysis {\bf 1} (1997), 435-441. \bibitem{b2} {D. D. Bainov, P. S. Simeonov}; \emph{Systems with Impulse Effect: Stability, Theory and Applications,} Ellis Horwood, Chichester, 1989. \bibitem{b3}{D. D. Bainov, P. S. Simeonov}; \emph{Theory of Impulsive Differential Equations: Periodic Solutions and Applications,} Longman, Harlow, 1993. \bibitem{b4} {D. D. Bainov, I. M. Stamova}; \emph{Lipschitz Stability of Impulsive Functional Differential Equations}, ANZIAM J. {\bf 42} (2001), 504-515. \bibitem{b5} {D. D. Bainov, I. M. Stamova}; \emph{Strong Stability of Impulsive Differential-Difference Equations}, PanAmerican Mathematical Journal {\bf 9} (1999), 87-95. \bibitem {h1} {J. K. Hale}; \emph{Theory of Functional Differential Equations}, Springer-Verlag, New York, Heidelberg, Berlin, 1977. \bibitem {h2} {J. K. Hale, V. Lunel}; \emph{Introduction to Functional Differential Equations}, Springer-Verlag, 1993. \bibitem {k1}{V. B. Kolmanovskii, V. R. Nosov}; \emph{Stability of Functional Differential Equations}, Academic Press, 1986. \bibitem{l1} {V. Lakshmikantham, D. D. Bainov and P. S. Simeonov}; \emph{Theory of Impulsive Differential Equations}, World Scientific, Singapore, 1989. \bibitem{l2}{V. Lakshmikantham, S. Leela and A. A. Martynyuk}; \emph{Stability Analysis of Nonlinear Systems}, Marcel Dekker Inc., New York, 1989. \bibitem {l3}{Z. Luo, J. Shen}; \emph{Stability Results for Impulsive Functional Differential Equations with Infinite Delays}, Journal of Computational and Applied Mathematics {\bf 131} (2001), 55-64. \bibitem{r1} {B. S. Razumikhin}; \emph{Stability of Systems with Retardation}, Nauka, Moscow, 1988 (in Russian). \bibitem{s1} {I. M. Stamova}; \emph{Stability Theorems of Perturbed Linear Impulsive Equations,} Mathematical Science Research Journal 6 {\bf 2} (2002), 96-103. \bibitem{s2} {I. M. Stamova, G. T. Stamov}; \emph{Lyapunov-Razumikhin Method for Impulsive Functional Differential Equations and Applications to the Population Dynamics}, Journal of Computational and Applied Mathematics {\bf 130} (2001), 163-171. \bibitem {y1} {J. Yan, J. Shen}; \emph{Impulsive Stabilization of Impulsive Functional Differential Equations by Lyapunov-Razumikhin Functions}; Nonlinear Analysis {\bf 37} (1999), 245-255. \end{thebibliography} \end{document}