﻿ 一类中立型马尔科夫跳跃系统的随机稳定性 Stochastic Stability of a Class of Neutral Markovian Jumping Systems

Vol. 08  No. 02 ( 2019 ), Article ID: 28972 , 9 pages
10.12677/AAM.2019.82033

Stochastic Stability of a Class of Neutral Markovian Jumping Systems

Bang Luo, Changchun Shen, Juan Li, Shouwei Zhou

School of Data Science and Information Engineering, Guizhou Minzu University, Guiyang Guizhou

Email: lb_waclj@126.com

Received: Jan. 30th, 2019; accepted: Feb. 14th, 2019; published: Feb. 22nd, 2019

ABSTRACT

Stability is one of the most important properties of dynamical system, which has important theoretical significance to solve practical problems. The existence of time-delays is the root of system performance difference and systematic instability, so it has been considered by many scientists and scholars at home and abroad. In this paper, the stochastic stability of a class of neutral Markovian jumping systems is considered. Firstly, the Lyapunov function is constructed, and the sufficient conditions of stochastic stability are obtained by using Jensen's inequality. Secondly, the LMI toolbox in Matlab is used to verify the correctness of the results. Finally, two examples are given to verify the validity of this method.

Keywords:Neutral System, Markovian Jumping Systems, Stochastic Stability, Jensen’s Inequality

Email: lb_waclj@126.com

1. 引言

17世纪初，Newton和Leibniz发明了微积分，同时也开创了微分方程的研究。1771年，Condorcet导出历史上第一个时滞微分方程，时滞微分方程就是时间滞后的微分方程，用于描述即依赖当前状态也依赖过去状态的微分方程。时滞微分方程所构建的时滞微分系统在现实世界中有着广泛的应用：经济系统、机械系统、计算机网络系统、制造系统等 [1] 。中立型时滞系统的研究起步较早，并且取得非常丰硕的研究成果。中立型时滞系统是存在于客观世界中的一种典型系统，时滞的存在是系统不稳定和性能变差的主要原因之一。因此，研究中立型时滞系统的稳定性是非常有必要的 [2] [3] 。文献 [4] 考虑用多积分的方法对中立型时滞系统进行研究，并且取得很好的结果。文献 [5] 中沈博士分别从“离散时变时滞”、“分布式时变时滞”、“非线性扰动和不确定时变时滞”和“离散延迟随时间变化”等几个方面对中立型时滞系统的稳定性进行深入考虑，并且取得了很好的结果。

2. 系统描述

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)-{C}_{\left({r}_{t},t\right)}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)={A}_{\left({r}_{t},t\right)}x\left(t\right)+{B}_{\left({r}_{t},t\right)}x\left(t-h\left(t\right)\right),\hfill \\ x\left({t}_{0}+\theta \right)=0,\forall \theta \in \left[-2\rho ,0\right].\hfill \end{array}$ (1)

$\begin{array}{l}0\le h\left(t\right)\le h,0<\stackrel{˙}{h}\left(t\right)\le {\mu }_{1}<1,\\ 0\le \tau \left(t\right)\le \tau ,0<\stackrel{˙}{\tau }\left(t\right)\le {\mu }_{2}<1.\end{array}$ (2)

${A}_{\left({r}_{t},t\right)},{B}_{\left({r}_{t},t\right)},{C}_{\left({r}_{t},t\right)}$ 是已知常数矩阵， $\left\{{r}_{t}\right\},t>0$ 在有限状态概率空间 $\xi =\left\{1,2,3,\cdots ,N\right\}$ 中取值， $\Lambda =\left({\lambda }_{ij}\right)\left(i,j\in \xi \right)$ ，具有以下性质

$P\left(r\left(t+\Delta \right)=j|r\left(t\right)=i\right)=\left\{\begin{array}{ll}{\lambda }_{ij}\Delta +o\left(\Delta \right),\hfill & i\ne j\hfill \\ 1+{\lambda }_{ij}\Delta +o\left(\Delta \right),\hfill & i=j\hfill \end{array}$ (3)

$\Lambda =\left[\begin{array}{cccc}{\lambda }_{11}& ?& {\lambda }_{13}& {\lambda }_{14}\\ ?& {\lambda }_{22}& ?& ?\\ {\lambda }_{31}& ?& {\lambda }_{33}& {\lambda }_{34}\\ {\lambda }_{41}& ?& ?& ?\end{array}\right]$ (4)

${U}_{k}^{i}:=\left\{j:对于j\in \xi ,{\lambda }_{ij}已知\right\},$

${U}_{uk}^{i}:=\left\{j:对于j\in \xi ,{\lambda }_{ij}未知\right\}.$

3. 引理及结论

$\epsilon \left\{{\int }_{0}^{\infty }{‖x\left(t\right)‖}^{2}\text{d}t|\phi ,{r}_{0}\right\}<\infty$ (5)

$LV\left({x}_{t},t,i\right)=\underset{\Delta \to {0}^{+}}{\mathrm{lim}}\frac{1}{\Delta }\left[E\left\{V\left({x}_{t+\Delta },t+\Delta ,{r}_{t+\Delta }\right)|{x}_{t},{r}_{t}=i\right\}-V\left({x}_{t},t,i\right)\right]$ (6)

$-h{\int }_{t-h}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right)W\stackrel{˙}{x}\left(s\right)\text{d}s\le {\left[\begin{array}{c}x\left(t\right)\\ x\left(t-h\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}-W& W\\ W& -W\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x\left(t-h\right)\end{array}\right]$ (7)

$\left(\alpha -\beta \right){\int }_{\beta }^{\alpha }{x}^{\text{T}}\left(s\right)Sx\left(s\right)\text{d}s\ge \left({\int }_{\beta }^{\alpha }{x}^{\text{T}}\left(s\right)\text{d}s\right)S\left({\int }_{\beta }^{\alpha }x\left(s\right)\text{d}s\right)$ (8)

${A}_{\left({r}_{t},t\right)}={A}_{i},{B}_{\left({r}_{t},t\right)}={B}_{i},{C}_{\left({r}_{t},t\right)}={C}_{i},{P}_{\left({r}_{t},t\right)}={P}_{i}.$

$\Phi =\left[\begin{array}{c}\begin{array}{ccccccccc}{\varphi }_{11}& {A}_{i}^{\text{T}}N& {\varphi }_{13}& 0& {W}_{1}& {W}_{2}& 0& 0& {\varphi }_{19}\\ *& {\varphi }_{22}& N{B}_{i}& 0& 0& 0& 0& 0& N{C}_{i}\\ *& *& {\varphi }_{33}& 0& 0& 0& 0& 0& -{B}_{i}^{\text{T}}N{C}_{i}\\ *& *& *& {\varphi }_{44}& 0& 0& 0& 0& 0\\ *& *& *& *& -{W}_{1}& 0& 0& 0& 0\\ *& *& *& *& *& -{W}_{2}& 0& 0& 0\\ *& *& *& *& *& *& {\varphi }_{77}& 0& 0\\ *& *& *& *& *& *& *& {\varphi }_{88}& 0\\ *& *& *& *& *& *& *& *& {\varphi }_{99}\end{array}\end{array}\right]<0$ (9)

$\begin{array}{l}\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)<0;\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)<0;\\ {P}_{j}-{P}_{i}\le 0,j\in {U}_{uk}^{i},j\ne i;{R}_{1j}-{R}_{1i}\le 0,j\in {U}_{uk}^{i},j\ne i;\\ {R}_{1j}-{R}_{1i}\ge 0,j\in {U}_{k}^{i},j=i;{R}_{2j}-{R}_{2i}\ge 0,j\in {U}_{k}^{i},j=i;\\ {R}_{2j}-{R}_{2i}\le 0,j\in {U}_{uk}^{i},j\ne i.\end{array}$

$\begin{array}{l}{\varphi }_{11}={P}_{i}{A}_{i}+{A}_{i}^{\text{T}}{P}_{i}+{R}_{1i}+{R}_{2i}+{T}_{1}+{T}_{2}-{W}_{1}-{W}_{2}-{A}_{i}^{\text{T}}N{A}_{i}+\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({P}_{j}-{P}_{i}\right),\\ {\varphi }_{13}={P}_{i}{B}_{i}-{A}_{i}^{\text{T}}N{B}_{i},{\varphi }_{19}={P}_{i}{C}_{i}-{A}_{i}^{\text{T}}N{C}_{i},\\ {\varphi }_{22}={h}^{2}{W}_{1}+{\tau }^{2}{W}_{2}-N+Q,{\varphi }_{33}=-\left(1-{\mu }_{1}\right){R}_{1i}-{B}_{i}^{\text{T}}N{B}_{i},\\ {\varphi }_{44}=-\left(1-{\mu }_{2}\right){R}_{2i},{\varphi }_{77}=-\left(1-{\mu }_{1}\right){T}_{1},\\ {\varphi }_{88}=-\left(1-{\mu }_{2}\right){T}_{2},{\varphi }_{99}=-{C}_{i}^{\text{T}}N{C}_{i}-\left(1-{\mu }_{2}\right)Q.\end{array}$

$V\left({x}_{t},t,{r}_{t}\right)=\underset{n=1}{\overset{5}{\sum }}{V}_{n}\left({x}_{t},t,{r}_{t}\right)$ (10)

${V}_{1}\left({x}_{t},t,{r}_{t}\right)={x}^{\text{T}}\left(t\right){P}_{\left({r}_{t},t\right)}x\left(t\right),$

${V}_{2}\left({x}_{t},t,{r}_{t}\right)={\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right){R}_{1}{}_{\left({r}_{t},t\right)}x\left(s\right)\text{d}s+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right){R}_{2}{}_{\left({r}_{t},t\right)}x\left(s\right)\text{d}s,$

${V}_{3}\left({x}_{t},t,{r}_{t}\right)={\int }_{t-h\left(t\right)-\tau }^{t}{x}^{\text{T}}\left(s\right){T}_{1}x\left(s\right)\text{d}s+{\int }_{t-\tau \left(t\right)-h}^{t}{x}^{\text{T}}\left(s\right){T}_{2}x\left(s\right)\text{d}s,$

${V}_{4}\left({x}_{t},t,{r}_{t}\right)={\int }_{t-\tau \left(t\right)}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right)Q\stackrel{˙}{x}\left(s\right)\text{d}s，$

${V}_{5}\left({x}_{t},t,{r}_{t}\right)={\int }_{-h}^{0}{\int }_{t+\theta }^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right)h{W}_{1}\stackrel{˙}{x}\left(s\right)\text{d}s\text{d}\theta +{\int }_{-\tau }^{0}{\int }_{t+\theta }^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right)\tau {W}_{2}\stackrel{˙}{x}\left(s\right)\text{d}s\text{d}\theta .$

$\begin{array}{c}L{V}_{1}\left({x}_{t},t,i\right)=2{x}^{\text{T}}\left(t\right){P}_{i}\left[{A}_{i}x\left(t\right)+{B}_{i}x\left(t-h\left(t\right)\right)+{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)\right]+{x}^{\text{T}}\left(t\right)\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{P}_{j}x\left(t\right)\\ ={x}^{\text{T}}\left(t\right)\left({P}_{i}{A}_{i}+{A}_{i}^{\text{T}}{P}_{i}\right)x\left(t\right)+2{x}^{\text{T}}\left(t\right){P}_{i}{B}_{i}x\left(t-h\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+2{x}^{\text{T}}\left(t\right){P}_{i}{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)+{x}^{\text{T}}\left(t\right)\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{P}_{j}x\left(t\right)\end{array}$

$\begin{array}{c}L{V}_{2}\left({x}_{t},t,i\right)={x}^{\text{T}}\left(t\right){R}_{1i}x\left(t\right)-\left(1-\stackrel{˙}{h}\left(t\right)\right){x}^{\text{T}}\left(t-h\left(t\right)\right){R}_{1i}x\left(t-h\left(t\right)\right)+{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{1j}\right)x\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t\right){R}_{2i}x\left(t\right)-\left(1-\stackrel{˙}{\tau }\left(t\right)\right){x}^{\text{T}}\left(t-\tau \left(t\right)\right){R}_{2i}x\left(t-\tau \left(t\right)\right)+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{2j}\right)x\left(s\right)\text{d}s\\ \le {x}^{\text{T}}\left(t\right)\left({R}_{1i}+{R}_{2i}\right)x\left(t\right)-\left(1-{\mu }_{1}\right){x}^{\text{T}}\left(t-h\left(t\right)\right){R}_{1i}x\left(t-h\left(t\right)\right)+{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{1j}\right)x\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(1-{\mu }_{2}\right){x}^{\text{T}}\left(t-\tau \left(t\right)\right){R}_{2i}x\left(t-\tau \left(t\right)\right)+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{2j}\right)x\left(s\right)ds\end{array}$

$\begin{array}{c}L{V}_{3}\left({x}_{t},t,i\right)={x}^{\text{T}}\left(t\right){T}_{1}x\left(t\right)-\left(1-\stackrel{˙}{h}\left(t\right)\right){x}^{\text{T}}\left(t-h\left(t\right)-\tau \right){T}_{1}x\left(t-h\left(t\right)-\tau \right)+{x}^{\text{T}}\left(t\right){T}_{2}x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(1-\stackrel{˙}{\tau }\left(t\right)\right){x}^{\text{T}}\left(t-\tau \left(t\right)-h\right){T}_{2}x\left(t-\tau \left(t\right)-h\right)\\ \le {x}^{\text{T}}\left(t\right)\left({T}_{1}+{T}_{2}\right)x\left(t\right)-\left(1-{\mu }_{1}\right){x}^{\text{T}}\left(t-h\left(t\right)-\tau \right){T}_{1}x\left(t-h\left(t\right)-\tau \right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(1-{\mu }_{2}\right){x}^{\text{T}}\left(t-\tau \left(t\right)-h\right){T}_{2}x\left(t-\tau \left(t\right)-h\right)\end{array}$ (11)

$\begin{array}{c}L{V}_{4}\left({x}_{t},t,i\right)={\stackrel{˙}{x}}^{\text{T}}\left(t\right)Q\stackrel{˙}{x}\left(t\right)-\left(1-\stackrel{˙}{\tau }\left(t\right)\right){\stackrel{˙}{x}}^{\text{T}}\left(t-\tau \left(t\right)\right)Q\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)\\ \le {\stackrel{˙}{x}}^{\text{T}}\left(t\right)Q\stackrel{˙}{x}\left(t\right)-\left(1-{\mu }_{2}\right){\stackrel{˙}{x}}^{\text{T}}\left(t-\tau \left(t\right)\right)Q\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)\end{array}$ (12)

$\begin{array}{c}L{V}_{5}\left({x}_{t},t,i\right)={h}^{2}{\stackrel{˙}{x}}^{\text{T}}\left(t\right){W}_{1}\stackrel{˙}{x}\left(t\right)-h{\int }_{t-h}^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right){W}_{1}\stackrel{˙}{x}\left(s\right)\text{d}s+{\tau }^{2}{\stackrel{˙}{x}}^{\text{T}}\left(t\right){W}_{2}\stackrel{˙}{x}\left(t\right)-\tau {\int }_{t-\tau }^{t}{\stackrel{˙}{x}}^{\text{T}}\left(s\right){W}_{2}\stackrel{˙}{x}\left(s\right)\text{d}s\\ \le {h}^{2}{\stackrel{˙}{x}}^{\text{T}}\left(t\right){W}_{1}\stackrel{˙}{x}\left(t\right)+{\left[\begin{array}{cc}x\left(t\right)& x\left(t-h\right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}-{W}_{1}& {W}_{1}\\ {W}_{1}& -{W}_{1}\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x\left(t-h\right)\end{array}\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\tau }^{2}{\stackrel{˙}{x}}^{T}\left(t\right){W}_{2}\stackrel{˙}{x}\left(t\right)+{\left[\begin{array}{cc}x\left(t\right)& x\left(t-\tau \right)\end{array}\right]}^{\text{T}}\left[\begin{array}{cc}-{W}_{2}& {W}_{2}\\ {W}_{2}& -{W}_{2}\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x\left(t-\tau \right)\end{array}\right]\\ ={\stackrel{˙}{x}}^{\text{T}}\left(t\right)\left({h}^{2}{W}_{1}+{\tau }^{2}{W}_{2}\right)\stackrel{˙}{x}\left(t\right)+{x}^{\text{T}}\left(t\right)\left(-{W}_{1}-{W}_{2}\right)x\left(t\right)+{x}^{\text{T}}\left(t-h\right){W}_{1}x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t\right){W}_{1}x\left(t-h\right)+{x}^{\text{T}}\left(t-h\right)\left(-{W}_{1}\right)x\left(t-h\right)+{x}^{\text{T}}\left(t-\tau \right){W}_{2}x\left(t\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t\right){W}_{2}x\left(t-\tau \right)+{x}^{\text{T}}\left(t-\tau \right)\left(-{W}_{2}\right)x\left(t-\tau \right)\end{array}$ (13)

$-\underset{j=1,j\ne i}{\overset{N}{\sum }}{\lambda }_{ij}=0$ 可得

$-{x}^{\text{T}}\left(t\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{P}_{i}\right)x\left(t\right)=0,\forall i\in \xi$ (14)

$-{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{1i}\right)x\left(s\right)\text{d}s=0,\forall i\in \xi$ (15)

$-{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\left(\underset{j=1}{\overset{N}{\sum }}{\lambda }_{ij}{R}_{2i}\right)x\left(s\right)\text{d}s=0,\forall i\in \xi$ (16)

$\begin{array}{c}L{V}_{1}\left({x}_{t},t,i\right)={x}^{\text{T}}\left(t\right)\left({P}_{i}{A}_{i}+{A}_{i}^{\text{T}}{P}_{i}\right)x\left(t\right)+2{x}^{\text{T}}\left(t\right){P}_{i}{B}_{i}x\left(t-h\left(t\right)\right)+2{x}^{\text{T}}\left(t\right){P}_{i}{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{x}^{\text{T}}\left(t\right)\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({P}_{j}-{P}_{i}\right)x\left(t\right)+{x}^{\text{T}}\left(t\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({P}_{j}-{P}_{i}\right)x\left(t\right)\end{array}$ (17)

$\begin{array}{c}L{V}_{2}\left({x}_{t},t,i\right)\le {x}^{\text{T}}\left(t\right)\left({R}_{2i}+{R}_{1i}\right)x\left(t\right)-\left(1-{\mu }_{1}\right){x}^{\text{T}}\left(t-h\left(t\right)\right){R}_{1i}x\left(t-h\left(t\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)x\left(s\right)\text{d}s+{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)x\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(1-{\mu }_{2}\right){x}^{\text{T}}\left(t-\tau \left(t\right)\right){R}_{2i}x\left(t-\tau \left(t\right)\right)+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)x\left(s\right)\text{d}s\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)x\left(s\right)\text{d}s\end{array}$ (18)

${\Upsilon }^{\text{T}}\left(-N\right)\Upsilon =0$ (19)

$\Upsilon =-\stackrel{˙}{x}\left(t\right)+{A}_{i}x\left(t\right)+{B}_{i}x\left(t-h\left(t\right)\right)+{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)$

$\begin{array}{l}LV\left({x}_{t},t,i\right)=\underset{n=1}{\overset{5}{\sum }}L{V}_{n}\left({x}_{t},t,i\right)\le {\xi }^{\text{T}}\left(t\right)\Phi \xi \left(t\right)+{x}^{\text{T}}\left(t\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({P}_{j}-{P}_{i}\right)x\left(t\right)+{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)x\left(s\right)\text{d}s\\ +{\int }_{t-h\left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)x\left(s\right)\text{d}s+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)x\left(s\right)\text{d}s+{\int }_{t-\tau \left(t\right)}^{t}{x}^{\text{T}}\left(s\right)\underset{j={U}_{uk}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)x\left(s\right)\text{d}s\end{array}$

${\xi }^{\text{T}}\left(t\right)={\left[\begin{array}{ccccccccc}x\left(t\right)& \stackrel{˙}{x}\left(t\right)& x\left(t-h\left(t\right)\right)& x\left(t-\tau \left(t\right)\right)& x\left(t-h\right)& x\left(t-\tau \right)& x\left(t-h\left(t\right)-\tau \right)& x\left(t-\tau \left(t\right)-h\right)& \stackrel{˙}{x}\left(t-\tau \left(t\right)\right)\end{array}\right]}^{\text{T}}$

$\epsilon \left\{{\int }_{0}^{\infty }{‖x\left(t\right)‖}^{2}\text{d}t|\phi ,{r}_{0}\right\}<\infty$ (20)

$\left\{\begin{array}{l}\stackrel{˙}{x}\left(t\right)-{C}_{\left({r}_{t},t\right)}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)={A}_{\left({r}_{t},t\right)}x\left(t\right)+{B}_{\left({r}_{t},t\right)}x\left(t-h\left(t\right)\right)+{D}_{\left({r}_{t},t\right)}{\int }_{t-r}^{t}x\left(s\right)\text{d}s,\hfill \\ x\left({t}_{0}+\theta \right)=0,\forall \theta \in \left[-2\rho ,0\right].\hfill \end{array}$ (21)

$\stackrel{˜}{\Phi }=\left[\begin{array}{c}\begin{array}{cccccccccc}{\stackrel{˜}{\varphi }}_{11}& {A}_{i}^{\text{T}}N& {\stackrel{˜}{\varphi }}_{13}& 0& {W}_{1}& {W}_{2}& 0& 0& {\stackrel{˜}{\varphi }}_{19}& {\stackrel{˜}{\varphi }}_{1,10}\\ *& {\stackrel{˜}{\varphi }}_{22}& N{B}_{i}& 0& 0& 0& 0& 0& N{C}_{i}& N{D}_{i}\\ *& *& {\stackrel{˜}{\varphi }}_{33}& 0& 0& 0& 0& 0& {\stackrel{˜}{\varphi }}_{39}& {\stackrel{˜}{\varphi }}_{3,10}\\ *& *& *& {\stackrel{˜}{\varphi }}_{44}& 0& 0& 0& 0& 0& 0\\ *& *& *& *& -{W}_{1}& 0& 0& 0& 0& 0\\ *& *& *& *& *& -{W}_{2}& 0& 0& 0& 0\\ *& *& *& *& *& *& {\stackrel{˜}{\varphi }}_{77}& 0& 0& 0\\ *& *& *& *& *& *& *& {\stackrel{˜}{\varphi }}_{88}& 0& 0\\ *& *& *& *& *& *& *& *& {\stackrel{˜}{\varphi }}_{99}& {\stackrel{˜}{\varphi }}_{9,10}\\ *& *& *& *& *& *& *& *& *& {\stackrel{˜}{\varphi }}_{10,10}\end{array}\end{array}\right]<0$ (22)

$\begin{array}{l}\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{1j}-{R}_{1i}\right)<0;\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({R}_{2j}-{R}_{2i}\right)<0;\\ {P}_{j}-{P}_{i}\le 0,j\in {U}_{uk}^{i},j\ne i;{R}_{1j}-{R}_{1i}\le 0,j\in {U}_{uk}^{i},j\ne i;\\ {R}_{1j}-{R}_{1i}\ge 0,j\in {U}_{k}^{i},j=i;{R}_{2j}-{R}_{2i}\ge 0,j\in {U}_{k}^{i},j=i;\\ {R}_{2j}-{R}_{2i}\le 0,j\in {U}_{uk}^{i},j\ne i.\end{array}$

$\begin{array}{l}{\stackrel{˜}{\varphi }}_{11}={P}_{i}{A}_{i}+{A}_{i}^{\text{T}}{P}_{i}+{R}_{1i}+{R}_{2i}+{T}_{1}+{T}_{2}-{W}_{1}-{W}_{2}-{A}_{i}^{\text{T}}N{A}_{i}+rS+\underset{j={U}_{k}^{i}}{\sum }{\lambda }_{ij}\left({P}_{j}-{P}_{i}\right),\\ {\stackrel{˜}{\varphi }}_{13}={P}_{i}{B}_{i}-{A}_{i}^{\text{T}}N{B}_{i},{\stackrel{˜}{\varphi }}_{19}={P}_{i}{C}_{i}-{A}_{i}^{\text{T}}N{C}_{i},\end{array}$

$\begin{array}{l}{\stackrel{˜}{\varphi }}_{1,10}={P}_{i}{D}_{i}-{A}_{i}^{\text{T}}N{D}_{i},{\stackrel{˜}{\varphi }}_{22}={h}^{2}{W}_{1}+{\tau }^{2}{W}_{2}-N+Q,\\ {\stackrel{˜}{\varphi }}_{33}=-\left(1-{\mu }_{1}\right){R}_{1i}-{B}_{i}^{\text{T}}N{B}_{i},{\stackrel{˜}{\varphi }}_{39}=-{B}_{i}^{T}N{C}_{i},{\stackrel{˜}{\varphi }}_{3,10}=-{B}_{i}^{\text{T}}N{D}_{i},\\ {\stackrel{˜}{\varphi }}_{44}=-\left(1-{\mu }_{2}\right){R}_{2i},{\stackrel{˜}{\varphi }}_{77}=-\left(1-{\mu }_{1}\right){T}_{1},{\stackrel{˜}{\varphi }}_{88}=-\left(1-{\mu }_{2}\right){T}_{2},\\ {\stackrel{˜}{\varphi }}_{99}=-{C}_{i}^{\text{T}}N{C}_{i}-\left(1-{\mu }_{2}\right)Q,{\stackrel{˜}{\varphi }}_{9,10}=-{C}_{i}^{\text{T}}N{D}_{i},\\ {\stackrel{˜}{\varphi }}_{9,10}=-{D}_{i}^{\text{T}}N{D}_{i}-\left(1/r\right)S.\end{array}$

${V}_{6}\left({x}_{t},t,{r}_{t}\right)={\int }_{-r}^{0}{\int }_{t+\theta }^{t}{x}^{\text{T}}\left(s\right)Sx\left(s\right)\text{d}s\text{d}\theta$

$\begin{array}{c}L{V}_{6}\left({x}_{t},t,i\right)=r{x}^{\text{T}}\left(t\right)Sx\left(t\right)-{\int }_{t-r}^{t}{x}^{\text{T}}\left(s\right)Sx\left(s\right)\text{d}s\\ \le r{x}^{\text{T}}\left(t\right)Sx\left(t\right)-\frac{1}{r}\left({\int }_{t-r}^{t}{x}^{\text{T}}\left(s\right)\text{d}s\right)S\left({\int }_{t-r}^{t}x\left(s\right)\text{d}s\right)\end{array}$ (23)

${\stackrel{˜}{\Upsilon }}^{\text{T}}\left(-N\right)\stackrel{˜}{\Upsilon }=0$ (24)

$\stackrel{˜}{\Upsilon }=-\stackrel{˙}{x}\left(t\right)+{A}_{i}x\left(t\right)+{B}_{i}x\left(t-h\left(t\right)\right)+{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)+{D}_{i}{\int }_{t-r}^{t}x\left(s\right)\text{d}s$

4. 数值仿真

$\stackrel{˙}{x}\left(t\right)-{C}_{i}\left(t-\tau \left(t\right)\right)={A}_{i}x\left(t\right)+{B}_{i}\left(t-h\left(t\right)\right)$

${\lambda }_{ij}=\left[\begin{array}{cc}-0.8& 0.8\\ 1.4& -1.4\end{array}\right],i,j\in \xi$

$\begin{array}{l}\begin{array}{r}\hfill {A}_{1}=\left[\begin{array}{cc}-2.2& 0.1\\ -0.2& -1.4\end{array}\right]\end{array},\text{\hspace{0.17em}}{A}_{2}=\left[\begin{array}{cc}-3& 0.2\\ 0.2& -1\end{array}\right],\text{\hspace{0.17em}}\begin{array}{r}\hfill {B}_{1}=\left[\begin{array}{cc}-0.2& 0\\ 0.5& -0.1\end{array}\right]\end{array},\\ {B}_{2}=\left[\begin{array}{cc}0.4& -0.2\\ 0.3& 0.1\end{array}\right],\text{\hspace{0.17em}}\begin{array}{r}\hfill {C}_{1}=\left[\begin{array}{cc}-0.1& -0.1\\ 0& 0\end{array}\right]\end{array},\text{\hspace{0.17em}}{C}_{2}=\left[\begin{array}{cc}0.1& 0\\ 0& -0.1\end{array}\right].\end{array}$

$\stackrel{˙}{x}\left(t\right)-{C}_{i}\stackrel{˙}{x}\left(t-\tau \left(t\right)\right)={A}_{i}x\left(t\right)+{B}_{i}x\left(t-h\left(t\right)\right)+{D}_{i}{\int }_{t-r}^{t}x\left(s\right)\text{d}s$

${\lambda }_{ij}=\left[\begin{array}{cccc}0.9& 1.2& ?& ?\\ 0.2& ?& 0.2& 1\\ ?& ?& -1.6& 1.2\\ -0.2& -0.3& ?& 1.8\end{array}\right]$

${A}_{1}=\left[\begin{array}{cc}-0.55& -0.68\\ 2.10& -2.34\end{array}\right],\text{\hspace{0.17em}}{A}_{2}=\left[\begin{array}{cc}-0.24& -0.50\\ 1.60& 1.60\end{array}\right],\text{\hspace{0.17em}}{A}_{3}=\left[\begin{array}{cc}-1.2& 0.1\\ -0.1& 1.6\end{array}\right]$

${B}_{1}=\left[\begin{array}{cc}-0.24& -0.50\\ 1.70& -1.90\end{array}\right],\text{\hspace{0.17em}}{B}_{2}=\left[\begin{array}{cc}-1.20& -1.90\\ 1.30& 0.18\end{array}\right],\text{\hspace{0.17em}}{B}_{3}=\left[\begin{array}{cc}1.0& -1.0\\ -1.0& 1.0\end{array}\right]$

${C}_{1}=\left[\begin{array}{c}-1\\ 1\end{array}\right],\text{\hspace{0.17em}}{C}_{2}=\left[\begin{array}{c}-1\\ 0\end{array}\right],\text{\hspace{0.17em}}{C}_{3}=\left[\begin{array}{c}0\\ 0\end{array}\right],\text{\hspace{0.17em}}{D}_{1}={D}_{2}={D}_{3}=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$

Figure 1. The state trajectories of a neutral Markov jumping system with discrete and time-varying delays

Figure 2. The state trajectory of a neutral Markov jumping system with time-varying delay and distributed delay

5. 结论

Stochastic Stability of a Class of Neutral Markovian Jumping Systems[J]. 应用数学进展, 2019, 08(02): 292-300. https://doi.org/10.12677/AAM.2019.82033

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