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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 262, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/262\hfil Practical stability]
{Practical stability of linear switched impulsive
system with time delay}

\author[S. Li, W. Feng \hfil EJDE-2014/262\hfilneg]
{Shao'e Li, Weizhen Feng}  % in alphabetical order

\address{Shao'e Li \newline
School of Mathematical Sciences,
South China Normal University, Guangzhou 510631, China}
\email{a15017509268@163.com}

\address{Weizhen Feng \newline
School of Mathematical Sciences,
South China Normal University, Guangzhou 510631, China}
\email{Fengweizhen2@126.com}

\thanks{Submitted August 25, 2014. Published December 17, 2014.}
\subjclass[2000]{34K45, 34K34, 34D99}
\keywords{Linear switched impulsive system with time delay; practical stability;
 \hfill\break\indent Halanay inequality}

\begin{abstract}
 This article concerns the study of practical stability of linear switched
 impulsive systems with time delay. By using Lyapunov functions and
 the extended Halanay inequality, we establish sufficient conditions for
 the practical stability and uniform practical stability of a linear switched
 impulsive system  with time delay. The last section provides some
 illustrative examples.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Recently, there has been considerable research on switched impulsive systems
with time delay. However, most of them is about Lyapunov stability \cite{x1,y1},
but not practical stability.  Li \cite{l2} clarified the different
definitions of practical stability, and gave some criteria for the practical
stability of switched impulsive system without time delay.
The book \cite{z1} provides conditions on practical stability of various systems,
including ordinary differential equations, impulsive differential equations,
functional differential equations.  But there has  been little study on
practical stability of switched impulsive systems with time delay.
In this article we fill this gap.

First, we introduce Halanay inequality (see Lemma \ref{lem3.1}).
From this  inequality, we gain of a upper estimate on a function  $u(t)$,
which decrease exponentially with time.
Then this estimate can be applied to the study of exponential stability,
boundedness,  practical stability, etc.
As in \cite{a1,c1,l3,s1, w1,w2}, utilizing an extended Halanay inequality, we study
Lyapunov stability and attractivity for delay differential systems,
impulsive systems  with delay, switched systems with delay and difference equations.

To adapt the extended Halanay inequality to linear switched impulsive system
with time delay, we establish multiple Lyapunov functions and revise some
conditions of the extended Halanay inequality. Also by utilizing the comparison
method and the method of segmentation, we settle the problem of discontinuity
caused by impulses and switches.
Then we give sufficient conditions for practical stability of linear
switched impulsive system with time delay, where the influence of delays,
impulses, and switches is considered. We strive to conclude the coupling
relation of the delay, impulses and the dwell-time.
 What is more, we distinguish between the restriction on the dwell-time of
every activation for good subsystems and  that for bad subsystems,
where the good subsystem denotes the one which is practically stable,
and the bad one just the opposite.
Lastly, we provide some illustrative examples and the simulations.

\section{Preliminaries}

It is convenient to establish some notation here.
Let $\mathbb{R}^{+}$ denote the set of all nonnegative real numbers,
 $\mathbb{R}^n$ the n-dimensional real space equipped with Euclidean
norm $|\cdot|$. Denote by $\mathbb{N}_{+}$ the set of all positive integers,
and $ \mathbb{N}=\mathbb{N}_{+}\cup \{0\}$. Let
$\Lambda=\{1,2,\dots, m\}$, where $m\in \mathbb{N}_{+}$.
If $M=(m_{ij})_{n\times m}$ is a matrix, we write the norm of M as
$|M|=\sqrt{\sum_{1\leq i\leq n,1\leq j\leq m}^{} m_{ij}^2}$, and the
transposition of $M$ as $M^T$. Denote by $\lambda_{\rm max}(M)$ the greatest
eigenvalue of $M$, and $\lambda_{\rm min}(M)$ the minimum eigenvalue. Set
$x(t^+)=\lim_{s\to t^+}x(s)$.
Let $r>0$, and  $PC([-r,0])$ be the Banach space of piecewise continuous
functions with supremum norm $\|\cdot\|$. If $x\in PC([t_0,+\infty))$, let
$$
\dot{x}(t)=\lim_{h\to 0^-}{\dfrac{x(t+h)-x(t)}{h}}.
$$

Consider $m$ subsystems with delay,
\begin{equation}\label{1}
\begin{gathered}
\dot{x}(t)=f_i(t,x(t),x(t-r)),\quad i=1,2,\dots,m,\,\; m\in \mathbb{N}_{+},\\
x_{t_0}=\varphi
\end{gathered}
\end{equation}
the switches
\begin{equation}\label{2}
S=\{(\tau_k,i_k): i_k\in\Lambda=\{1,2,\dots, m\},\tau_k >0,\;k\in\mathbb{N}_{+}\},
\end{equation}
and the impulses
\begin{equation}\label{3}
x(t_k^+)=I_k (x(t_k)),\quad k=1,2,\dots,
\end{equation}
where $x\in \mathbb{R}^{n}$,
 $\varphi\in C([-r,0],\mathbb{R})$,
$f_i\in C(\mathbb{R}^+\times \mathbb{R}^{n}\times \mathbb{R}^{n},\mathbb{R}^{n})$,
$i\in\Lambda$, $I_k\in C(\mathbb{R}^{n},\mathbb{R}^{n})$,
$k\in\mathbb{N}_+$.
Here $\tau_k>0$ denotes switching intervals.
For any $t_0\in\mathbb{R}^{+}$,  $t_k=t_0+\sum_{i=1}^{k}\tau_i$ denotes
switching instants, which satisfies $ \lim_{k\to+\infty}t_{k}=+\infty$.
We assume that $f_{i}(t,0)=0$, $I_k(0)=0$ for any
$t\geq 0,k\in \mathbb{N}_{+},i\in\Lambda$.

According to \eqref{1}-\eqref{3}, we write switched impulsive systems with
 time delay as:
\begin{equation}\label{z1}
\begin{gathered}
\dot{x}(t)=f_{i_k}(t,x(t),x(t-r)), \quad t\in (t_{k-1},t_k]\\
x(t_k^+)=I_k(x(t_k)),\quad k=\mathbb{N}_+\\
x_{t_0}=\varphi.
\end{gathered}
\end{equation}


\begin{remark} \label{rmk2.1} \rm
We assume throughout this paper that solution of \eqref{z1} is unique
and of global existence \cite{l1,y1}.
\end{remark}


\begin{definition} \label{def2.1}  \rm
 Given $(\lambda,A)$ with $0<\lambda<A$,  system \eqref{z1} is said to be
\begin{itemize}
\item[(i)] $\lambda$-$ A$-practically stable, if  $\|\varphi\|<\lambda$
implies $|x(t,t_0^+,x_0)|<A$ for all $t\geq t_0$, and some $t_0\in\mathbb{R}^{+}$.

\item[(ii)] $\lambda$-$ A$-uniformly practically stable, if
$\|\varphi\|<\lambda$ implies $|x(t,t_0^+,x_0)|<A$, for all $t\geq t_0$, and
 every $t_0\in\mathbb{R}^{+}$.
\end{itemize}
\end{definition}

\section{Lemmas}

In this section, we provide some results that are needed in Section 4.

\begin{lemma}[Halanay inequality \cite{c1}] \label{lem3.1}
 If $r\geq0$, $a>b>0$, $u(t)$ is a continuous function satisfying
 $u(t)\geq 0$, and
\[
D^+u(t)\leq -au(t)+b\sup_{-r\leq\theta\leq0}u(t+\theta),\quad t\geq t_0,
\]
then $u(t)\leq \sup_{-r\leq\theta\leq0}u(t_0+\theta)e^{-\mu(t-t_0)}$, $t\geq t_0$,
where $\mu>0$ and $\mu-a+be^{\mu r}=0$.
\end{lemma}

\begin{lemma} \label{lem3.2}
Consider  $(\lambda,A)$ with $0<\lambda<A$. If $2a+b^2<-1$, then the system
\begin{equation}\label{y32}
\begin{gathered}
\dot{x}(t)=ax(t)+bx(t-r)\\
x_{t_0}=\varphi
\end{gathered}
\end{equation}
is $\lambda$-$ A$-practically stable.
\end{lemma}

\begin{proof}
Let $x(t)$ denote the solution of \eqref{y32}, and define $V(t)=x^2(t)$. Then
\begin{align*}
\dot{V}(t)
&=2x(t)\dot{x}(t)\\
&=2x(t)[ax(t)+bx(t-r)]\\
&\leq (2a+b^2)x^2(t)+x^2(t-r)\\
&\leq (2a+b^2)V(x(t))+\sup_{\theta\in [-r,0]}^{}V(t+\theta).
\end{align*}
By the Halanay inequality and $2a+b^2<-1$,
$$
V(x(t))\leq\sup_{\theta\in [-r,0]}^{}V(t_0+\theta)\,e^{-u(t-t_0)},\quad t\geq t_0,
$$
where $u>0$ and $u+2a+b^2+e^{ur}=0$.
Consequently, when $\|\varphi\|<\lambda$,
$$
|x(t)|=V^{1/2}(t)\leq\sup_{\theta\in [-r,0]}^{}V^{1/2}(t_0+\theta)<\lambda <A.
$$
The proof is complete.
\end{proof}

\begin{lemma}[\cite{l3}] \label{lem3.3}
 Consider the system
\begin{equation}\label{y4}
\begin{gathered}
D^+f(t)\leq pf(t)+q\bar{f}(t),\quad t\in[t_0,+\infty)
 \backslash\{t_k,k\in \mathbb{N}_{+}\}\\
f(t_k)\leq d_kf(t_k^-),\quad k\in \mathbb{N}_{+},
\end{gathered}
\end{equation}
where
\begin{equation}\label{y1}
\begin{gathered}
t_k\in\mathbb{R}^{+},\quad  t_{k+1}>t_k,\; k\in\mathbb{N},\;
 \lim_{k\to+\infty}t_{k}=+\infty,\\
p\in\mathbb{R},\quad  q\geq0,\; r>0,\; \delta>1,\\
f\in PC(\mathbb{R},\mathbb{R}^{+}),\quad
\bar{f}(t)=\sup\{f(s): t-r\leq s\leq t \}.
\end{gathered}
\end{equation}
Assume that
\begin{gather}\label{y2}
p+q\delta<\frac{\ln\delta}{\sigma},\quad \text{where }
\sigma=\sup\{t_{n+1}-t_n: n\in\mathbb{N}\}<\infty; \\
\label{y3}
0<\lambda<\frac{\ln \delta}{\sigma}-p-q\delta e^{\lambda r}.
\end{gather}
Then let $f\in PC(\mathbb{R},\mathbb{R}^{+})$ be the solution of \eqref{y4},
and define
\begin{equation}\label{y5}
g(t)=\begin{cases}
f(t)e^{\lambda(t-t_0)},&t> t_0,\\
f(t), &t_0-r\leq t\leq t_0.
\end{cases}
\end{equation}
If $t_n\leq t_*< t^*< t_{n+1}$ for some $n\in\mathbb{N}$, and
$\delta g(t)\geq g(s)$ for any $s\in [t_0-\tau,t^*]$ and $t\in [t_*,t^*]$, then
$\delta>g(t^*)/g(t_*)$.
\end{lemma}

Now, we adapt the conclusion of Lemma \ref{lem3.3} to the linear switched impulsive
 system with time delay.

\begin{lemma}[Extended inequality]  \label{lem3.4}
Replace \eqref{y4} in Lemma \ref{lem3.3} by
\begin{equation}\label{y41}
\begin{gathered}
D^-f(t)\leq p_{i_k}f(t)+q_{i_k}\bar{f}(t),\quad t\in (t_{k-1},t_k]\\
f(t_k^+)\leq d_kf(t_k),\quad  k\in \mathbf{N_+},
\end{gathered}
\end{equation}
where ${i_k}\in\Lambda, k\in\mathbb{N}_+$, $p_{i_b}=p, q_{i_b}=q$, and
$b$ is a given positive integer. Let $f(t)$ be the solution of \eqref{y41}.
And suppose there is a $\lambda>0$ such that \eqref{y1}, \eqref{y3} and
\eqref{y5} hold. If $t_{b-1}\leq t_*< t^*\leq t_{b}$, and
$\delta g(t)\geq g(s)$ for all $s\in [t_b-r,t^*],t\in (t_*,t^*]$. Then
$\delta>g(t^*)/g(t_*^+)$.
\end{lemma}

\begin{proof}
For any $t\in[t_0,+\infty)$, we can find a $u_t\in[t-r,t]$ such that
$\bar{f}(t)=f(u_t)$. Note that $e^{\lambda (t-t_0)}\leq 1$, for every
 $t\in [t_0-r,t_0]$. Then,
\begin{equation}\label{yinli34ty}
f(t)e^{\lambda (t-t_0)}\leq g(t), \quad t\in[t_0-r,+\infty).
\end{equation}
Consider $t\in (t_*,t^*]$, then
\begin{equation} \label{yinli34te}
\begin{aligned}
D^-g(t)
&=(D^-f(t))e^{ \lambda(t-t_0)}+\lambda f(t)e^{\lambda(t-t_0)}\\
&\leq (p_{i_b}f(t)+q_{i_b}\bar{f}(t))e^{ \lambda(t-t_0)}
 +\lambda f(t)e^{\lambda(t-t_0)}\\
&= (pf(t)+q\bar{f}(t))e^{ \lambda(t-t_0)}+\lambda f(t)e^{\lambda(t-t_0)}\\
&= (\lambda+p)f(t)e^{\lambda(t-t_0)}+qf(u_t)e^{\lambda(u_t-t_0)}e^{\lambda(t-u_t)},
\quad t\in(t_*,t^*].
\end{aligned}
\end{equation}
From \eqref{yinli34ty}, \eqref{yinli34te} and the assumption in the lemma, we have
\begin{equation} \label{yinli34ts}
\begin{aligned}
D^-g(t)&\leq (\lambda+p)f(t)e^{\lambda(t-t_0)}+qg(u_t)e^{\lambda(t-u_t)}\\
&\leq (\lambda+p)g(t)+q\delta g(t)e^{\lambda \tau}\\
&=(\lambda+p+q\delta e^{\lambda \tau})g(t),\quad t\in(t_*,t^*].
\end{aligned}
\end{equation}
By \eqref{yinli34ts} and \eqref{y3}, we have
\[
\int_{t_*}^{t^*}\frac{dg(t)}{g(t)}\leq \int_{t_*}^{t^*}
(\lambda+p+q\delta e^{\lambda \tau})dt
=(\lambda+p+q\delta e^{\lambda \tau})(t^*-t_*)<\ln \delta.
\]
Note that $g(t_*)\neq g(t_*^+)$, if $t_*=t_b$. Then
\[
\int_{t_*}^{t^*}\frac{dg(t)}{g(t)}=\ln g(t^*)-\ln g(t_*^+)
=\ln(\frac{g(t^*)}{g(t_*^+)}).
\]
It follows that $\delta>g(t^*)/g(t_*^+)$.
\end{proof}

\begin{lemma}[Comparison theorem] \label{lem3.5}
Consider two systems:
\begin{equation}\label{y7}
\begin{gathered}
\dot{x}(t)=f_{i_k}(t,x(t),x(t-r)),\quad t\in(t_{k-1},t_k]\\
x(t_{k}^+)=c_k x(t_{k}),\quad k\in \mathbb{N}_{+}\\
x_{t_0}=\varphi_1,
\end{gathered}
\end{equation}
and
\begin{equation}\label{y8}
\begin{gathered}
\dot{y}(t)=g_{i_k}(t,y(t),y(t-r)):=a_{i_k}y(t)+b_{i_k}y(t-r),\quad
 t\in(t_{k-1},t_k]\\
y(t_{k}^+)=d_k y(t_{k}),\quad k\in \mathbb{N}_{+}\\
y_{t_0}=\varphi_2,
\end{gathered}
\end{equation}
where $a_{i_k}\in \mathbb{R}$; $x,y,r,c_k,d_k,b_{i_k}\in \mathbb{R}^+$;
$f_{i_k}(t,u,v), g_{i_k}(t,u,v): \mathbb{R}^+\times\mathbb{R}^+
\times\mathbb{R}^+\to \mathbb{R}$ are  continuous functions with
$k\in\mathbb{N}_+$ and ${i_k}\in\Lambda$.
Here $\varphi_1, \varphi_2\in C([-r,0],\mathbb{R}^+)$,
$ f_i(t,0,0)=0$, $i\in\Lambda$. If for every
$u,v\in\mathbb{R}^+$, $t\in(t_k,t_{k+1}]$ and $s\in[-r,0]$, we have
\[
f_{i_k}(t,u,v)\leq g_{i_k}(t,u,v),\quad
c_k\leq d_k, \quad \varphi_1(s)\leq \varphi_2(s),
\]
then, $x(t)\leq y(t)$ for each $t>t_0$, where $x(t)$ and $y(t)$
are the solutions of \eqref{y7} and \eqref{y8} respectively.
\end{lemma}

\begin{proof}
(I) when $t\in(t_0,t_1]$, we have $x(t)\leq y(t)$ by comparison theorem
of functional differential equation \cite{s1}.

(II) Assume that $x(t)\leq y(t)$, where $t\in(t_j,t_{j+1}]$ an $j=0,1,2,\dots,k-1$.
Then, we need to prove
\begin{equation}\label{xg1}
x(t)\leq y(t), \quad t\in(t_k,t_{k+1}].
\end{equation}
Firstly, we claim that $x(t)\leq y(t)$ for each $t\in(t_k,t_{k+1}]$,
if
$$
f_{i_{k+1}}(t,u,v)<g_{i_{k+1}}(t,u,v),\quad
c_k\leq d_k, \quad
x(s)\leq y(s),
$$
for
$ t\in(t_k,t_{k+1}],\;s\in(t_k-r,t_{k}]$.
Otherwise, there is $\bar{t}\in(t_k,t_{k+1}]$, such that
$x(\bar{t})>y(\bar{t})$. Define
$$
t^*=\inf\{t: x(t)>y(t), t\in(t_k,t_{k+1}]\}.
$$
Because $x(t)$ and $y(t)$ are continuous on $(t_k,t_{k+1}]$, and
$x(t_k^+)=c_k x(t_k)\leq d_k y(t_k)=y(t_k^+)$, we have
$$
x(t^*)=y(t^*),\quad x(t)\leq y(t), \quad t\in[t_k-r,t^*],
$$
where $t^*\in[t_k,t_{k+1})$. Hence, $\dot{x}(t^*)\geq\dot{y}(t^*)$.
On the other hand, if $t^*\in(t_k,t_{k+1})$,
$$
f_{i_{k+1}}(t^*,x(t^*),x(t^*-r)) <g_{i_{k+1}}(t^*,x(t^*),x(t^*-r))\leq
g_{i_{k+1}}(t^*,y(t^*),y(t^*-r)).
$$
 Namely, $\dot{x}(t^*)<\dot{y}(t^*)$, if $t^*\in(t_k,t_{k+1})$.
Also if $t^*=t_k$, we can have $\dot{x}(t^*)<\dot{y}(t^*)$ similarly.
This contradiction proves the result in this case.

Then we need to study system \eqref{y8} on $(t_k,t_{k+1}]$.
Taking $t_k$ as the initial time, we rewrite system \eqref{y8} as
\begin{equation}\label{xg2}
\begin{gathered}
\dot{y}(t)=g(t,y(t),y(t-r)),\quad t\in(t_{k},t_{k+1}]\\
y(t_{k}^+)=d_k y(t_{k})\\
y_{t_k}=\varphi_3,
\end{gathered}
\end{equation}
where $g(t,y(t),y(t-r))=a_{i_{k+1}}y(t)+b_{i_{k+1}}y(t-r)$,
$\varphi_3(s)= y(t_k+s), s\in[-r,0]$. Denote by $\tilde{y}(t)$
 the solution of \eqref{xg2}. Obviously, $y(t)=\tilde{y}(t)$ if
$t\in[t_k-r,t_{k+1}]$. If $g$ in \eqref{xg2}is replaced by $g+\frac{1}{n}$
for any $n\in\mathbb{N}_+$, then we rewrite the solution as
 $y_{n}(t)$ respectively.
By the results of previous paragraph, we  conclude that
$$
x(t)\leq y_{n}(t), \quad t\in(t_k,t_{k+1}].
$$
So,  to prove \eqref{xg1}, we need only to prove that
\begin{equation}\label{bjdl3}
{y}_{n}(t)\to y(t), \quad \text{as } n\to +\infty,
\end{equation}
for every $t\in(t_k,t_{k+1}]$. Define $z_{n}(t)=y_{n}(t)-\tilde{y}(t)$, then
\begin{gather*}
\dot{z}_{n}(t)=a_{i_{k+1}}z_{n}(t)+b_{i_{k+1}}z_{n}(t-r)+\frac{1}{n},\quad
 t\in(t_{k},t_{k+1}]\\
z_{n_{t_k}}=\varphi_4(s),
\end{gather*}
where $\varphi_4(s)= 0$, $s\in[-r,0]$. By
\cite[Theorem 2.2 Chap. 2]{h1}], we have
${z}_{n}(t)\to 0$ as $n\to +\infty$, for each $t\in(t_k,t_{k+1}]$.
Namely, \eqref{bjdl3} is true.
So, $x(t)\leq y(t)$, for each $t\in(t_k,t_{k+1}]$. By mathematical induction,
$x(t)\leq y(t)$, if $ t>t_0$.
\end{proof}

\begin{lemma}[\cite{w1}] \label{lem3.6}
Consider the system
\begin{equation}\label{y6}
\begin{gathered}
\dot{x}(t)=a(t)x(t)+b(t)x(t-r),\\
x_{t_0}=\phi,
\end{gathered}
\end{equation}
where $a(t),b(t)\in C(\mathbb{R}^+,R)$, $r>0$ is a constant.
Assume $-\frac{1}{2r}\leq a(t)+b(t+r)\leq -rb^2(t+r)$.
Let $x(t)=x(t,t_0,\phi)$ be the solution of \eqref{y6} on $[t_0,+\infty)$. Then
\begin{gather*}
|x(t)|\leq \|\phi\|\Big(1+\int_{t_0}^{t_0+r/2}|b(u)|du\Big)
e^{\int_{t_0}^{t}a(s)ds},\quad t\in (t_0,t_0+r/2);\\
|x(t)|\leq \sqrt{6V(t_0)} e^{\frac{1}{2}\int_{t_0}^{t-r/2}[a(s)+b(s+r)]ds},\quad
t\in [t_0+r/2,+\infty).
\end{gather*}
where $V(t_0)=\big[x(t_0)+\int_{t_0-r}^{t_0}b(s+r)x(s)ds\big]^2
+\int_{-r}^{0}\int_{t_0+s}^{t_0}b^2(z+r)x^2(z)\,dz\,ds$.
\end{lemma}

\begin{corollary}\label{coro3.1}
If we add an impulse $x(t_0^+)=d_0 x(t_0)$ at the initial time $t_0$,
and replace the initial function $\phi$ by $\varphi\in PC(-r,0)$ in
Lemma \ref{lem3.5}, then the conclusion becomes
\begin{gather*}
|x(t)|\leq \max\{|x(t_0^+)|,\|\varphi\|\}\Big(1+\int_{t_0}^{t_0+r/2}|b(u)|du\Big)
 e^{\int_{t_0}^{t}a(s)ds},\quad t\in (t_0,t_0+r/2);\\
|x(t)|\leq \sqrt{6V(t_0^+)} e^{\frac{1}{2}\int_{t_0}^{t-r/2}[a(s)+b(s+r)]ds},
\quad t\in [t_0+r/2,+\infty),
\end{gather*}
where
\[
V(u)=\big[x(u)+\int_{u-r}^{u}b(s+r)x(s)ds\big]^2
+\int_{-r}^{0}\int_{u+s}^{u}b^2(z+r)x^2(z)\,dz\,ds.
\]
\end{corollary}

\begin{remark} \label{rmk3.1}   \rm
Since the proof of Lemma \ref{lem3.6} is not dependent on the continuity of the
initial function, we can prove Corollary \ref{coro3.1} similarly.
\end{remark}

\section{Practical stability results}

Now, we are ready to give results on practical stability of the systems,
including one-dimensional systems and n-dimensional ones.
Firstly, consider the one-dimensional system with constant coefficients.
That is, $f_{i_k}(t,x(t),x(t-r))=a_{i_k}x(t)+b_{i_k}x(t-r)$ in \eqref{z1}.
\begin{equation}\label{c1}
\begin{gathered}
\dot{x}(t)=a_{i_k}x(t)+b_{i_k}x(t-r),\quad t\in(t_{k-1},t_k]\\
x(t_{k}^+)=d_k x(t_{k}),\quad k\in \mathbb{N}_{+}\\
x_{t_0}=\varphi,
\end{gathered}
\end{equation}
where $a_{i_k},b_{i_k}\in \mathbb{R}$,
$r,d_k, t_0\in \mathbb{R}^+$, ${i_k}\in \Lambda$,
$k\in\mathbb{N}_+$, $\varphi\in C([-r,0],\mathbb{R})$.

The subsystem of \eqref{c1} is
\begin{equation}\label{cz1}
\begin{gathered}
\dot{x}(t)=a_{i}x(t)+b_{i}x(t-r)\\
x_{t_0}=\varphi,
\end{gathered}
\end{equation}
where $i\in\Lambda$. If $2a_i+b_i^2<-1$, then the subsystem is
practical stable by Lemma \ref{lem3.2}, and we call it a good subsystem.
Otherwise, we can not guarantee practical stability of it.
So we call it a bad subsystem. In order to guarantee practical
stability of \eqref{c1}, it is sensible that there would be stricter
restriction on the dwell time of bad subsystems than on that of good ones,
as Theorem \ref{thm4.1} shows. For convenience, we assume that the first $m_1$
subsystems are good subsystems, and the rest are bad ones.


\begin{theorem} \label{thm4.1}
Consider $(\lambda,A)$ with $0<\lambda<A$. If there exist constants
$\delta_1,\delta_2>1$ and $\beta>0$ which satisfy
\begin{gather*}
\tilde{\delta}_k=\begin{cases}
\delta_1,& \text{if }i_k\leq m_1\\
\delta_2,& \text{if }i_k> m_1;
\end{cases}
\quad \beta<\frac{\ln\delta_i}{\sigma_i}-p_i-\delta_ie^{\beta r},\quad i=1,2;
\\
\prod_{j=0}^{k}(\tilde{\delta}_{j+1} \tilde{d}_j^2)e^{-\beta(t_k-t_0)}
\leq (\frac{A}{\lambda})^2,\quad k\in \mathbb{N},
\end{gather*}
where $m_1\in\Lambda$, $\Lambda_1:=\{1,2,\dots,m_1\}$,
$\Lambda_2:=\{m_1+1, m_1+2,\dots,m\}$,
\begin{gather*}
2a_i+b_i^2<-1,\; i\in\Lambda_1; \quad
2a_i+b_i^2\geq-1,\; i\in\Lambda_2;\\
p_1=\max\{2a_i+b_i^2: i\in\Lambda_1\},\quad
p_2=\max\{2a_i+b_i^2: i\in\Lambda_2\};\\
\sigma_1=\sup\{t_{k}-t_{k-1}: i_k\in\Lambda_1\},\quad
\sigma_2=\sup\{t_{k}-t_{k-1}: i_k\in\Lambda_2\};\\
\tilde{d}_0=1,\quad  \tilde{d}_k=\max\{d_k, (\tilde{\delta}_{k+1})^{-1/2}\},
\; k\in \mathbb{N}_+,
\end{gather*}
then system \eqref{c1} is $\lambda$-$ A$-uniformly practically stable.
\end{theorem}

\begin{proof}
Let $x(t)$ be the solution of \eqref{cz1} and set the function $V(t)=x^2(t)$.
Then the derivative of $V(t)$ with respect to each subsystem is:
\begin{align*}
\dot{V(t)}&=2x(t)\dot{x(t)}\\
&=2x(t)[ax(t)+bx(t-r)]\\
&\leq (2a_i+b_i^2)x(t)^2+x(t-r)^2\\
%&=(2a_i+b_i^2)V(t)+V(t-r)\\
&\leq (2a_i+b_i^2)V(t)+\sup_{\theta\in [-r,0]}^{}V(t_0+\theta).
\end{align*}
For any $t_0\in \mathbb{R}^+$, $\|\varphi\|<\lambda$, we have:
\begin{gather*}
\sup_{t\in [t_0-r,t_0]}^{}V(t)=\|\varphi\|^2<\lambda ^2;\\
V(t_k^+)=x^2(t_k^+)=d_k^2x^2(t_k)=d_k^2V(t_k),\quad k\in \mathbb{N}_+.
\end{gather*}
Define
\[
g_1(t)=\begin{cases}
V(t)e^{\beta(t-t_0)}, & t\in(t_0,+\infty]\\
V(t), & t\in[t_0-r,t_0].
\end{cases}
\]

\noindent\textbf{Case 1:}
For any $k\in \mathbb{N}_+$, $d_k\geq (\tilde{\delta}_{k+1})^{-1/2}$.

(I) Consider the condition $t\in (t_0,t_1]$. Then we have
$$
V(t_0^+)=d_0^2V(t_0)<\tilde{\delta}_1d_0^2
\sup_{t_0-r\leq s\leq t_0 }^{}V(s):= \alpha_0,
$$
where $d_0=1$. Note that
\[
g_1(t)< \tilde{\delta}_1d_0^2\sup_{t_0-r\leq s\leq t_0 }^{}V(s)= \alpha_0
\]
 for each $t\in[t_0-r,t_0]$ and $g_1(t)$ is continuous on $[t_0-r,t_1]$.
 We claim that $g_1(t)\leq\alpha_0$, for any $t\in[t_0-r,t_1]$.
If not, there is a $\tilde{t}_1\in(t_0,t_1]$ such that $g_1(\tilde{t}_1)>\alpha_0$.
Define
\begin{gather*}
t_1^*=\inf\{t\in(t_0,\tilde{t}_1]: g_1(t)>\alpha_0\},\\
t_{1*}=\sup\{t\in[t_0,t_1^*]: g_1(t)\leq d_0^2\sup_{t_0-r\leq s\leq t_0 }^{}V(s)\}
\end{gather*}
Hence, $t_0\leq t_{1*}<t_1^*< t_1$ and
$\tilde{\delta}_1 g_1(t)\geq g_1(s)$ for all $s\in [t_0-r,t_1^*]$,
$t\in [t_{1*},t_1^*]$. By Lemma \ref{lem3.4},
$\tilde{\delta}_1>\frac{g(t_1^*)}{g(t_{1*})}=\tilde{\delta}_1$.
This contradiction proves that
$$
g_1(t)\leq\alpha_0,\quad V(t)\leq\alpha_0e^{-\beta(t-t_0)},\quad t\in(t_0,t_1].
$$

(II) Assume that $V(t)\leq\alpha_{i}e^{-\beta(t-t_0)}$ for each
$t\in(t_i,t_{i+1}]$, where
$$
\alpha_{i}= \prod_{j=0}^{i}(\tilde{\delta}_{j+1} d_j^2)\sup_{t_0-r\leq s
\leq t_0 }^{}V(s), \quad i=0,1,\dots,k-1.
$$
Below we prove $V(t)\leq\alpha_{k}e^{-\beta(t-t_0)}$ for each $t\in(t_k,t_{k+1}]$.
Note that
$$
V(t_k^+)=d_k^2V(t_k)\leq d_k^2\alpha_{k-1}e^{-\beta(t_k-t_0)}
<\tilde{\delta}_{k+1}d_k^2\alpha_{k-1} e^{-\beta(t_k-t_0)}
:= \alpha_k e^{-\beta(t_k-t_0)}.
$$
Thus, $g_1(t_k^+)\leq d_k^2\alpha_{k-1}<\alpha_{k}$. Because
 $\{\alpha_k\}$ is nondecreasing, we have $g_1(t)\leq \alpha_k$ for each
$ t\in[t_k-r,t_k]$. We claim that $g_1(t)\leq\alpha_k$, if $t\in[t_k-r,t_{k+1}]$.
Otherwise, by the continuity of $g_1(t)$ on $(t_k,t_{k+1}]$, there is a
$\tilde{t}_k\in(t_k,t_{k+1}]$ such that $g_1(\tilde{t}_k)>\alpha_k$.
Define
\begin{gather*}
t_k^*=\inf\{t\in(t_k,\tilde{t}_k]: g_1(t)>\alpha_k\},\\
E_k= \{t\in(t_k,t_k^*]: g_1(t)\leq d_k^2\alpha_{k-1}\},\\
t_{k*}= \begin{cases}
t_k, & \text{if } E_k=\emptyset\\
\sup E_k, &\text{if }E_k\neq\emptyset.
\end{cases}
\end{gather*}
Hence, $t_k\leq t_{1*}<t_1^*\leq t_{k+1}$ and
 $\tilde{\delta}_{k+1} g_1(t)\geq g_1(s)$ for any
$s\in [t_k-r,t_k^*]$ and $t\in (t_{k*},t_k^*]$. By Lemma \ref{lem3.4},
$\tilde{\delta}_{k+1}>\frac{g(t_k^*)}{g(t_{k*}^+)}=\tilde{\delta}_{k+1}$.
This leads to a contradiction. So,
$$
g_1(t)\leq\alpha_k,\quad V(t)\leq\alpha_ke^{-\beta(t-t_0)},\quad t\in(t_k,t_{k+1}].
$$
By mathematical induction, $V(t)\leq\alpha_{k}e^{-\beta(t-t_0)}$ for every
$t\in(t_k,t_{k+1}]$ and $k\in \mathbb{N}$.
Since $d_k=\tilde{d}_k=\max\{d_k, (\tilde{\delta}_{k+1})^{-1/2}\}$, we have
$$
|x(t)|= V^{1/2}(t)\leq[\alpha_{k}e^{-\beta(t_k-t_0)}]^{1/2}
<\Big[\lambda^2 \prod_{j=0}^{k}(\tilde{\delta}_{j+1}
\tilde{d}_j^2)e^{-\beta(t_k-t_0)}\Big]^{1/2}\leq A,
$$
for every $t\in(t_k,t_{k+1}]$, $k\in \mathbb{N}$.
\smallskip

\noindent\textbf{Case 2:}
There is some $k_0\in \mathbb{N}_+$, such that
 $d_{k_0}< (\tilde{\delta}_{k_0+1})^{-1/2}$. We establish a new system
\begin{equation}\label{gzc1}
\begin{gathered}
\dot{y}(t)=(2a_{i_k}+b_{i_k}^2)y(t)+y(t-r),\quad t\in(t_{k-1},t_k]\\
y(t_{k}^+)=\tilde{d}_k^2 y(t_{k}),\quad k\in \mathbb{N}_{+}\\
y_{t_0}=\varphi^2,
\end{gathered}
\end{equation}
where $\tilde{d}_k=\max\{d_k, (\tilde{\delta}_{k+1})^{-1/2}\}$.
By Lemma \ref{lem3.5} and the results of Case 1,
$$
|x(t)|=V^{1/2}(t)\leq y^{1/2}(t)<A.
$$
So, system \eqref{c1} is $\lambda$-$ A$-uniformly practically stable.
\end{proof}

\begin{corollary} \label{coro4.1}
Consider $(\lambda,A)$ with $0<\lambda<A$. Assume that  there exist constants $\delta>1$
and $\beta>0$ satisfying:
\begin{gather*}
\beta<\frac{\ln\delta}{\sigma}-p-\delta e^{\beta r};\\
\delta^{k+1} \prod_{j=0}^{k}\tilde{d}_j^2e^{-\beta(t_k-t_0)}
\leq (\frac{A}{\lambda})^2,\quad k\in \mathbb{N},
\end{gather*}
where $p=\max\{2a_i+b_i^2: i\in\Lambda\}$,
$ d_0=1$, $\tilde{d}_k=\max\{d_k, (\tilde{\delta}_{k+1})^{-1/2}\}$,
$\sigma=\sup\{t_{k+1}-t_k: k\in\mathbb{N}\}$. Then system \eqref{c1} is
 $\lambda$-$ A$-uniformly practically stable.
\end{corollary}


Below we  study the one-dimensional system with variable coefficients.
Namely, $f_{i_k}(t,x(t),x(t-r))=a_{i_k}(t)x(t)+b_{i_k}(t)x(t-r)$ in \eqref{z1}.
\begin{equation}\label{b1}
\begin{gathered}
\dot{x}(t)=a_{i_k}(t)x(t)+b_{i_k}(t)x(t-r), \quad t\in(t_{k-1},t_k]\\
x(t_{k}^+)=d_k x(t_{k}),\quad k\in \mathbb{N}_{+}\\
x_{t_0}=\varphi,
\end{gathered}
\end{equation}
where $a_{i_k}(t), b_{i_k}(t):\mathbb{R}^+\to\mathbb{R}$ are continuous
functions, $d_k, r, t_0\in\mathbb{R}^+$, $k\in\mathbb{N}_+$.

\begin{theorem} \label{thm4.2}
Consider $(\lambda,A)$ with $0<\lambda<A$. If there exist
$t_0\in\mathbb{R}^{+}$ and $\sigma>0$ such that
\begin{gather*}
2a_{i_k}(t)+b^2_{i_k}(t)+1\leq-\sigma<0,\,t\in (t_{k-1},t_k],\\
 \prod_{i=1}^{k}\tilde{d}_i\leq\frac{A}{\lambda},\quad k\in\mathbb{N}_{+},
\end{gather*}
where $\tilde{d}_i=\max\{d_i,1\}$, then system \eqref{b1}
is $\lambda$-$ A$-practically stable.
\end{theorem}

\begin{proof}
For any $\varphi\in C([-r,0],\mathbb{R})$ and $\|\varphi\|<\lambda$,
denote by $x(t)$ the solution of \eqref{b1}. Set the function $V(t)=x^2(t)$.
Then the derivative of $V(t)$ with respect to each subsystem is:
\begin{align*}
\dot{V}(t)&=2x(t)\dot{x(t)}\\
&=2x(t)[a_i(t)x(t)+b_i(t)x(t-r)]\\
&\leq [2a_i(t)+b_i^2(t)]x(t)^2+x(t-r)^2\\
&\leq[2a_i(t)+b_i^2(t)]V(t)+\sup\{V(s): s\in [t-r,t]\}.
\end{align*}
Hence, $\dot{V}(t)\leq (2a_{i_k}(t)+b_{i_k}^2(t))V(t)+\sup \{V(s): s\in[t-r,t]\}$,
for each $t\in(t_{k-1},t_k],\,\,k\in \mathbb{N}_{+}$. Define
$$
G=\sup \{V(s): s\in[t_0-r,t_0]\}=\sup \{\varphi^2(s): s\in[-r,0]\}<\lambda^2.
 $$
For any given $\varepsilon\in(1,2)$, we have:

(I) Note that $V(t)$ is continuous on $(t_0-r,t_1]$ and
$V(t_0)\leq G<\varepsilon G:= \alpha_0$. Then we are to prove that
 $V(t)< \alpha_0$, for each $t\in(t_0,t_1]$.
If not, there is a $\bar{t}_0\in (t_0,t_1]$ such that
$V(t)<\alpha_0$ for each $t\in (t_0,\bar{t}_0)$ and $V(\bar{t}_0)=\alpha_0$.
 Hence, $\dot{V}(\bar{t}_0)\geq0$. But,
\begin{align*}
\dot{V}(\bar{t}_0)&\leq (2a_{i_1}(\bar{t}_0)+b_{i_1}^2(\bar{t}_0))V(\bar{t}_0)
 +\sup_{s\in [\bar{t}_0-r,\bar{t}_0]}^{} V(s)\\
&= [2a_{i_1}(\bar{t}_0)+b_{i_1}^2(\bar{t}_0)+1]\alpha_0\\
&\leq -\sigma \alpha_0< 0.
\end{align*}
This contradiction proves $V(t)<\alpha_0$ for $t\in (t_0,t_1]$.

(II)  Assume that
\[
V(t)<\prod_{i=0}^{j-1}\tilde{d}_i^2 \varepsilon G:= \alpha_{j-1}
\]
 for each $t\in (t_{j-1},t_j]$ and $j=1,2,\dots,k$, where
$\tilde{d}_0=1$. Note that $V(t)$ is continuous on $(t_k,t_{k+1}]$
and $V(t_k^+)\leq (\tilde{d}_kx(t_k))^2=\tilde{d}_k^2V(t_k)
<\tilde{d}_k^2\alpha_{k-1}= \alpha_k$. Then we  need to prove
\begin{equation}\label{thset1}
V(t)<\prod_{i=1}^{k}\tilde{d}_i^2\varepsilon G
:= \alpha_k,\quad t\in (t_k,t_{k+1}].
\end{equation}
If not,  there exists a $\bar{t}_k\in (t_k,t_{k+1}]$ such that
$V(t)<\alpha_k$ for each $t\in (t_k,\bar{t}_k)$ and $V(\bar{t}_k)=\alpha_k$.
Hence, $\dot{V}(\bar{t}_k)\geq0$. $\{\alpha_k\}$ is nondecreasing, so
$V(t)\leq \alpha_k$ for each $t\in[t_0-r,\bar{t}_k]$. It follows that
 \begin{align*}
 \dot{V}(\bar{t}_k)
&\leq \Big(2a_{i_{k+1}}(\bar{t}_k)+b_{i_{k+1}}^2(\bar{t}_k)\Big)V(\bar{t}_k)
 +\sup_{s\in [\bar{t}_k-r,\bar{t}_k]}^{} V(s)\\
&= [2a_{i_{k+1}}(\bar{t}_k)+b_{i_{k+1}}^2(\bar{t}_k)+1]\alpha_k\\
&\leq -\sigma \alpha_k< 0.
 \end{align*}
This contradiction proves \eqref{thset1}. By mathematical induction, we have
$$
V(t)<\alpha_k=\prod_{i=0}^{k}d_i^2\cdot\varepsilon G.
$$
for any $t\in (t_k,t_{k+1}],\,k\in \mathbb{N}$. Furthermore,
\begin{align*}
V(t)\leq\prod_{i=0}^{k}d_i^2 G<\prod_{i=0}^{k}d_i^2 \lambda^2\leq A^2,\quad
\forall t\in (t_k,t_{k+1}],\; k\in \mathbb{N}.
\end{align*}
Namely, $|x(t)|<A$ for $t\in [t_0,+\infty)$. So system \eqref{b1} is
$\lambda$-$ A$-practically stable.
\end{proof}

\begin{remark} \label{rmk4.1}   \rm
We can loosen the restriction on $d_k$ in Theorem \ref{thm4.2}, if the dwell
 time $\tau_k$ is big enough, $k\in \mathbb{N}_+$. As shown is Theorem \ref{thm4.3}.
\end{remark}

\begin{theorem} \label{thm4.3}
Consider $(\lambda,A)$ with $0<\lambda<A$. Assume that there are constants
$t_0\in\mathbb{R}^{+}$ and $ \sigma>0$ such that
$$
2a_{i_k}(t)+b^2_{i_k}(t)+1\leq-\sigma<0,\quad t\in (t_{k-1},t_k];
$$
and suppose
$$
\tau_k>r,\quad \prod_{i=1}^{k}\tilde{d}_i^2 e^{-u(t_k-t_0-kr)}
\leq\frac{A^2}{\lambda^2},\quad k\in\mathbb{N}_{+},
$$
where $\tau_k=t_k-t_{k-1}$, $\tilde{d}_i=max\{d_i,1\}$, $u-(1+\sigma)+e^{ur}=0$.
Then system \eqref{b1} is $\lambda$-$ A$-practically stable.
\end{theorem}

\begin{proof}
For any $\varphi\in C([-r,0],\mathbb{R})$ and $\|\varphi\|<\lambda$,
denote by $x(t)$ the solution of \eqref{b1}.
Set the function $V(t)=x^2(t)$. Then the derivative of $V(t)$ with respect
to each subsystem is:
\begin{align*}
\dot{V}(t)
&=2x(t)\dot{x(t)}\\
&=2x(t)[a_i(t)x(t)+b_i(t)x(t-r)]\\
&\leq [2a_i(t)+b_i^2(t)]x(t)^2+x(t-r)^2\\
&=[2a_i(t)+b_i^2(t)]V(t)+V(t-r).
\end{align*}
Obviously, we have
\begin{gather*}
V(t_{k}^+)=[d_k x(t_k)]^2=d_k^2 V(t_{k}),\quad k\in \mathbb{N}_{+},\\
2a_{i_k}(t)+b^2_{i_k}(t)+1\leq-\sigma<0,\,\,t\in (t_{k-1},t_k].
\end{gather*}
Hence,
\begin{gather*}
\dot{V}(t)\leq -(1+\sigma)V(t)+V(t-r),\quad t\in(t_{k-1},t_k]\\
V(t_{k}^+)=d_k^2 V(t_{k}),\quad k\in \mathbb{N}_{+}\\
V_{t_0}=\varphi^2.
\end{gather*}
 Define
$$
G=\sup \{V(s): s\in[t_0-r,t_0]\}=\sup \{\varphi^2(s): s\in[-r,0]\}<\lambda^2.
 $$
Below we  prove that
$V(t)\leq  \prod_{i=0}^{k}\tilde{d}_i^2 Ge^{-u(t-t_0-kr)}$ for $t\in(t_k,t_{k+1}]$,
where $\tilde{d}_0=1$, and $u-(1+\sigma)+e^{ur}=0$.

(I) If $t\in(t_0,t_1]$, we establish a comparison system:
\begin{gather*}
\dot{W}_0(t)= -(1+\sigma)W_0(t)+W_0(t-r),\quad t\in(t_{0},t_1]\\
W_0(t_{0}^+)=\tilde{d}_0^2 V(t_{0})\\
{W_0}_{t_0}=\tilde{d}_0^2\varphi^2,
\end{gather*}
where $\tilde{d}_0=1$. From  Lemma \ref{lem3.5}, we have
$V(t)\leq W_0(t)$ for $t\in(t_0,t_1]$. Furthermore,
$$
W_0(t)\leq \tilde{d}_0^2Ge^{-u(t-t_0)},\quad t\in(t_0,t_1],
$$
by Halanay inequality. So, $V(t)\leq \tilde{d}_0^2Ge^{-u(t-t_0)}$ for
each $t\in(t_0,t_1]$.

(II) Assume that $V(t)\leq \prod_{i=0}^{j-1}\tilde{d}_i^2 Ge^{-u(t-t_0-(j-1)r)}$
for each $t\in (t_{j-1},t_j]$ and $j=1,2,\dots,k$.  Below we prove that
\begin{equation}\label{bj2}
V(t)\leq\prod_{i=1}^{k}\tilde{d}_i^2 Ge^{-u(t-t_0-kr)},\,t\in (t_k,t_{k+1}].
\end{equation}
Consider \eqref{b1} on $(t_k,t_{k+1}]$, and take $t_k$ as the initial time.
Then we establish a comparison system
\begin{gather*}
\dot{W}_k(t)= -(1+\sigma)W_k(t)+W_k(t-r),\quad t\in(t_{k},t_{k+1}]\\
W_k(t_{k}^+)=\tilde{d}_k^2 V(t_{k})\\
{W_k}_{t_k}=\tilde{d}_k^2V^2(t-t_k).
\end{gather*}
By Lemma \ref{lem3.5}, $V(t)\leq W_k(t)$ for every $t\in(t_k,t_{k+1}]$.
 And $W_k(t)$ is continuous on $[t_k-r,t_{k+1}]$, because
$t_k-t_{k-1}>r$. From Halanay  inequality,
\begin{align*}
W_k(t)
&\leq \Big\{\tilde{d}_k^2\cdot\prod_{i=0}^{k-1}\tilde{d}_i^2
 Ge^{-u(t_k-t_0-kr)}\Big\}e^{-u(t-t_k)}\\
&= \prod_{i=1}^{k}\tilde{d}_i^2 Ge^{-u(t-t_0-kr)},\quad t\in(t_k,t_{k+1}].
\end{align*}
Hence,
\[
V(t)\leq \prod_{i=1}^{k}\tilde{d}_i^2 Ge^{-u(t-t_0-kr)},\quad
t\in(t_k,t_{k+1}].
\]
 By mathematical induction,
$$
V(t)\leq  \prod_{i=1}^{k}\tilde{d}_i^2
Ge^{-u(t-t_0-kr)}<\lambda^2\prod_{i=1}^{k}\tilde{d}_i^2
e^{-u(t_k-t_0-kr)}\leq A^2,\quad t\in(t_k,t_{k+1}],\; k\in\mathbb{N}.
$$
Namely, $|x(t)|<A$, $t\in [t_0,+\infty)$. The proof is complete.
\end{proof}

\begin{remark} \label{rmk4.2}   \rm
In Theorem \ref{thm4.2}, it requests that $a_{i_k}(t)\leq0$ if $t\in (t_{k-1},t_k]$;
However, Theorems \ref{thm4.4} and \ref{thm4.5} establish criterions,
 where $a_{i_k}(t)$  do not have to be negative.
\end{remark}

\begin{theorem} \label{thm4.4}
Consider $(\lambda,A)$ with $0< \lambda<A$. Assume that there exist constants
$t_0\in \mathbb{R}^+$ and $\delta>0$ such that:
\begin{gather*} %(A1)
\Big(1+\int_{t_0}^{t_0+r/2}|b_{i_1}(u)|du\Big)
e^{\int_{t_0}^{t}a_{i_1}(s)ds}\leq\frac{A}{\lambda},\quad t\in (t_0,t_0+r/2];\\
\Big[1 +\int_{t_0-r}^{t_0}|b_{i_1}(s+r)|ds\Big]^2
+\int_{-r}^{0}\int_{t_0+s}^{t_0}b_{i_1}^2(z+r)\,dz\,ds
\leq\frac{A^2}{6\lambda^2};
\end{gather*}
\begin{gather*}
-\frac{1}{2r}\leq g_{i_k}(t)\leq -rb_{i_k}^2(t+r),\quad t\in (t_{k-1},t_k]; \\
d_k\leq \frac{\delta}{Ah_k(t_k)},\quad
t_k-t_{k-1}\geq 2r,\quad k\in\mathbb{N}_+;
\end{gather*}
%\item[(A2.2)]
for any $k\in \mathbb{N}_+$,
\begin{gather*}
\max\{d_k\cdot h_k(t_k),h_k(t_k-r)\}
\Big(1+\int_{t_k}^{t_k+\frac{r}{2}}|b_{i_{k+1}}(u)|du\Big)
e^{\int_{t_k}^{t}a_{i_{k+1}}(s)ds}\leq1,\\
t\in (t_k,t_k+\frac{r}{2}];\\
\Big[\delta +A\int_{t_k-r}^{t_k}|b_{i_{k+1}}(s+r)|h_k(s)ds\Big]^2
+A^2\int_{-r}^{0}\int_{t_k+s}^{t_k}b_{i_{k+1}}^2(z+r)
h_k^2(z)\,dz\,ds\leq\frac{A^2}{6};
\end{gather*}
where $g_i(t)=a_i(t)+b_i(t+r)$,
$h_k(t)=e^{\frac{1}{2}\int_{t_{k-1}}^{t-0.5r}[a_{i_k}(s)+b_{i_k}(s+r)]ds}$,
$i\in\Lambda$, $k\in\mathbb{N}_+$.
Then system \eqref{b1} is $\lambda$-$ A$-practically stable.
\end{theorem}

\begin{proof}
For any $\varphi\in C([-r,0],\mathbb{R}), \|\varphi\|<\lambda$, let $x(t)$
be the solution of \eqref{b1}.
For each $k\in\mathbb{N}_{+}$, we define a function:
$$
V_k(u)=\Big[x(u)+\int_{u-r}^{u}b_{i_k}(s+r)x(s)ds\Big]^2
+\int_{-r}^{0}\int_{u+s}^{u}b_{i_k}^2(z+r)x^2(z)\,dz\,ds,
$$
$u\in (t_{k-1},t_{k}]$.
(I)  when $t\in (t_0,t_1]$, by Lemma \ref{lem3.6},
\begin{align*}
|x(t)|&\leq \|\varphi\| (1+\int_{t_0}^{t_0+r/2}|b_{i_1}(u)|du)
 e^{\int_{t_0}^{t}a_{i_1}(s)ds}\\
&< \lambda(1+\int_{t_0}^{t_0+r/2}|b_{i_1}(u)|du)e^{\int_{t_0}^{t}a_{i_1}(s)ds}\\
&\leq A,\quad t\in (t_0,t_0+r/2];
\end{align*}
\begin{align*}
|x(t)|&\leq \sqrt{6V(t_0)}e^{\frac{1}{2}
 \int_{t_0}^{t-r/2}a_{i_1}(s)+b_{i_1}(s+r)ds}\\
&= \Big\{6\Big[x(t_0)+\int_{t_0-r}^{t_0}b_{i_1}(s+r)x(s)ds\Big]^2\\
&\quad +6\int_{-r}^{0} \int_{t_0+s}^{t_0}b_{i_1}^2(z+r)x^2(z)\,dz\,ds\Big\}^{1/2}h_1(t)\\
&<\Big\{6[\lambda+\lambda \int_{t_0-r}^{t_0}|b_{i_1}(s+r)|ds]^2
+6\int_{-r}^{0}\int_{t_0+s}^{t_0}b_{i_1}^2(z+r)\lambda^2
\,dz\,ds\Big\}^{1/2}h_1(t)\\
&\leq  A h_1(t),\quad t\in(t_0+r/2,t_1].
\end{align*}
So, $|x(t)|<A$, if $t\in(t_0,t_1]$.

(II) Assume that $|x(t)|<A$, if $t\in(t_j,t_j+\frac{r}{2}]$, and
$|x(t)|<Ah_{j+1}(t)$, if $t\in[t_j+\frac{r}{2},t_{j+1}]$,
where $j=0,1,\dots,k-1$. Then we need to prove that
\begin{equation} \label{t431}
|x(t)|<A,\;t\in(t_k,t_k+\frac{r}{2}];\quad
|x(t)|<Ah_{k+1}(t),\; t\in[t_k+\frac{r}{2},t_{k+1}].
\end{equation}
From Corollary \ref{coro3.1},
\begin{align*}
|x(t)|
&\leq \max\Big\{d_k x(t_k),\sup_{s\in[t_k-r,t_k]}^{}\{x(s)\}\Big\}
 \Big(1+\int_{t_k}^{t_k+r/2}|b_{i_{k+1}}(u)|du\Big)e^{\int_{t_k}^{t}a_{k+1}(s)ds}\\
&< A  \max\{d_k h_k(t_k),h_k(t_k-r)\}\Big(1+\int_{t_k}^{t_k+r/2}|b_{i_{k+1}}(u)|du\Big)e^{\int_{t_k}^{t}a_{i_{k+1}}(s)ds}\\
&\leq A,\quad t\in (t_k,t_k+r/2];
\end{align*}
\begin{align*}
|x(t)|^2
&\leq 6V_{k+1}(t_k^+)e^{\int_{t_k}^{t-r/2}a_{i_{k+1}}(s)+b_{i_{k+1}}(s+r)ds}\\
&= 6\Big\{\Big[x(t_k^+)+\int_{t_k-r}^{t_k}b_{i_{k+1}}(s+r)x(s)ds\Big]^2\\
&\quad +\int_{-r}^{0}\int_{t_k+s}^{t_k}b_{i_{k+1}}^2(z+r)x^2(z)\,dz\,ds\Big\}
 h_{k+1}^2(t)\\
&< 6\Big\{\Big[\delta+\int_{t_k-r}^{t_k}|b_{i_{k+1}}(s+r)|Ah_{k}(s) ds\Big]^2\\
&\quad +\int_{-r}^{0}\int_{t_k+s}^{t_k}b_{i_{k+1}}^2(z+r)A^2h_{k}^2(z) \,dz\,ds\Big\}
 h_{k+1}^2(t)\\
&\leq A^2h_{k+1}^2(t), \quad t\in(t_k+r/2,t_{k+1}].
\end{align*}
Hence, \eqref{t431} holds. By  mathematical induction,
$|x(t)|<A,\,\,t\geq t_0$. Namely, system \eqref{b1} is $\lambda$-$ A$-practically
stable.
\end{proof}

\begin{corollary} \label{coro4.2}
Consider $(\lambda,A)$ with $0< \lambda<A$. Assume that $a_i(t)=a_i$, $b_i(t)=b_i$,
 where $a_i, b_i\in\mathbb{R}, i\in\Lambda$. If there is a $\delta>0$ and the
following assumptions are satisfied: %$(A1)$
\[
(1+\frac{1}{2}|b_{i_1}|r)e^{a_{i_1}u}\leq\frac{A}{\lambda},\quad
u\in (0,\frac{1}{2}];\quad
\frac{3}{2}b_{i_1}^2r^2+2|b_{i_1}|r+1\leq\frac{A^2}{6\lambda^2};
\]
%(A2.1)
\[
-\frac{1}{2r}\leq c_{i}\leq -rb_{i}^2,\quad i\in\Lambda; \quad
d_k\leq \frac{\delta}{Aw_k(t_k)},\quad \tau_k\geq 2r,\quad k\in\mathbb{N}_+;
\]
for any $k\in \mathbb{N}_+$ and $u\in (0,\frac{1}{2}]$,
$(1+\frac{1}{2}|b_{i_{k+1}}|r)e^{\frac{1}{2}c_{i_k}(\tau_k-1.5r)
+u a_{i_{k+1}}}\leq1$;
\[
\Big[\delta +A|b_{i_{k+1}}|\int_{t_k-r}^{t_k}w_k(s)ds\Big]^2+A^2b_{i_{k+1}}^2
\int_{t_k-r}^{t_k}(z-t_k+r)w_k^2(z)dz\leq\frac{A^2}{6};
\]
where $c_i=a_i+b_i, w_k(t)=e^{\frac{1}{2}c_{i_k}(t-t_{k-1}-0.5r)}$,
$\tau_k=t_k-t_{k-1}$, $i\in\Lambda$, $k\in\mathbb{N}_+$, then system
\eqref{b1} is $\lambda$-$ A$-uniformly practically stable.
\end{corollary}


\begin{theorem} \label{thm4.5}
Consider $(\lambda,A)$ with $0< \lambda<A$. Assume that there are constants
 $t_0\in\mathbb{R}^+$, $\beta>0$ and $\delta>0$ such that
\begin{itemize}
\item[(i)] $|b_{i_k}(t)|\leq ru_{i_k}(t)$ for $t\in (t_{k-1},t_k]$;
 $d_k\leq \frac{\delta}{A}$;

\item[(ii)] $\big(1+r^2u_{i_1}(t_0)\big)g_1(t)\leq \frac{A}{\lambda}$ for
$t\in(t_0,t_1]$;

\item[(iii)] $\big[\delta+Ar^2u_{i_{k+1}}(t_k^+)\big]g_{k+1}(t)\leq A$ for
$t\in(t_k,t_{k+1}]$, $k\in \mathbb{N}_{+}$;
where
\[
u_{i}(t)=\frac{e^{\int_{0}^{t}a_{i}(s)ds}}{1+r\int_{t}^{t+\beta}
e^{\int_{0}^{u}a_{i}(s)ds}du},\quad
g_k(t)=e^{\int_{t_{k-1}}^{t}[a_{i_{k}}(s)+ru_{i_{k}}(s)]ds}, \quad
i\in\Lambda,\; k\in \mathbb{N}_{+}.
\]
\end{itemize}
Then system \eqref{b1} is $\lambda$-$ A$-practically stable.
\end{theorem}

\begin{proof}
For any $\varphi\in C([-r,0],\mathbb{R}),\|\varphi\|<\lambda$,
let $x(t)$ be the solution of \eqref{b1}.
Define
$$
V_i(t)=|x(t)|+ru_i(t)\int_{t-h}^{t}|x(s)|ds,\quad i\in\Lambda.
$$
For any $k\in\mathbb{N}_+$ and $t\in (t_{k-1},t_{k}]$, we have
\begin{align*}
\dot{u}_{i_k}(t)
&=  \frac{a_{i_k}(t)e^{\int_{0}^{t}a_{i_k}(s)ds}}{1+r\int_{t}^{t+\beta}
e^{\int_{0}^{u}a_{i_k}(s)ds}du}
-\frac{e^{r\int_{0}^{t}a_{i_k}(s)ds}}{(1+r\int_{t}^{t+\beta}
e^{\int_{0}^{u}a_{i_k}(s)ds}du)^2}\\
&\quad \times \Big(e^{\int_{0}^{t+\beta}a_{i_k}(s)ds}
 -e^{\int_{0}^{t}a_{i_k}(s)ds}\Big)\\
&= a_{i_k}(t)u_{i_k}(t)-ru_{i_k}^2(t)(e^{\int_{t}^{t+\beta}a_{i_k}(s)ds}-1)\\
&\leq a_{i_k}(t)u_{i_k}(t)+ru_{i_k}^2(t);
\end{align*}
\begin{align*}
&\dot{V}_{i_k}(t)\\
&\leq [a_{i_k}(t)+ru_{i_k}(t)]|x(t)|+r\dot{u}_{i_k}(t)
 \int_{t-h}^{t}|x(s)|ds+(|b_{i_k}(t)|-ru_{i_k}(t))|x(t-r)|\\
&\leq [a_{i_k}(t)+ru_{i_k}(t)]\Big(|x(t)|+ru_{i_k}(t)
 \int_{t-h}^{t}|x(s)|ds\Big)-[a_{i_k}(t)+ru_{i_k}(t)]r\\
&\quad\times u_{i_k}(t)\int_{t-h}^{t}|x(s)|ds
 +r\cdot[a_{i_k}(t)u_{i_k}(t)+ru_{i_k}^2(t)]\int_{t-h}^{t}|x(s)|ds\\
&\leq [a_{i_k}(t)+ru_{i_k}(t)]V(t).
\end{align*}
So, $|x(t)|\leq V_{i_k}(t)\leq V_{i_k}(t_k^+)
 e^{\int_{t_k}^{t}[a_{i_k}(s)+ru_{i_k}(s)]ds}$, $t\in(t_{k-1},t_{k}]$.

(I) If $t\in (t_0,t_1]$, then
\begin{align*}
|x(t)|
&\leq V(t_0^+)e^{\int_{t_0}^{t}a_{i_1}(s)+ru_{i_1}(s)ds}\\
&=\Big(|x(t_0)|+ru_{i_1}(t_0)\int_{t_0-r}^{t}|x(s)|ds\Big)g_1(t)\\
&<\Big(\lambda+ru_{i_1}(t_0)\int_{t_0-r}^{t}\lambda ds\Big)g_1(t)
\leq A.
\end{align*}

(II) Assume that $|x(t)|<A$, if $t\in (t_i-1,t_i]$, $i=1,2,\dots,k$.
If $t\in (t_k,t_{k+1}]$, then
\begin{align*}
|x(t)|&\leq V(t_k^+)e^{\int_{t_k}^{t}a_{i_{k+1}}(s)+ru_{i_{k+1}}(s)ds}\\
&=\Big(|x(t_k^+)|+ru_{i_{k+1}}(t_k)\int_{t_k-r}^{t}|x(s)|ds\Big)g_{k+1}(t)\\
&<\Big(A  \frac{\delta}{A}+ru_{i_{k+1}}(t_k)\int_{t_k-r}^{t}A ds\Big)g_{k+1}(t)
\leq A.
\end{align*}
By mathematical induction, $|x(t)|<A$, $t\geq t_0$. The proof is complete.
\end{proof}

\begin{corollary} \label{coro4.3}
Assume that $a_{i_k}(t)\equiv 0$, $k\in\mathbb{N}_+$. And suppose there
exist constants $t_0\geq0$, and $\beta, \delta>0$, such that
\begin{gather*}
|b_{i_k}(t)|\leq \frac{h}{1+\beta r};\quad
d_k\leq\frac{\delta}{A};\\
\big(1+\frac{r^2}{1+\beta r}\big)e^{\frac{r(t_1-t_0)}{1+\beta r}}
\leq\frac{A}{\lambda};\quad
\big(\delta+\frac{Ar^2}{1+\beta r}\big)e^{\frac{r(t_{k+1}-t_k)}{1+\beta r}}
\leq A,\quad k\in \mathbb{N}_+.
\end{gather*}
Then system \eqref{b1} is $\lambda$-$ A$-practically stable.
\end{corollary}

Lastly, we consider the $n$-dimensional system with constant coefficients.
That is, $f_{i_k}(t,x(t),x(t-r))=A_{i_k}x(t)+B_{i_k}x(t-r)$ in \eqref{z1}.
\begin{equation}\label{g1}
\begin{gathered}
\dot{x}(t)=A_{i_k}x(t)+B_{i_k}x(t-r),\quad t\in(t_{k-1},t_k]\\
x(t_{k}^+)=d_k x(t_{k}),\quad k\in \mathbb{N}_{+}\\
x_{t_0}=\varphi,
\end{gathered}
\end{equation}
where $x\in \mathbb{R}^{n}$, $d_k\in \mathbb{R}_{+}$,
$\varphi\in C([-r,0],\mathbb{R})$, $A_i$ and $B_i$ are $n\times n$ matrices,
 $k\in\mathbb{N}_{+}$, $i\in\Lambda$.

\begin{theorem} \label{thm4.6}
Consider $(\lambda,A)$ with $0< \lambda<A$. Assume that there is a $\eta>0$,
such that the  linear matrix inequality with respect to symmetric matrices
 $\{P_i>0\}_{i=1}^{m}$
\begin{equation}\label{g2}
\begin{pmatrix}
A_i^TP_i+P_iA_i+(1+\eta)P_i & P_iB_i  \\
B_i^TP_i & -P_i  \\
\end{pmatrix}
<0,\quad i\in\Lambda
\end{equation}
is solvable. And suppose that there exist constants $\delta>1$ and
$\beta>0$ such that
\begin{gather*}
\beta<\frac{\ln\delta}{\sigma}+1+\eta-\delta e^{\beta r},\quad
d_k^2\geq \frac{1}{\delta},\\
\delta^{k+1}\chi^k\prod_{i=0}^{k}d_i^2
\lambda_{\rm max}(P_{i_1})e^{-\beta(t_k-t_0)}
\leq\lambda_{\rm min}(P_{i_{k+1}})\frac{A^2}{\lambda^2},\,\,k\in\mathbb{N},
\end{gather*}
where $d_0=1$, $\sigma=\sup\{t_{n+1}-t_n: n\in\mathbb{N}\}<+\infty$,
$\chi=\max\big\{\frac{\lambda_{\rm max}(P_i)}{\lambda_{\rm min}(P_j)}
: i,j\in\Lambda, i\neq j\big\}$. Then system \eqref{g1} is $\lambda$-$ A$-uniformly
practically stable.
\end{theorem}

\begin{proof}
By \eqref{g2}, we have
$$
A_i^TP_i+P_iA_i+P_iB_iP_i^{-1}B_i^TP_i+(1+\eta)P_i<0, \quad i\in\Lambda.
$$
For any $\varphi\in C([-r,0],\mathbb{R}), \|\varphi\|<\lambda$ and
$t_0\geq0$, let $x(t)$ be the solution of \eqref{g1}.
Establish a function $V_i=x(t)^TP_ix(t)$ with respect to the $i$-th
subsystem of \eqref{g1}, and define
$\overline{V}_i(t)=\sup_{-r\leq \theta\leq 0}V_i(t+\theta)$, $i\in\Lambda$.
It follows that
\begin{align*}
\dot{V}_i(t)
&=2x^T(t)P_i\dot{x}(t)\\
&=2x^T(t)P_i(A_{i}x(t)+B_{i}x(t-r))\\
&\leq x^T(t)(A_i^TP_i+P_iA_i+P_iB_iP_i^{-1}B_i^TP_i)x(t)+x^T(t-r)
P_ix(t-r)\\
&\leq-(1+\eta)V_i(t)+\overline{V}_i(t),\,\,t\geq t_0.
\end{align*}
At the switching time,
$$
V_{i_{k+1}}(t_k^+)=x^T(t_k^+)P_{i_{k+1}}x(t_k^+)
=d_k^2x^T(t_k)P_{i_{k+1}}x(t_k)=d_k^2V_{i_{k+1}}(t_k),\quad k\in\mathbb{N}_+.
$$
If $t\in[t_0-r,t_0]$, it is easy to verify that
$V_{i_1}(t)\leq \lambda_{\rm max}(P_{i_1})|x(t)|^2$. Hence,
$$
\overline{V}_{i_1}(t_0)\leq\lambda_{\rm max}(P_{i_1})
\|\varphi\|^2<\lambda^2\cdot\lambda_{\rm max}(P_{i_1}).
$$

(I) If $t\in(t_0,t_1]$, we claim that
\begin{equation}\label{dingli6z}
V_{i_1}(t)\leq \delta d_0\overline{V}_{i_1}(t_0) e^{-\beta(t-t_0)}
:= \alpha_0 e^{-\beta(t-t_0)}.
\end{equation}
If not, define
\[
g_0(t)=\begin{cases}
V_{i_1}(t)e^{\beta (t-t_0)}, &t\in(t_0,t_1]\\
V_{i_1}(t), &t\in[t_0-r,t_0].
\end{cases}
\]
Then, there is a $\tilde{t}_0\in(t_0,t_1]$, such that
$g_0(\tilde{t}_0)>\alpha_0$. Define
\begin{gather*}
t_0^*=\inf\{t\in(t_0,t_1]: g_0(t)>\alpha_0\}; \\
t_{0*}=\sup\{t\in[t_0,t_0^*]: g_0(t)\leq\overline{V}_{i_1}(t_0)\}.
\end{gather*}
Then $t_0\leq t_{0*}<t_0^*< t_1$, and $\delta g_0(t)\geq \alpha_0\geq g_0(s)$
for any $s\in[t_0-r,t_0^*],\,t\in[t_{0*},t_0^*]$.
From Lemma \ref{lem3.4}, $\delta>\frac{g_0(t_0^*)}{g_0(t_{0*})}=\delta$.
This contradiction proves \eqref{dingli6z}.

(II) Assume that $V_{i_{j+1}}(t)\leq \alpha_je^{-\beta(t-t_0)}$, where
$t\in(t_j,t_{j+1}]$, $j=0,1,\dots,k-1$, and
$$
\alpha_j=\delta^{j+1}\chi^j\prod_{i=0}^{j}d_i^2\cdot\overline{V}_{i_1}(t_0).
$$
 Below, we are to prove $V_{i_{k+1}}(t)\leq \alpha_ke^{-\beta(t-t_0)}$
for every $t\in(t_k,t_{k+1}]$.
Note that $V_{i_{k+1}}(t)\leq \chi V_{i_{j+1}}(t)
\leq \chi \alpha_{j}e^{-\beta(t-t_0)}$, if $t\in(t_{j},t_{j+1}]$, $j=0,1,\dots,k-1$.
$$
V_{i_{k+1}}(t_k^+)=d_k^2V_{i_{k+1}}(t_k)
\leq\chi d_k^2V_{i_{k}}(t_k)\leq\chi d_k^2\alpha_{k-1}e^{-\beta(t_k-t_0)}.
$$
Define
\[
g_k(t)=\begin{cases}
V_{i_{k+1}}(t)e^{\beta (t-t_0)}, &t\in(t_0,t_{k+1}]\\
V_{i_{k+1}}(t), &t\in[t_0-r,t_0].
\end{cases}
\]
Because $\alpha_i\leq\alpha_j$ for any $i\leq j$, we have
$g_k(t)\leq \delta (d_k^2\chi \alpha_{k-1}):=\alpha_k$, if
 $t\in[t_k-r,t_k]$. We claim that $g_k(t)\leq\alpha_k$ for
each $t\in(t_k,t_{k+1}]$. Otherwise, there is a
$\tilde{t}_k\in(t_k,t_{k+1}]$, such that $g_k(\tilde{t}_k)>\alpha_k$. Define
\begin{gather*}
t_k^*=\inf\{t\in(t_k,\tilde{t}_k]: g_k(t)>\alpha_k\};\\
E_k=\{t\in(t_k,t_k^*]: g_k(t)\leq \alpha_{k-1}d_k^2\chi \};\\
t_{k*}=\begin{cases}
t_k,&\text{if } E_k=\emptyset\\
\sup E_k,&\text{if }E_k\neq\emptyset.
\end{cases}
\end{gather*}
Hence, $t_k\leq t_{k*}<t_k^*\leq t_{k+1}$, and
$\delta g_k(t)\geq \alpha_k\geq g_k(s)$, for any
$s\in[t_k-r,t_k^*]$, $t\in(t_{k*},t_k^*]$. By Lemma \ref{lem3.4},
$\delta>\frac{g_k(t_k^*)}{g_k(t_{k*}^+)}=\delta$.
This contradiction proves $g_k(t)\leq \alpha_k$, for each
$t\in(t_k,t_{k+1}]$. That is, $V_{i_{k+1}}(t)\leq \alpha_ke^{-\beta(t-t_0)}$,
if $t\in(t_k,t_{k+1}]$. By mathematical induction,
$$
V_{i_{k+1}}\leq \alpha_ke^{-\beta(t-t_0)},\quad
 t\in(t_k,t_{k+1}], \,\,k\in\mathbb{N}.
$$
It is easy to verify that
$\lambda_{\rm min}(P_{i_{k+1}})|x(t)|^2\leq V_{{i_{k+1}}}(t)$. Furthermore,
\begin{align*}
|x(t)|
&\leq  \sqrt{\frac{1}{\lambda_{\rm min}(P_{i_{k+1}})}V_{{i_{k+1}}}(t)}\\
&\leq \Big[\frac{1}{\lambda_{\rm min}(P_{i_{k+1}})}\delta^{k+1}\chi^k
 \prod_{i=0}^{k}d_i^2e^{-\beta (t_k-t_0)}\overline{V}_{i_1}(t_0)\Big]^{1/2}\\
<&\Big[\frac{\delta^{k+1}\chi^k}{\lambda_{\rm min}(P_{i_{k+1}})}
 \prod_{i=0}^{k}d_i^2 e^{-\beta (t_k-t_0)}\lambda_{\rm max}(P_{i_1})
 \lambda^2\Big]^{1/2}\\
&\leq  A,\quad  t\in(t_k,t_{k+1}],\; k\in\mathbb{N}.
\end{align*}
So, system \eqref{g1} is $\lambda$-$ A$-uniformly practically stable.
\end{proof}

\begin{corollary} \label{coro4.4}
Consider $(\lambda,A)$ with $0< \lambda<A$. Suppose that there exist constants
$\eta>0$, $\delta>1$, and $\beta>0$, such that the
 linear matrix inequality with respect to symmetric matrices $\{P_i>0\}_{i=1}^{m}$
\begin{equation}\label{g4}
\begin{pmatrix}
A_i^TP_i+P_iA_i+(1+\eta)P_i & P_iB_i  \\
B_i^TP_i & -P_i  \\
\end{pmatrix}
<0,\quad i\in\Lambda,
\end{equation}
has a common solution $P_i=P$, $i\in\Lambda$;
and the following assumptions are satisfied:
\begin{gather*}
\beta<\frac{\ln\delta}{\sigma}+1+\eta-\delta e^{\beta r},
\quad d_k^2\geq \frac{1}{\delta},\\
\delta^{n+1}\prod_{k=0}^{n}d_k^2\lambda_{\rm max}(P)e^{-\beta(t_n-t_0)}
 \leq\lambda_{\rm min}(P)\frac{A^2}{\lambda^2},\quad n\in\mathbb{N},
\end{gather*}
where $d_0=1$, $\sigma=\sup\{t_{n+1}-t_n: n\in\mathbb{N}\}$.
 Then  system\eqref{g1} is $\lambda$-$ A$-uniformly practically stable.
\end{corollary}

\begin{remark} \label{rmk4.3}   \rm
According to \cite{c1}, the system is exponentially stable if \eqref{g2} is true.
Then, Theorem \ref{thm4.6} and Corollary \ref{coro4.3} tells us that the switched system
with delay can keep practically stable under some impulse perturbation,
if the subsystem are of some good quality, such as exponential stability.
That is to say, the restriction on impulses is loose in this case.
\end{remark}

\section{Examples}

In this section, several examples are given to illustrate our theorems.

\begin{example} \label{examp5.1} \rm
Consider  system \eqref{c1}, where
\begin{gather*}
\dot{x}(t)=a_1x(t)+b_1x(t-r)=-x(t)+\frac{1}{2}x(t-0.25);\\
\dot{x}(t)=a_2x(t)+b_2x(t-r)=\frac{3}{8}x(t)+\frac{1}{2}x(t-0.25);\\
S=\{(0.25,2),(2,1),(0.25,2),\dots\};\\
d_k=\begin{cases}
\frac{\sqrt{5} e^{0.1}}{4} &\text{if $k$ is even}\\
\frac{\sqrt{6}e^{0.0125}}{3} &\text{if $k$ is odd}.
\end{cases}
\end{gather*}
 Given $\lambda=1$, $A=1.8$, we set $p_1=-1.75$, $p_2=1$,
$\delta_1=1.5$, $\delta_2=3.2$, $\sigma_1=2$, $\sigma_2=0.25$,
$\beta=0.1$. According to Theorem \ref{thm4.1}, it is easy to verify that 
the system is  $\lambda$-$ A$-uniformly practically stable 
(see Figure \ref{fig1}(a)).
\end{example}

\begin{figure}[htb] % two figures side by side
\begin{center}
\includegraphics[width=0.48\textwidth]{fig1a}  %li1099.jpg
\includegraphics[width=0.48\textwidth]{fig1b}   \\ %li2095.jpg
(a)\hfil (b)
\end{center}
\caption{(a) $t_0=0$, $\varphi=0.99$; (b)  $\varphi=0.95$}
\label{fig1} 
\end{figure}

\begin{example} \label{examp5.2} \rm
Consider  system \eqref{b1}, where
\begin{gather*}
\dot{x}(t)=a_1(t)x(t)+b_1(t)x(t-r)=(-0.6+\frac{\sin^4(t)}{8})x(t)
+\frac{1}{2}\sin^2(t)x(t-\frac{\pi}{6}),\\
\dot{x}(t)=a_2(t)x(t)+b_2(t)x(t-r)=(-0.6+\frac{\cos^4(t)}{8})x(t)
+\frac{1}{2}\cos^2(t)x(t-\frac{\pi}{6});\\
S=\{(\frac{\pi}{2},1),(\frac{\pi}{2},2),(\frac{\pi}{2},1),(\frac{\pi}{2},2),\dots\};
d_1=1.5,\quad d_k=\frac{k^2}{k^2-1},\quad k\geq 2.
\end{gather*}
Given $\lambda=1$, $A=3$, we set $t_0=\frac{7\pi}{4}$, $\sigma=0.075$.
According to Theorem \ref{thm4.2}, it is easy to verify that the system is
$\lambda$-$ A$-practically stable (see Figure \ref{fig1}(b)).
\end{example}

\begin{example} \label{examp5.3} \rm
Consider  system \eqref{b1}, where
\begin{gather*}
\dot{x}(t)=a_1(t)x(t)+b_1(t)x(t-r)=-\sin(\pi t)x(t)
+\frac{1}{4}e^{\frac{\cos(\pi t)-1}{\pi}}x(t-\frac{1}{2}),\\
\dot{x}(t)=a_2(t)x(t)+b_2(t)x(t-r)
=-\sin(2\pi t)x(t)+\frac{1}{4}e^{\frac{\cos(2\pi t)-1}{2\pi}}x(t-\frac{1}{2});\\
S=\{(2,1),(2,2),(2,1),(2,2),\dots \};\quad
d_k=\frac{9}{20},\quad k\in\mathbb{N}_{+}.
\end{gather*}
Given $\lambda=1$, $A=2$, we set $t_0=0$, $\delta=0.9$, $\beta=2$. Then
\begin{align*}
u_1(t)
&= \frac{e^{\int_{0}^{t}-\sin(\pi s)ds}}{1+\frac{1}{2}\int_{t}^{t+2}
 e^{\int_{0}^{u}-\sin(\pi s)ds}du}\\
&= \frac{e^{[\cos(\pi t)-1]/\pi}}{1+\frac{1}{2}\int_{t}^{t+2}
 e^{[\cos(\pi u)-1]/\pi}du}\\
&\geq \frac{e^{[\cos(\pi t)-1]/\pi}}{1+\frac{1}{2}(e^0+e^{\frac{-1}{\pi}})}\\
&\geq \frac{1}{2}e^{\frac{\cos(\pi t)-1}{\pi}}
= \frac{1}{r}b_1(t).
\end{align*}
\begin{align*}
u_1(t)
&= \frac{e^{[\cos(\pi t)-1]/\pi}}{1+\frac{1}{2}\int_{t}^{t+2}
 e^{[\cos(\pi u)-1]/\pi}du}\\
&\leq  \frac{e^{[\cos(\pi t)-1]/\pi}}{1+\frac{1}{2}(e^{\frac{-2}{\pi}}
 +e^{\frac{-1}{\pi}})}\\
&\leq 0.6142e^{\frac{\cos(\pi t)-1}{\pi}}\leq\frac{5}{8}
e^{\frac{\cos(\pi t)-1}{\pi}}.
\end{align*}
\begin{align*}
u_2(t)&= \frac{e^{\int_{0}^{t}-\sin(2\pi s)ds}}{1+\frac{1}{2}\int_{t}^{t+2}
 e^{\int_{0}^{u}-\sin(2\pi s)ds}du}\\
&= \frac{e^{[\cos(2\pi t)-1]/(2\pi)}}{1+\frac{1}{2}\int_{t}^{t+2}
 e^{[\cos(2\pi u)-1]/(2\pi)}du}\\
&\geq \frac{e^{[\cos(2\pi t)-1]/(2\pi)}}{1+\frac{1}{2}(e^0+e^{\frac{-1}{2\pi}})}\\
&\geq \frac{1}{2}e^{\frac{\cos(2\pi t)-1}{2\pi}}
= \frac{1}{r}b_2(t).
\end{align*}
\begin{align*}
u_2(t)
&= \frac{e^{[\cos(2\pi t)-1]/(2\pi)}}{1+\frac{1}{2}\int_{t}^{t+2}
e^{[\cos(2\pi u)-1]/(2\pi)}du}\\
&\leq  \frac{e^{[\cos(2\pi t)-1]/(2\pi)}}{1+\frac{1}{2}
(e^{\frac{-2}{2\pi}}+e^{\frac{-1}{2\pi}})}\\
&\leq \frac{100}{179}e^{\frac{2\cos(\pi t)-1}{2\pi}}.
\end{align*}
Consequently, if $t\in(t_0,t_1]$,
\begin{align*}
\Big(1+r^2u_{i_1}(t_0)\Big)e^{\int_{t_0}^{t}a_{i_1}(s)+ru_{i_1}(s)}ds
&\leq  [1+\frac{1}{4}\times \frac{5}{8}]
 e^{\int_{0}^{2}\frac{1}{2}\times\frac{5}{8}e^{\frac{\cos(\pi s)-1}{\pi}}ds}\\
&\leq \frac{37}{32}e^{\frac{5}{16}(e^0+e^{\frac{-1}{\pi}})}\\
&\leq  2=\frac{A}{\lambda};
\end{align*}
if $t\in(t_k,t_{k+1}],\, k\in\mathbb{N}_{+}$,
\begin{align*}
&\Big[\delta+Ar^2u_1(t_k^+)\Big]e^{\int_{t_k}^{t}[a_1(s)+ru_1(s)]ds}\\
&\leq  [0.9+2\times\frac{1}{4}\times 0.6142]
 e^{\int_{t_k}^{t_k+2}\frac{1}{2}\times\frac{5}{8}e^{\frac{\cos(\pi s)-1}{\pi}}ds}\\
&\leq (0.9+0.3071)e^{\frac{5}{16}(e^0\times\frac{2}{3}+e^{\frac{-1}{2\pi}}\times\frac{1}{3}+e^{\frac{-1}{\pi}}\times\frac{1}{3}+e^{\frac{-3}{2\pi}}\times\frac{2}{3})}\\
&\leq  2=A.
\end{align*}
\begin{align*}
\Big[\delta+Ar^2u_2(t_k^+)\Big]e^{\int_{t_k}^{t}[a_2(s)+ru_2(s)]ds}
&\leq  [0.9+2\times\frac{1}{4}\times \frac{100}{179}]
 e^{\int_{t_k}^{t_k+2}\frac{1}{2}\times\frac{100}{179}
 e^{\frac{\cos(2\pi s)-1}{2\pi}}ds}\\
&\leq (0.9+\frac{50}{179})e^{\frac{50}{179}(e^0+e^{\frac{-1}{2\pi}})}\\
&\leq  2=A.
\end{align*}
And $d_k=9/20=\delta/A$ yields $d_k\leq \delta/A$.
According to Theorem \ref{thm4.5}, it is easy to verify that the system is
 $\lambda$-$ A$-practically stable (see Figure \ref{fig2}).
\end{example}

\begin{figure}[htb] % two figures  in a row
\begin{center}
\includegraphics[width=0.48\textwidth]{fig2a} % li3099.jpg
\includegraphics[width=0.48\textwidth]{fig2b} \\ % li308.jpg}
(a)\hfil (b) 
\end{center}
\caption{(a) $\varphi=0.99$; (b) $\varphi=0.8$}
\label{fig2}
\end{figure}

\begin{example} \label{examp5.4} \rm
Consider  system \eqref{g1}, where
\begin{gather*}
A_1=\begin{pmatrix}
-1 & 0  \\
0 & -2  \\
\end{pmatrix} , \quad
A_2=\begin{pmatrix}
-4 & 0  \\
0 & -3 \\
\end{pmatrix}, \quad
B_1=\begin{pmatrix}
0 & 0  \\
0 & 1  \\
\end{pmatrix}, \quad
B_2=\begin{pmatrix}
2 & 0  \\
0 & 2 \\
\end{pmatrix}.
\\
S=\{(2,1),(2,2),(2,1),(2,2),\dots \};\quad
d_1=\frac{20e^{0.3}}{11},\quad d_k=\frac{e^{0.3}}{\sqrt{1.1}},\,\,k\geq 2.
\end{gather*}
$r=1$. Given $\lambda=1, A=2$, we set
 $\eta=0.9$, $\delta=1.1$, $\sigma=2$, $\beta=0.3$,
$P_1=P_2=\begin{pmatrix}
1 & 0  \\
0 & 1 \\
\end{pmatrix}$.
According to Theorem \ref{thm4.6}, it is easy to verify that the system is
 $\lambda$-$ A$-practically stable.
\end{example}

\subsection*{Acknowledgments}
This work is supported by grant S2012010010034 from the Guangdong Natural
Science Foundation of China, and by the Key Support Subject
(Applied Mathematics) of South China Normal University.

\begin{thebibliography}{00}


\bibitem{a1} R. P. Agarwal, Y.H. Kim and S.K. Sen;
\emph{Advanced discrete Halanay-type inequalities: stability of difference equations},
Journal of Inequalities and Applications, 2009.

\bibitem{c1} Shen Cong, Shumin Fei;
\emph{Exponential Stability of Switched Systems with Delay:a Multiple Lyapunov Function Approach},
Acta Automatica Sinica, 33(2007), 985-988.

\bibitem{h1} J.K.Hale, S.M. Verduyn Lunel;
\emph{Introduction to Functional Differential Equations}, Springer New York, 1993.

\bibitem{l1} V. Lakshmikantham, Xinzhi Liu;
\emph{Impulsive hybrid systems and stability theory}, Dynamic Systems and Applications, 7(1998), 1-10.

\bibitem{l2} Shao'e Li, Weizhen Feng;
\emph{Practical Stability of Time-dependent Impulsive Switched Systems}, Journal of Zhongkai University of Agriculture and Engineering,
27(2014), 36-44.

\bibitem{l3} Xiaodi Li, Martin Bohner;
\emph{An impulsive delay differential inequality and applications},
Computers and Mathematics with Applications, 64(2012), 1875-1881.

\bibitem{s1}  Hal Smith;
\emph{An Introduction to Delay Differential Equations with Applications to the Life Sciences},
Springer New York, 2011.

\bibitem{w1} Tingxiu Wang;
\emph{Inequalities and stability for a linear scalar functional differential equation},
J.Math.Anal.Appl., 298(2004), 33-44.

\bibitem{w2} Liping Wen, Yuexin Yu;
\emph{Generalized Halanay inequalities for dissipativity of Volterra functional differential equations},
J.Math.Anal.Appl., 347(2008), 169-178.

\bibitem{x1} Honglei Xu, Xinzhi Liu;
\emph{Delay independent stability criteria of impulsive switched systems with time-invariant delays},
Mathematical and Computer Modelling, 47(2008), 372-379.

\bibitem{y1} Chunde Yang, Wei Zhu;
\emph{Stability analysis of impulsive switched systems with time delays},
Mathematical and Computer Modelling, 50(2009), 1188-1194.

\bibitem{y2} Sun Ye, Anthony N. Michel, Guisheng Zhai;
\emph{Stability of discontinuous retarded functional differential equations with applications}, IEEE Transactions on Automatic Control, 50(2005), 1090¨C1105.

\bibitem{z1} A. A. M, Zhenqi Sun;
\emph{Practical Stability and Application}, Science Press, Beijing, 2004.

\end{thebibliography}


\end{document}