随机切换非线性系统是一类重要的混杂动态系统,许多实际问题都可以用这类系统进行描述.众所周知,时滞在系统中无处不在,例如物理系统、信息科学、生物系统^[1-3].在实际系统中,时滞不仅存在于系统的状态变量中,而且存在于系统状态的导数项中,这样的系统称为中立型系统.近年来,由于中立型随机切换非线性系统在机械工程、电气工程和汽车工业等许多领域有着广泛应用^[4-6],因此,得到了越来越多的关注.

众所周知,稳定性是系统理论研究的主要问题之一.在时滞系统稳定性的研究中, Lyapunov-Razumikhin方法是最有效的研究方法之一. Razumikhin在文献[7-8]中研究确定性时滞系统的稳定性时最早提出了这种方法.随后,毛等将这种方法推广到了随机泛函微分系统^[9],在研究随机时滞微分系统的$P$阶矩指数稳定和几乎必然指数稳定时,毛将这种方法应用到随机泛函微分方程^[10]和中立型随机泛函微分方程^[11]中.此后, Razumikhin方法得到了快速的发展,并在其他随机泛函微分系统中得到了广泛的应用,如无限时滞随机泛函微分系统^[12-14],脉冲随机时滞微分系统^[15-17].近年来,一些研究者通过引入$\psi $函数把指数稳定性推广到了$\psi ^{\gamma }$稳定性^[18-19].

在上述文献中,研究者主要研究了随机时滞非线性系统和中立型随机非线性系统的$P$阶矩稳定性和几乎必然稳定性.然而,对于中立型随机切换非线性系统的研究却很少.文献[20]采用多Lyapunov函数方法研究了随机切换非线性系统的稳定性.在本文中,我们主要研究中立型随机切换非线性系统的$P$阶矩$\psi ^{\gamma }$稳定性和几乎必然$\psi ^{\gamma }$稳定性.基于Lyapunov方法的Razumikhin定理给出中立型随机切换非线性系统稳定的充分条件,采用Lyapunov-Razumikhin方法建立并证明中立型随机切换非线性系统的$P$阶矩$\psi ^{\gamma }$稳定性,采用Holder不等式, Burkholder-Davis-Gundy不等式和Borel-Cantelli引理建立并证明中立型随机切换非线性系统的几乎必然$\psi ^{\gamma }$稳定性.

本文的主要内容安排如下:第2节给出问题的描述及本文所用到的预备知识;第3节利用Lyapunov-Razumikhin型方法给出并证明中立型随机切换非线性系统$P$阶矩$\psi ^{\gamma }$稳定与几乎必然$\psi ^{\gamma }$稳定的主要结果;第4节通过一个算例来说明本文主要结果的有效性;第5节为本文的结论.

首先给出本文用到的一些记号. ${\mathbb{R}}^n $表示$n$维欧氏空间; ${\mathbb{R}}_+$表示所有的非负实数集; ${\mathbb{R}}^{n\times m}$表示$n\times m$阶实矩阵;若$A$是一个矩阵,它的范数$\|A\|$定义为$ \|A\|=\sup \{|Ax|:|x|=1\}$; $C([-\tau , 0];{\mathbb{R}}^n )$表示一族范数为$ \|\varphi \|= \sup\limits_{\theta \leq 0}|\varphi (\theta )|$的连续函数$\varphi:[-\tau , 0]\rightarrow {\mathbb{R}}^n $的集合; $|\cdot|$表示向量的欧氏范数;令$p \geq 1$,对任意的$t \geq 0$, $L^{p}_{{\cal F}_{t}}([-\tau , 0];{\mathbb{R}}^n )$表示一族$ {\cal F}_{t}$可测的$C([-\tau , 0];{\mathbb{R}}^n )$值随机变量$\varphi =\{\varphi (\theta ):-\tau \leq \theta \leq 0 \}$的全体，且$\varphi$满足$ \sup\limits_{\theta \leq 0}E|\varphi (\theta )|^{p}<\infty$; $C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$是一族$ {\cal F}_{0}$可测的$C([-\tau , 0];{\mathbb{R}}^n )$值有界随机变量; $ {\cal C}^{1, 2}([-\tau , \infty )\times {\mathbb{R}}^n ; {\mathbb{R}}_+)$表示所有在$[-\tau , \infty )\times {\mathbb{R}}^n $上的非负函数$V(t, x(t))$的全体，其中$V(t, x(t))$作为$x$的函数二阶连续可微，作为$t$的函数一阶连续可微; $a \vee b$表示$a$和$b$的最大值; $E(\cdot)$表示随机变量的期望.

考虑一族中立型随机切换非线性系统

(2.1) $\left\{ \begin{array}{ll}d(x(t)-u_{\sigma(t)}(x_{t}))=f_{\sigma (t)}(t, x_{t}){\rm d}t+g_{\sigma (t)}(t, x_{t}){\rm d}w(t), \\ x_{0}=\xi, \end{array} \right.$

其中$\sigma (t): [t_{0}, \infty)\rightarrow N$是切换信号, $\{t_{1}<t_{2}<\cdots <t_{k}<\cdots \}$是切换序列,在区间$[t_{k}, t_{k+1})$内,第$i_{k}$个子系统活跃,其中$t_{k}$是切换时刻, $i_{k}\in {\cal N}, \ k=0, 1, 2, \cdots$.系统(2.1)是由一系列子系统$d(x(t)-u_{i}(x_{t}))=f_{i}(t, x_{t}){\rm d}t+g_{i}(t, x_{t}){\rm d}w(t)$和作用在它上的切换信号$\sigma (t)$构成. $\xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$, $x_{t}=\{x(t+\theta ): -\tau \leq \theta \leq 0\}\in C([-\tau , 0];{\mathbb{R}}^n )$，$\tau $是一个有限数, $w(t)$是一个$m$维标准布朗运动. $(\Omega, {\cal F}, \{{\cal F}_{t}\}, P)$是一个以${\cal F}_{t}$为滤波的完备概率空间且满足通常的条件(即它是递增且右连续的, ${\cal F}_{0}$包含所有的P-0集).函数$u:C([-\tau , 0];{\mathbb{R}}^n )\rightarrow {\mathbb{R}}^n $, $f:{\mathbb{R}}_+\times C([-\tau , 0]; {\mathbb{R}}^n )\rightarrow {\mathbb{R}}^n $, $g:{\mathbb{R}}_+\times C([-\tau , 0];{\mathbb{R}}^n )\rightarrow R^{n\times m}$均连续可测,对于$\forall t\geq t_{0}$令$u(0)=0, \ f(t, 0)=0, \ g(t, 0)=0, t\in {[t_{0}, \infty)}$.

对于解的存在唯一性我们作如下假设

假设2.1^[18]  函数$f_{i}(t, \varphi )$和$g_{i}(t, \varphi )$均满足Lipschitz条件和线性增长条件,即对$\forall t\geq0$, $\forall \varphi , \phi \in C([-\tau , 0];{\mathbb{R}}^n )$,存在一族常数$L_{i}>0$使得

$ |f_{i}(t, \varphi )-f_{i}(t, \phi )|^{2}\vee|g_{i}(t, \varphi )-g_{i}(t, \phi )|^{2}\leq L_{i}\|\varphi -\phi \|^{2} , $

对$\forall t\geq0$, $\forall \varphi \in C([-\tau , 0];{\mathbb{R}}^n )$,存在一族常数$K_{i}>0$使得

$ |f_{i}(t, \varphi )|^{2}\vee|g_{i}(t, \varphi )|^{2}\leq K_{i}(1+\|\varphi \|^{2}), $

且函数$u_{i}(\varphi)$均满足压缩性条件,即存在一族常数$\kappa_{i} \in (0, 1)$,对$\forall \varphi , \phi \in C([-\tau , 0];{\mathbb{R}}^n )$,有

$|u_{i}(\varphi)-u_{i}(\phi)|\leq \kappa _{i}\|\varphi - \phi\|.$

定义2.1^[18]   $\psi(t)\in C^{1}([-\tau , \infty];{\mathbb{R}}_+)$称为$\psi $ -型函数,如果满足以下条件

(1)  该函数单调递减且连续可微;

(2)  $\psi (0)=1$且当$t\rightarrow \infty$时,有$\psi (t)\rightarrow0$;

(3)  $\psi_{1}(t)=\psi'(t)/\psi(t)<0$;

(4)  对于所有的$t, s\geq 0, \ \psi (t+s)\geq \psi (t)\psi (s)$.

定义2.2  对于$p>0$,中立型随机切换非线性系统(2.1)称为$P$阶矩$\psi ^{\gamma }$稳定,若存在正常数$\gamma $和函数$\psi(\cdot)$,使得

(2.2) $\limsup\limits_{t \rightarrow \infty}\frac{\ln E|x(t, \xi )|^{p}}{|\ln \psi (t)|}\leq -\gamma . $

当$p=2$时,称为均方$\psi ^{\gamma }$稳定,当$\psi (t)=e^{-t}$时,称为$P$阶矩指数稳定,当$\psi (t)=(1+t)^{-1}$时,称为$P$阶矩多项式稳定.

定义2.3  中立型随机切换非线性系统(2.1)称为几乎必然$\psi ^{\gamma }$稳定,若存在正常数$\bar{\gamma}$和函数$\psi(\cdot)$,使得

(2.3) $\limsup\limits _{t\rightarrow\infty}\frac{\ln |x(t, \xi )|}{|\ln \psi (t)|}\leq -\bar{\gamma} , {\rm a.s..}$

当$\psi (t)=e^{-t}$时,称为几乎必然指数稳定,当$\psi (t)=(1+t)^{-1}$时,称为几乎必然多项式稳定.

在给出本文引理之前,先介绍本文所用的$ {\rm{It\hat o}} $公式.

对于系统(2.1),任给函数$V(t, x)\in {\cal C}^{1, 2}([-\tau , \infty )\times {\mathbb{R}}^n ; {\mathbb{R}}_+)$,定义微分算子${\cal L}V(t, \varphi ): {\mathbb{R}}_+\times C([-\tau , 0];{\mathbb{R}}^n )$如下

$ \begin{eqnarray*} \label{eq:4} {\cal L}V(t, \varphi )&=& V_{t}(t, \varphi (0)-u(\varphi))+V_{x}(t, \varphi (0)-u(\varphi) )f(t, \varphi ) \\ &&+\frac{1}{2}{\rm tr}(g^{T}(t, \varphi )V_{xx}(t, \varphi (0)-u(\varphi))g(t, \varphi )). \end{eqnarray*}$

为便于本文定理的推导,记$x_{\sigma(t)}(t)=x(t)-u_{\sigma(t)}(x_{t})$,

$h(t)=\psi^{-\gamma}(t)E|x(t)|^{p}, H(t)= \sup\limits_{s \leq t} h(s);$

$h_{\sigma(t)}(t)=\psi^{-\gamma}(t)E|x_{\sigma(t)}(t)|^{p}, H_{\sigma(t)}(t)= \sup\limits_{s \leq t}h_{\sigma(t)}(s); $

$ \tilde{h}(t)=\max h_{\sigma(t)}(t), \tilde{H}(t)=\max H_{\sigma(t)}(t).$

引理2.1^[14]  对于任意常数$p\geq 1$, $\gamma \geq 0$,当$\gamma \in (0, \mu)$时,满足

(2.4) $E|u_{\sigma (t)}(\varphi)|^{p}\leq \kappa^{p} \sup\limits_{\theta \leq 0}\psi^{-\mu}(\theta)E|\varphi(\theta)|^{p}, $

则对于任意初始值$\xi$满足$ \sup\limits_{\theta \leq0}\psi ^{-\gamma}(\theta)E|\xi(\theta)|^{p}< \infty$时,有

$H(t)\leq \frac{ \sup\limits _{\theta \leq0}\psi ^{-\gamma}(\theta)E|\xi(\theta)|^{p}}{1-\kappa} \vee \frac{\tilde{H}(t)}{(1-\kappa)^{p}}.$

证  由$x_{\sigma(t)}(t)$的定义可知

$h(t)=\psi^{-\gamma}(t)E|x(t)|^{p}=\psi^{-\gamma}(t)E|x_{\sigma(t)}(t) + u_{\sigma(t)}(x_{t})|^{p}.$

由基本不等式,条件(2.4)和函数$\psi (\cdot)$的性质可知,对$\forall t\geq 0$, $\kappa \in (0, 1)$,有

$\begin{eqnarray*}h(t)& \leq&\psi^{-\gamma}(t)((1-\kappa)^{1-p}E|x_{\sigma(t)}(t)|^{p}+\kappa^{1-p}| u_{\sigma(t)}(x_{t})|^{p}) \\&\leq&\psi^{-\gamma}(t) ((1-\kappa)^{1-p}E|x_{\sigma(t)}(t)|^{p}+ \kappa \sup _{\theta \leq0}\psi ^{-\gamma}(\theta)E|x(t+\theta)|^{p})\\&=&(1-\kappa)^{1-p}h_{\sigma (t)}(t)+ \kappa \sup _{\theta \leq0}\psi ^{-\gamma}(t)\psi ^{-\gamma}(\theta)E|x(t+\theta)|^{p}\\&\leq&(1-\kappa)^{1-p}H_{\sigma (t)}(t)+ \kappa \sup _{\theta \leq0}\psi ^{-\gamma}(t+\theta)E|x(t+\theta)|^{p}\\&\leq&(1-\kappa)^{1-p}\tilde{H}(t)+ \kappa \sup _{\theta \leq0}\psi ^{-\gamma}(t+\theta)E|x(t+\theta)|^{p}\\&\leq&(1-\kappa)^{1-p}\tilde{H}(t)+ \kappa H(t).\end{eqnarray*}$

由$H(t)$的定义可知

$H(t) \leq \sup\limits _{s\leq 0}h(s) \vee \sup\limits _{0 \leq s \leq t}h(s)\leq \sup\limits _{s\leq 0}h(s) \vee ((1-\kappa)^{1-p}\tilde{H}(t)+ \kappa H(t)).$

由$\xi \in L_{{\cal F}_{0}}^{p}([-\tau , 0];{\mathbb{R}}^n )$,知$H(t)<\infty$.由$ \sup\limits_{\theta \leq 0}h(\theta)= \sup\limits_{\theta \leq0}\psi ^{-\gamma}(\theta)E|\xi(\theta)|^{p}< \infty$,可得

$H(t)\leq \frac{\sup\limits _{\theta \leq0}\psi ^{-\gamma}(\theta)E|\xi(\theta)|^{p}}{1-\kappa} \vee \frac{\tilde{H}(t)}{(1-\kappa)^{p}}.$

证毕.

引理2.2^[9]  (Burkholder-Davis-Cundy不等式) 当$0<p<\infty$时,有$g\in L^{2}([0, T];{\mathbb{R}}^{n\times m})$,则存在正常数$c_{p}$和$C_{p}$,使得

$c_{p}E \biggl(\int_{0}^{T}|g(t)|^{2}{\rm d}t \biggr)^{\frac{p}{2}}\leq E \biggl( \sup\limits_{0\leq t \leq T} \biggl|\int _{0}^{t}g(s){\rm d}w(s)\biggr|^{p}\biggr)\leq C_{p}E \biggl(\int _{0}^{T}|g(t)|^{2}{\rm d}t \biggr)^{\frac{p}{2}}, $

其中

$c_{p}=\Big(\frac{p}{2}\Big)^{p}, C_{p}=\Big(\frac{32}{p}\Big)^{\frac{p}{2}}, \mbox{若} \ p\in (0, 2);$

$c_{p}=1, C_{p}=4, \mbox{若} \ p=2; $

$c_{p}=(2p)^{\frac{p}{2}}, C_{p}=\Big(\frac{p^{p+1}}{2(p-1)^{p-1}}\Big)^{\frac{p}{2}}, \mbox{若} \ p>2.$

引理2.3^[9]  (Borel-Cantelli引理)

(1)  若$\{A_{k}\}\subset {\cal F}$且满足$\sum\limits _{k=1}^{\infty}P(A_{k})<\infty$,则有

$P \Big(\limsup\limits_{k\rightarrow \infty}A_{k} \Big)=0.$

即存在集合$\Omega _{o}\in {\cal F}$满足$P(\Omega _{o})=1$和一个整数值随机向量$k_{o}$,使得对于所有的$\omega \in \Omega_{o}$,当$k\geq k_{o}(\omega )$时,有$\omega tin A_{k}$.

(2)  若序列$\{A_{k}\}\subset {\cal F}$是独立的且满足$\sum\limits_{k=1}^{\infty}P(A_{k})=\infty$,则有

$P \Big(\limsup\limits_{k\rightarrow \infty}A_{k} \Big)=1.$

即存在集合$\Omega_{\theta }\in {\cal F}$满足$P(\Omega _{\theta })=1$,使得对于所有的$\omega \in \Omega_{\theta }$,存在一个子序列$\{A_{k_{i}}\}$,使得$\omega $属于每个$A_{k_{i}}$.

本节采用Razumikhin技术和Lyapunov函数建立中立型随机切换非线性系统的$P$阶矩$\psi ^{\gamma }$稳定与几乎必然$\psi ^{\gamma }$稳定的Razumikhin型理论.假设切换信号$\sigma(t)$是右连续的.

定理3.1  对于中立型随机切换非线性系统(2.1),若假设2.1成立,当$p\geq1$时,存在常数$\kappa \in (0, 1)$, $u\geq 0$,使得对$\forall \varphi \in L_{{\cal F}_{t}}^{p}([-\tau, 0];{\mathbb{R}}^n )$,有

(3.1) $ \begin{eqnarray} \label{eq:6}E|u_{i}(\varphi)|^{p}\leq \kappa^{p} \sup\limits_{\theta \leq 0}\psi^{-\mu}(\theta)E|\varphi(\theta)|^{p}, \end{eqnarray} $

且存在正常数$b_{i}, c_{i}, \mu $和一族Lyapunov函数$V_{i}(t, x)\in {\cal C}^{1, 2}([-\tau , \infty )\times {\mathbb{R}}^n ; {\mathbb{R}}_+)$及函数$\zeta(t) \in {\cal C}({\mathbb{R}}_+, {\mathbb{R}}_+)$,满足$\psi^{-\mu}(t)\zeta(t) \in L^{1}({\mathbb{R}}_+, {\mathbb{R}}_+)$,且

(3.2) $b_{i}|x|^{p}\leq V_{i}(t, x)\leq c_{i}|x|^{p}, $

(3.3) $E {\cal L}V_{i}(t, \varphi )\leq \zeta(t)+\mu \psi _{1}(t)E V_{i}(t, \varphi (0)-u_{i}(\varphi)), $

对$\forall t\geq 0, \forall \theta \in [-\tau , 0]$和$\varphi \in L^{p}_{{\cal F}_{t}}([-\tau , 0];{\mathbb{R}}^n )$,满足

(3.4) $E V_{i}(t+\theta , \varphi (\theta ))\leq q E V_{i}(t, \varphi (0)-u_{i}(\varphi)), $

其中$q\geq \frac{c}{b} (1-\kappa)^{-p} \psi ^{-\mu }(-\theta )$.在每一个切换时刻$t_{k}, \ (k=1, 2, \cdots)$,有

(3.5) $E V_{\sigma (t_{k+1})}(t_{k}, x_{\sigma (t_{k+1})}(t_{k}))\leq d_{k} E V_{\sigma (t_{k})}(t_{k}, x_{\sigma (t_{k})}(t_{k})), $

其中$0\leq d_{k} \leq \psi ^{\gamma }(t_{k+1}-t_{k})$.

则对任意初始值$\xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$,在区间$[t_{0}, \infty)$上,中立型随机切换非线性系统(2.1)存在唯一解$x(t)=x(t, \xi )$,使得系统(2.1)是$P$阶矩$\psi ^{\gamma }$稳定的,且有

(3.6) $E|x(t, \xi )|^{p}\leq \left(\frac{c(1+\kappa)^{p}}{b(1-\kappa)^{p}} \sup\limits_{\theta\leq0}\psi^{-\mu}(\theta)E|\xi(\theta)|^{p} +\frac{1}{b(1-\kappa)^{p}}|\zeta(t)\psi^{-\mu}(t)| \right)\psi^{\mu}(t).$

证  给定任意初始值$\xi \in C^{b}_{\mathcal {F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$,记$x(t)=x(t, \xi)$.对$\forall \gamma \in (0, \mu )$,只要(3.6)式成立,则令$\gamma \rightarrow \mu $,即可使定理得证.

给定任意时刻$t$和切换信号$\sigma (t)$,假设$t_{k}$是时刻$t$之前的最后一个切换时刻,即在区间$[t_{k}, t)$上没有切换发生.

对$\forall \gamma \in (0, \mu )$,只需证

(3.7) $ h(t)\leq\frac{c(1+\kappa)^{p}}{b(1-\kappa)^{p}} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p} + \frac{1}{b(1-\kappa)^{p}}\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s , $

即等价于证

(3.8) $H(t)\leq \frac{c(1+\kappa)^{p}}{b(1-\kappa)^{p}} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+ \frac{1}{b(1-\kappa)^{p}}\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s .$

由引理2.1可知,只需证

$ b \tilde{H}(t)\leq c(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+ \int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s , $

即等价于证

$ b \tilde{h}(t)\leq c(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+ \int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s . $

由(3.2)式可知

$ \psi^{-\gamma}(t) EV_{\sigma(t)}(t, x_{\sigma(t)}(t)) \leq c_{\sigma(t)}(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p} +\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s, $

即等价于证对$\forall t\geq 0$,有

(3.9) $ \begin{eqnarray}W_{\sigma(t)}(t)&=& \psi^{-\gamma}(t) EV_{\sigma(t)}(t, x_{\sigma(t)}(t)) - \int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \\& \leq & c_{\sigma(t)} (1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}=c_{\sigma(t)}\beta . \end{eqnarray}$

下证对$\forall t \geq 0$,有(3.9)式成立.

由$W_{\sigma (t)}(t)$的连续性, (3.1)式, (3.2)式和基本不等式得

$ \begin{eqnarray*} W_{\sigma(t)}(0)&=&EV_{\sigma(t)}(0, x_{\sigma(t)}(0)) \leq c_{\sigma(t)}E|x_{\sigma(t)}(0)|^{p} =c_{\sigma(t)}E|\xi(0)-u_{\sigma(t)}(\xi)|^{p}\\& \leq&c_{\sigma(t)} \biggl(\biggl(1-\frac{\kappa}{1+\kappa}\biggr)^{1-p} E|\xi(0)|^{p}+ \biggl(\frac{\kappa}{1+\kappa}\biggr)^{1-p}E|u_{\sigma(t)}(\xi)|^{p}\biggr)\\ &\leq&c_{\sigma(t)}\biggl(\biggl(1-\frac{\kappa}{1+\kappa}\biggr)^{1-p} E|\xi(0)|^{p}+ \biggl(\frac{\kappa}{1+\kappa}\biggr)^{1-p}\kappa ^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta) E|\xi(\theta)|^{p}\biggr)\\ &\leq&c_{\sigma(t)}(1+\kappa)^{p-1}(E|\xi(0)|^{p}+ \kappa \sup\limits_{\theta\leq0}\psi^{-\gamma} (\theta)E|\xi(\theta)|^{p})\\ &\leq&c_{\sigma(t)} (1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}\\ &=& c_{\sigma(t)}\beta.\end{eqnarray*}$

首先证

(3.10) $ W_{\sigma(t_{1})}(t) \leq c_{\sigma(t_{1})}\beta . $

采用反证法.假设(3.10)式不成立,由$W_{\sigma (t_{1})}(t)$的连续性可知,存在最小的$t^{*}\geq 0$,对$\forall t \in [0, t^{*}]$,有$W_{\sigma (t_{1})}(t)\leq c_{1}\beta$且$W_{\sigma (t_{1})}(t^{*})= c_{1}\beta$,对于充分小的$\delta $,有$W_{\sigma (t_{1})}(t^{*}+\delta )>W_{\sigma (t_{1})}(t^{*})$.则对$\forall \theta \in [-\tau , 0]$,若$t^{*}+\theta <0$, $t^{*}+\theta \in [-\tau , 0]$,由(3.2)式可得

(3.11) $ \begin{eqnarray}&& E V _{\sigma (t_{1})}(t^{*}+\theta, x_{\sigma (t_{1})}(t^{*}+\theta)) \\&\leq &c_{\sigma (t_{1})}E|x_{\sigma (t_{1})}(t^{*}+\theta)|^{p}=c_{\sigma (t_{1})}\psi^{\gamma}(t^{*}+\theta)h_{\sigma(t_{1})}(t^{*}+\theta) \\ & \leq &c_{\sigma (t_{1})}\psi^{\gamma}(t^{*}+\theta) \biggl(\frac{c_{\sigma (t_{1})}(1+\kappa)^{p}}{b_{\sigma (t_{1})}(1-\kappa)^{p}} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+\frac{1}{b_{\sigma (t_{1})}(1-\kappa)^{p}}\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr) \\&= &\frac{c_{\sigma (t_{1})}}{b_{\sigma (t_{1})}}(1-\kappa)^{-p} \biggl(c_{\sigma (t_{1})}(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr) \psi^{\gamma}(t^{*}+\theta) \\&\leq& \frac{c_{\sigma (t_{1})}}{b_{\sigma (t_{1})}}(1-\kappa)^{-p} \biggl(W_{\sigma(t_{1})}(t^{*})+\int _{0}^{t^{*}}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr)\psi^{\gamma}(t^{*}+\theta)\\& \leq &\frac{c_{\sigma (t_{1})}}{b_{\sigma (t_{1})}}(1-\kappa)^{-p} \psi^{-\gamma}(t^{*}) E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*}))\psi^{\gamma}(t^{*}+\theta)\\&\leq &\frac{c_{\sigma (t_{1})}}{b_{\sigma (t_{1})}}(1-\kappa)^{-p} \psi^{-\gamma}(-\theta) E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*})) \\ & \leq& q E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*})), \end{eqnarray} $

其中$q\geq \frac{c_{\sigma (t_{1})}}{b_{\sigma (t_{1})}}(1-\kappa)^{-p} \psi^{-\gamma}(-\theta)$.若$t^{*}+\theta \geq 0$,由于$t^{*}+\theta \in [0, t^{*}]$,故$W_{\sigma(t_{1})}(t^{*}+\delta )\leq c_{\sigma(t_{1})}\beta $.因此,有

(3.12) $ \begin{eqnarray} E V _{\sigma (t_{1})}(t^{*}+\theta , x_{\sigma (t_{1})}(t^{*}+\theta))& \leq &c_{\sigma (t_{1})} E|\xi (t^{*}+\theta)|^{p} \\ & \leq &\psi^{\gamma}(t^{*}+\theta)c_{\sigma (t_{1})}(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma} (\theta)E|\xi(\theta)|^{p}\\& =& \psi^{\gamma}(t^{*}+\theta)W_{\sigma (t_{1})}(t^{*}) \\ & \leq &\psi^{\gamma}(t^{*}+\theta) \psi^{-\gamma}(t^{*}) E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*})) \\&\leq &\psi^{-\gamma}(-\theta) E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*}))\\&\leq& q E V_{\sigma (t_{1})}(t^{*} , x_{\sigma (t_{1})}(t^{*})). \end{eqnarray} $

故,对$\forall \theta \in [-\tau , 0]$,有

$ E V_{\sigma (t_{1})}(t^{*}+\theta , x_{\sigma (t_{1})}(t^{*}+\theta)) \leq q E V_{\sigma (t_{1})}(t^{*}, x_{\sigma (t_{1})}(t^{*})).$

由(3.3)式得

$E {\cal L}V_{\sigma (t_{1})}(t^{*}, x_{t^{*}}) \leq \zeta(t)+\gamma \psi_{1}(t^{*}) E V_{\sigma (t_{1})}(t^{*}, x_{\sigma (t_{1})}(t^{*})).$

对$W_{\sigma (t_{1})}(t)$应用$ {\rm{It\hat o}} $,得

$\begin{eqnarray*} && W_{\sigma (t_{1})} (t^{*}+\delta )-W_{\sigma (t_{1})}(t^{*}) \\&=&\int_{t^{*}}^{t^{*}+\delta }\psi ^{-\gamma }(t)(E{\cal L}V_{\sigma (t_{1})}(t, x_{\sigma (t_{1})}(t)) -\gamma \psi_{1}(t)E V_{\sigma (t_{1})}(t, x_{\sigma (t_{1})}(t))-\zeta(t)){\rm d}t . \end{eqnarray*} $

由(3.3)式可知

$ W_{\sigma (t_{1})}(t^{*}+\delta )-W_{\sigma (t_{1})}(t^{*})\leq 0, $

这与假设矛盾.因此,对$\forall t\geq 0$, (3.10)式均成立.即对于$k=1$,有(3.9)式成立.

令$c=\max\{c_{i}\}$,假设对于$k=1, 2, \cdots , m, \ (m \in N, m\geq 1)$, (3.9)式均成立,即

(3.13) $ W_{\sigma(t_{k})}(t)= \psi^{-\gamma}(t)EV_{\sigma(t_{k})}(t, x_{\sigma(t_{k})}(t))-\int_{0}^{t}\zeta(s) \psi^{-\gamma}(s){\rm d}s \leq c_{\sigma(t_{k})}\beta, \ \ t \in [t_{k-1}, t_{k}).$

下证

(3.14) $W_{\sigma(t_{m+1})}(t) \leq c_{\sigma(t_{m+1})}\beta, \ \ t \in [t_{m}, t_{m+1}).$

令

(3.15) $ E V_{\sigma (t_{k})}(t_{k}, x_{\sigma (t_{k})}(t_{k})) \leq c_{\sigma (t_{k})} \beta \psi ^{\gamma }(t_{k})\leq c \beta \psi ^{\gamma }(t_{k}), \ \ k=1, 2, \cdots , m. $

采用反证法.假设(3.14)式不成立,由(3.5)和(3.15)式,可得

(3.16) $\begin{eqnarray} E V _{\sigma (t_{m+1})}(t_{m}, x_{\sigma (t_{m+1})}(t_{m}))& \leq &d_{m}E V_{\sigma (t_{m})}(t_{m}, x_{\sigma (t_{m})}(t_{m})) \\ & \leq& d_{m}c\beta\psi ^{\gamma }(t_{m})\\& \leq&c \beta \psi ^{\gamma }(t_{m})\psi ^{\gamma }(t_{m+1}-t_{m}) \\ & \leq& c \beta \psi ^{\gamma }(t_{m+1}) , \end{eqnarray} $

即

(3.17) $ W_{\sigma(t_{m+1})}(t_{m}) \leq c_{\sigma(t_{m+1})}\beta. $

由$W_{\sigma (t_{m+1})}(t)$的连续性知,存在最小的$t^{*} \in (t_{m}, t_{m+1})$,对$\forall t \in [t_{m}, t^{*}]$,有$W_{\sigma (t_{m+1})}(t) \leq c_{\sigma (t_{m+1})}\beta$且$W_{\sigma (t_{m+1})}(t^{*})= c_{\sigma (t_{m+1})}\beta$,对于充分小的$\delta $,有$W_{\sigma (t_{m+1})}(t^{*}+\delta )>W_{\sigma (t_{m+1})}(t^{*})$.因此,对$\forall \theta \in [-\tau , 0]$,若$t^{*}+\theta <t_{m}$,由(3.16)式可得

(3.18) $ \begin{eqnarray}&& E V _{\sigma (t_{m+1})}(t^{*}+\theta, x_{\sigma (t_{m+1})}(t^{*}+\theta)) \leq c_{\sigma (t_{m+1})}\psi^{\gamma} (t^{*}+\theta)h_{\sigma(t_{m+1})}(t^{*}+\theta) \\ &\leq&c_{\sigma (t_{m+1})}\psi^{\gamma}(t^{*}+\theta) \biggl(\frac{c_{\sigma (t_{m+1})}(1+\kappa)^{p}}{b_{\sigma (t_{m+1})} (1-\kappa)^{p}} \sup\limits_{\theta\leq0}\psi^{-\gamma} (\theta) E|\xi(\theta)|^{p} \\&&+ \frac{1}{b_{\sigma (t_{m+1})}(1-\kappa)^{p}}\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr) \\& =&\frac{c_{\sigma (t_{m+1})}}{b_{\sigma (t_{m+1})}}(1-\kappa)^{-p} \biggl(c_{\sigma (t_{m+1})}(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p}+ \int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr) \psi^{\gamma}(t^{*}+\theta) \\& \leq&\frac{c_{\sigma (t_{m+1})}}{b_{\sigma (t_{m+1})}}(1-\kappa)^{-p} \biggl(W_{\sigma (t_{m+1})}(t^{*}) +\int_{0}^{t^{*}}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr) \psi^{\gamma}(t^{*}+\theta)\\ &\leq&\frac{c_{\sigma (t_{m+1})}}{b_{\sigma (t_{m+1})}}(1-\kappa)^{-p} \psi^{-\gamma}(t^{*}) E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})) \psi^{\gamma}(t^{*}+\theta)\\ &\leq&\frac{c_{\sigma (t_{m+1})}}{b_{\sigma (t_{m+1})}}(1-\kappa)^{-p} \psi^{-\gamma}(-\theta) E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})) \\ &\leq&q E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})), \end{eqnarray} $

其中$q\geq \frac{c_{\sigma (t_{m+1})}}{b_{\sigma (t_{m+1})}}(1-\kappa)^{-p} \psi^{-\gamma}(-\theta)$.若$t^{*}+\theta \geq t_{m}$,由$t^{*}+\theta \in [t_{m}, t^{*}]$,可知$W_{\sigma (t_{m+1})}(t^{*}+\delta )\leq c_{\sigma (t_{m+1})}\beta$,于是,有

(3.19) $\begin{eqnarray}&&E V _{\sigma (t_{m+1})}(t^{*}+\theta , x_{\sigma (t_{m+1})}(t^{*}+\theta)) \\&\leq &c_{\sigma (t_{m+1})} E|\xi (t^{*}+\theta)|^{p} \leq c_{\sigma(t_{m+1})}\psi^{\gamma}(t^{*}+\theta) \sup\limits_{\theta\leq0}\psi^{-\gamma}(\theta)E|\xi(\theta)|^{p} \\&\leq& \psi^{\gamma}(t^{*}+\theta)c_{\sigma (t_{m+1})}(1+\kappa)^{p} \sup\limits_{\theta\leq0}\psi^{-\gamma} (\theta)E|\xi(\theta)|^{p} = \psi^{\gamma}(t^{*}+\theta)W_{\sigma (t_{m+1})}(t^{*})\\ & =& \psi^{\gamma}(t^{*}+\theta) \biggl(\psi^{-\gamma}(t^{*}) E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*}))-\int _{0}^{t}\zeta(s)\psi^{-\gamma}(s){\rm d}s \biggr)\\ & \leq& \psi^{\gamma}(t^{*}+\theta) \psi^{-\gamma}(t^{*}) E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})) \\&\leq &\psi^{-\gamma}(-\theta) E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})) \\&\leq& q E V_{\sigma (t_{m+1})}(t^{*} , x_{\sigma (t_{m+1})}(t^{*})) . \end{eqnarray} $

故,对$\forall \theta \in [-\tau , 0]$,有

$E V_{\sigma (t_{m+1})}(t^{*}+\theta , x_{\sigma (t_{m+1})}(t^{*}+\theta)) \leq q E V_{\sigma (t_{m+1})}(t^{*}, x_{\sigma (t_{m+1})}(t^{*})) .$

由(3.3)式,可得

$ E {\cal L}V_{\sigma (t_{m+1})}(t^{*}, x_{t^{*}}) \leq \zeta(t)+\gamma \psi_{1}(t^{*}) E V_{\sigma (t_{m+1})}(t^{*}, x_{\sigma (t_{m+1})}(t^{*})) .$

对$W_{\sigma (t_{m+1})}(t)$应用$ {\rm{It\hat o}} $公式,得

$ \begin{eqnarray*} & &W_{\sigma (t_{m+1})}(t^{*}+\delta )-W_{\sigma (t_{m+1})}(t^{*})\\ &=& \int_{t^{*}}^{t^{*}+\delta }\psi ^{-\gamma }(t)(E{\cal L}V_{\sigma (t_{m+1})}(t, x_{\sigma (t_{m+1})}(t))-\gamma \psi_{1}(t)E V_{\sigma (t_{m+1})}(t, x_{\sigma (t_{m+1})}(t))-\zeta(t)){\rm d}t.\end{eqnarray*} $

由(3.3)式,可知

$ W_{\sigma (t_{m+1})}(t^{*}+\delta )-W_{\sigma (t_{m+1})}(t^{*})\leq 0 .$

这与假设矛盾.故对$\forall t\geq 0$, (3.14)式均成立.

由数学归纳法,对$\forall k \in N$,有(3.9)式成立.因此,系统(2.1)是$P$阶矩$\psi ^{\gamma }$稳定的.

注3.1  定理3.1给出并证明了中立型随机切换非线性系统(2.1)的$P$阶矩$\psi^{\gamma}$稳定的充分条件.虽然Razumikhin定理在研究中立型随机系统的指数稳定^{[6, 9, 11]}和$\psi^{\gamma}$稳定^{[9, 13-14, 18]}中得到了广泛的应用,但通过定理3.1,我们将$\psi^{\gamma}$稳定性从随机非线性系统推广到了随机切换非线性系统.

定理3.2  对于中立型随机切换非线性系统(2.1),假设定理3.1的所有条件均成立,即存在常数$p \geq 2, \beta, \mu >0$,使得

(3.20) $ E|x(t)|^{p}\leq \beta \psi ^{\mu}(t), \ \ \forall t\geq 0, $

且存在常数$K_{i}>0$,使得

(3.21) $E|f_{i}(t, x_{t})|^{p}\vee E|g_{i}(t, x_{t})|^{p}\leq K_{i} \sup\limits_{\theta\leq0}\psi ^{-\mu}(\theta) E|x(t+\theta)|^{p}.$

对$\forall \xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$,若存在$\gamma \in (0, \mu)$,使得

(3.22) $\sum\limits_{k=0}^{\infty} \psi ^{\mu-\gamma}(t_{k})<\infty, $

则对任意初始值$\xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$和$\forall \gamma \in (0, \mu)$,在区间$[t_{0}, \infty)$内,中立型随机切换非线性系统(2.1)存在唯一解$x(t)=x(t, \xi )$,使得系统(2.1)是几乎必然$\psi ^{\gamma }$稳定的,且有

(3.23) $\limsup\limits_{t\rightarrow \infty}\frac{\ln |x(t, \xi )|}{|\ln \psi(t)|}\leq -\frac{\gamma}{p}, \ \ {\rm a.s.} .$

证  对任意初始值$\xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$,令$x(t)=x(t, \xi )$.要证(3.23)式成立,即证

(3.24) $\psi^{-\gamma}(t)|x(t)|^{p}\leq D, \ \ {\rm a.s. , }$

其中$D$是一个常数.记$ x_{\sigma(t)}(t)=x(t)-u_{\sigma(t)}(x_{t})$, $\tilde{x}(t)=\max x_{\sigma(t)}(t)$.由(3.1)式和引理2.1,得

(3.25) $ \sup\limits_{s\leq t}\psi^{-\gamma}(s)|x(s)|^{p} \leq\frac{ \sup\limits_{\theta \leq0}\psi ^{-\gamma}(\theta)E|\xi(\theta)|^{p}}{1-\kappa} \vee \frac{ \sup\limits_{s \leq t}\psi ^{-\gamma}(s)E|\tilde{x}(s)|^{p}}{(1-\kappa)^{p}} .$

由于$\xi \in C^{b}_{{\cal F}_{0}}([-\tau , 0];{\mathbb{R}}^n )$,所以有$ \sup_{\theta \leq 0}|\xi(\theta)|^{p}<\infty$.因此,要证(3.24)式成立,只需证$\psi^{-\gamma}(t)E|\tilde{x}(t)|^{p}$是几乎必然有界的.

选取充分小的$\delta $使得$0<\delta <t_{k}-t_{k-1}$,对于固定的$\delta $,令$k_{\delta }=[\frac{t_{k}-t_{k-1}}{\delta }]\in N$,其中$[x]$是不超过$x$的最大整数.则对$\forall t \in [t_{k-1}, t_{k})$,存在正整数$i$满足$1 \leq i \leq k_{\delta }+1$,使得当$t_{k-1}+(i-1)\delta \leq t \leq t_{k-1}+i\delta $时,对$\forall t \in [t_{k-1}, t_{k})$, $k \in N$,有

(3.26) $I_{t_{k}}=E\Big( \sup\limits_{t_{k-1}\leq t\leq t_{k}}|\tilde{x}(t)|^{p}\Big) \leq \sum\limits_{i=1}^{k_{\delta }+1} E\Big( \sup\limits_{t_{k-1}+(i-1)\delta \leq t\leq t_{k-1}+i\delta }|\tilde{x}(t)|^{p}\Big) =\sum\limits_{i=1}^{k_{\delta }+1} I_{i} .$

由$\tilde{x}(t)$的定义和系统(2.1),对任意的$i$,当$1\leq i \leq k_{\delta }+1$, $k \in N$时,有

(3.27) $\begin{eqnarray} I_{i}&=& E \Big( \sup\limits_{t_{k-1}+(i-1)\delta \leq t\leq t_{k-1}+i\delta }|\tilde{x}(t)|^{p} \Big)\\& =& E \biggl( \sup\limits_{t_{k-1}+(i-1)\delta \leq t\leq t_{k-1}+i\delta} \biggl|\tilde{x}(t_{k-1}+(i-1)\delta) +\int_{t_{k-1}+(i-1)\delta}^{t}f_{\sigma (s)}(s, x_{s}){\rm d}s\\&&+\int_{t_{k-1}+(i-1)\delta}^{t}g_{\sigma (s)}(s, x_{s}){\rm d}w(s)\biggr|^{p}\biggr) \\& \leq&c \biggl( E|\tilde{x}(t_{k-1}+(i-1)\delta)|^{p} + E \biggl(\int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}|f_{\sigma (s)}(s, x_{s})|{\rm d}s \biggr)^{p} \\&& + E \biggl( \sup\limits_{t_{k-1}+(i-1)\delta \leq t \leq t_{k-1}+i\delta} \biggl|\int _{t_{k-1}+(i-1)\delta}^{t}g_{\sigma (s) }(s, x_{s}){\rm d}w(s) \biggr|^{p} \biggr) \biggr) , \end{eqnarray}$

其中$c=3^{p-1}$.由(3.1)和(3.20)式,得

(3.28) $ \begin{eqnarray} && E|\tilde{x} (t_{k-1}+(i-1)\delta)|^{p} \leq E|x(t_{k-1}+(i-1)\delta)|^{p}+E|u_{\sigma(t)}(x_{t_{k-1}+(i-1)\delta})|^{p} \\&\leq &\beta\psi^{\mu}(t_{k-1}+(i-1)\delta)+\kappa^{p} \sup\limits_{t\leq t_{k-1}+(i-1)\delta} \psi^{-\mu}(t-t_{k-1}-(i-1)\delta)E|x(t)|^{p} \\ & \leq& \beta\psi^{\mu}(t_{k-1}+(i-1)\delta)+\kappa^{p} \sup\limits_{t\leq t_{k-1}+(i-1)\delta} \psi^{-\mu}(t-t_{k-1}-(i-1)\delta)\beta\psi^{\mu}(t) \\&\leq &\beta\psi^{\mu}(t_{k-1}+(i-1)\delta)+\kappa^{p}\beta \sup\limits_{t\leq t_{k-1}+(i-1)\delta} \psi^{\mu}(t_{k-1}+(i-1)\delta) \\ & \leq &\beta(1+\kappa^{p})\psi^{\mu}(t_{k-1}+(i-1)\delta) .\end{eqnarray}$

由Holder不等式, (3.20)式, (3.21)式和$\psi(t)$的性质,得

(3.29) $\begin{eqnarray} && E \biggl( \int _{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}|f_{\sigma (t) }(t, x_{t})|{\rm d}t \biggr)^{p} \leq \int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}E|f_{\sigma (t)}(t, x_{t})|^{p}{\rm d}t \\ & \leq &\int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}K_{\sigma (t)} \sup\limits_{s\leq t}\psi^{-\mu}(s-t)E |x(s)|^{p}{\rm d}t \\ & \leq &\int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}K_{\sigma (t)} \sup\limits_{s\leq t}\psi ^{-\mu}(s-t)\beta \psi ^{\mu}(s){\rm d}t \\&\leq &K_{\sigma (t)}\beta \int _{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}\psi ^{\mu}(t){\rm d}t\\ & \leq &K_{\sigma (t)} \beta \delta \psi^{\mu}(t_{k-1}+(i-1)\delta) . \end{eqnarray}$

类似地,由Holder不等式,引理2.2和(3.21)式,得

(3.30) $ \begin{eqnarray} &&E \biggl( \sup\limits_{t_{k-1}+(i-1)\delta\leq t \leq t_{k-1}+i\delta} \biggl|\int_{t_{k-1}+(i-1)\delta}^{t}g_{\sigma (s)}(s, x_{s}){\rm d}w(s) \biggr|^{p} \biggr)\\&\leq &C_{p}E \biggl(\int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta}|g_{\sigma (t)}(t, x_{t})|^{2}{\rm d}t \biggr)^{\frac{p}{2}} \leq C_{p}\int_{t_{k-1}+(i-1)\delta}^{t_{k-1}+i\delta} E|g_{\sigma (t)}(t, x_{t})|^{p}{\rm d}t\\&\leq &C_{p} K_{\sigma (t)} \beta \delta \psi^{\mu}(t_{k-1}+(i-1)\delta) , \end{eqnarray} $

其中$C_{p}$是一个仅与$p$有关的正常数.把(3.28)-(3.30)式代入(3.27)式,得

(3.31) $ \begin{eqnarray}I_{i}& \leq&c (\beta(1+\kappa^{p})\psi^{\mu}(t_{k-1}+(i-1)\delta) +K_{\sigma (t)} \beta \delta \psi^{\mu}(t_{k-1}+(i-1)\delta) \\&&+C_{p} K_{\sigma (t)} \beta \delta \psi^{\mu}(t_{k-1}+(i-1)\delta))\\& =& c \beta((1+\kappa^{p})+K_{\sigma (t)}\delta(1+C_{p}))\psi^{\mu}(t_{k-1}+(i-1)\delta) .\end{eqnarray} $

把(3.31)式代入(3.26)式得

$\begin{eqnarray*} I_{t_{k}}&=&E \Big( \sup\limits_{t_{k-1}\leq t\leq t_{k}}|\tilde{x}(t)|^{p}\Big)\leq \sum\limits_{i=1}^{k_{\delta }+1}I_{i}\\& \leq&\sum\limits_{i=1}^{k_{\delta }+1} c \beta((1+\kappa^{p})+K_{\sigma (t)}\delta(1+C_{p}))\psi^{\mu}(t_{k-1}+(i-1)\delta)\\ &\leq&c \beta((1+\kappa^{p})+K_{\sigma (t)}\delta(1+C_{p}))k_{\delta}\psi^{\mu}(t_{k-1}).\end{eqnarray*}$

令$K=\max \{K_{i}\}$,有

$I_{t_{k}} \leq C \psi^{\mu}(t_{k-1}) , $

其中$C=c\beta k_{\delta}((1+\kappa^{p})+K\delta(1+C_{p}))$.由Chebyshev不等式可知

$\begin{eqnarray*} && P\Big( \sup\limits_{t_{k-1}\leq t\leq t_{k}}|\tilde{x}(t)|^{p}\geq \psi^{\gamma}(t_{k})\Big)\leq I_{t_{k}}\psi^{-\gamma}(t_{k}) \\ & \leq& C\psi^{\mu}(t_{k-1})\psi^{-\gamma}(t_{k})\leq C\psi^{\mu-\gamma}(t_{k-1})\psi^{-\gamma}(t_{k}-t_{k-1}) .\end{eqnarray*}$

由(3.22)式和引理2.3,可知,当$t_{k}\rightarrow \infty$, $t_{k-1}\leq t \leq t_{k}$时,有

(3.32) $|\tilde{x}(t)|^{p}<\psi^{\gamma}(t_{k})<\psi^{\gamma}(t), $

即$\psi ^{-\gamma}(t)E|\tilde{x}(t)|^{p}$是几乎必然有界的.证毕.

注3.2  定理3.2给出并证明了中立型随机切换非线性系统(2.1)的几乎必然$\psi^{\gamma}$稳定的充分条件.在定理3.1的基础上增加条件(3.21)和(3.22), $P$阶矩$\psi^{\gamma}$稳定可推出几乎必然$\psi^{\gamma}$稳定.定理3.2的证明依赖于几乎必然有界，这种方法比较容易实现.

本节通过一个具体的算例验证上一节主要定理的有效性.

考虑一族中立型随机切换非线性系统

${\rm d}(x(t)-u_{\sigma(t)}(x_{t}))=f_{\sigma (t)}(t, x_{t}){\rm d}t+g_{\sigma (t)}(t, x_{t}){\rm d}w(t), $

其中$\sigma (t): [t_{0}, \infty)\rightarrow \{1, 2\}$是切换信号. $\{t_{1}<t_{2}< \cdots <t_{k}<\cdots \}$是切换序列,在区间$[t_{k}, t_{k+1})$内,第$i_{k}$个子系统活跃,其中$t_{k}$是切换时刻, $k=0, 1, \cdots$, $i_{k}\in \{1, 2\}$.

选取$\kappa=0.75$, $d_{k}=0.15$, $t_{k+1}-t_{k}=0.5$, $\tau =0.1$, $q=1.6$,对任意的$ \zeta(t)>0$,取$\psi (t)=\frac{e^{-t}}{1+t}, \ t\geq 0$,则有$\psi (0)=1, \psi (\infty)\rightarrow 0$, $\psi '(t)=-\frac{2+t}{(1+t)^{2}} e^{-t}$, $\psi_{1}(t)=-(1+\frac{1}{1+t})$, $-2\leq \psi _{1}(t)\leq -1$.

当$\sigma (t)=1$时,对于第一个子系统,选取函数$f_{1}(t, x)=-3\frac{x^{2}(t)-3}{3+x(t)} -{\rm sgn}(\frac{x^{2} (t)-3}{3+x(t)})+0.625x(t-0.1)$, $g_{1}(t, x)=\sqrt{\frac{2x^{2}(t)-6}{3+x(t)}}$, $u_{1}(x_{t})=3\frac{1+x(t)}{3+x(t)}$;当$\sigma (t)=2$时,对于第二个子系统,选取函数$f_{2}(t, x)=-7(\frac{x^{2}(t)-x(t)-2}{x(t)+2})+0.625x(t-0.1) $, $g_{2}(t, x)=\sqrt{5}x(t-0.1)$, $u_{2}(x_{t})=\frac{x(t)+3}{x(t)+2}$.

对于第一个子系统,选取$V_{1}(t, x)=8x^{2}$,有$x^{2}\leq V_{1}(t, x) \leq 8x^{2}$.若$x\neq 0$,则

$ \begin{eqnarray*} E {\cal L}V_{1}(t, x_{t})&=&E \biggl(16(x(t)-u_{1}(x_{t}))\biggl(-3\frac{x^{2}(t)-3}{3+x(t)}-{\rm sgn }\biggl( \frac{x^{2}(t)-3}{3+x(t)} \biggr)+0.625x(t-0.1)\biggr) \\ && +8\biggl(\sqrt{\frac{2x^{2}(t)-6}{3+x(t)}} \biggr)^{2} \biggr)\\& =& -48E|x(t)-u_{1}(x_{t})|^{2}-16E|x(t)-u_{1}(x_{t})|\\&&+10E|x(t)-u_{1}(x_{t})|E|x(t-0.1)| +16E|x(t)-u_{1}(x_{t})| \\& \leq&-48E|x(t)-u_{1}(x_{t})|^{2}+10q E|x(t)-u_{1}(x_{t})|^{2} \\ &=&-32E|x(t)-u_{1}(x_{t})|^{2} \leq 2\psi _{1}(t)E V_{1}(t, x-u_{1})\\ &\leq&\zeta(t)+2\psi _{1}(t)E V_{1}(t, x-u_{1}).\end{eqnarray*} $

若$x=0$,则有$V_{1}(t, x)=0, \ E {\cal L}V_{1}(t, x)=0$.

对于第二个子系统,选取$V_{2}(t, x)=x^{2}$,有$\frac{1}{2}x^{2}\leq V_{2}(t, x) \leq x^{2}$.若$x\neq 0$,则

$ \begin{eqnarray*} E {\cal L}V_{2}(t, x_{t})&=&E \biggl(2(x(t)-u_{2}(x_{t}))\biggl(-7 \biggl(\frac{x^{2}(t)-x(t)-2}{x(t)+2} \biggr)+0.625x(t-0.1)\biggr) \\ && +(\sqrt{5}x(t-0.1))^{2} \biggr) \\ &=& -14E|(x(t)-u_{2}(x_{t}))|^{2}+1.25E|(x(t)-u_{2}(x_{t}))|E|x(t-0.1)| \\&&+5E|x(t-0.1)|^{2}\\& \leq&-14E|(x(t)-u_{2}(x_{t}))|^{2}+1.25q E|(x(t)-u_{2}(x_{t}))|^{2}+5q E|(x(t)-u_{2}(x_{t}))|^{2}\\& =& -4E|(x(t)-u_{2}(x_{t}))|^{2} \leq 2\psi _{1}(t)E V_{2}(t, x-u_{2})\\ &\leq&\zeta(t)+2\psi _{1}(t)E V_{2}(t, x-u_{2}) .\end{eqnarray*}$

若$x=0$,则有$V_{2}(t, x)=0, \ E {\cal L}V_{2}(t, x)=0$.

由定理3.1,选取$p=2, \ \gamma =2$,则有$E {\cal L}V_{i}(t, x)\leq \zeta(t)+\gamma \psi _{1}(t)E V_{i}(t, x-u_{t})$, $i=1, 2$,即定理3.1的所有条件均满足.所以中立型随机切换非线性系统是$P$阶矩$\psi ^{\gamma }$稳定的.切换信号和状态轨迹的图像如图 1, 图 2.

注4.1  本算例构造了一个中立型随机切换非线性系统,通过MATLAB仿真说明了定理主要结果的有效性. 图 1描述了切换信号随时间的变化, 图 2描述了状态轨迹随时间的变化.本例说明中立型随机切换非线性系统是$P$阶矩$\psi ^{\gamma }$稳定的.

本文究了中立型随机切换非线性系统的$P$阶矩$\psi ^{\gamma }$稳定与几乎必然$\psi ^{\gamma }$稳定.首先,利用Lyapunov-Razumikhin方法给出了系统稳定的充分条件,其次,通过一个具体的算例说明了所构造定理的有效性.下一步,我们将对脉冲中立型随机切换非线性系统的$\psi ^{\gamma }$稳定性进行研究.