Robust Fuzzy Model-Based $H_2/H_\infty$ Control for Markovian Jump Systems with Random Delays and Uncertain Transition Probabilities

Abstract

This paper studies the mixed $H_2/H_\infty$ control for Takagi–Sugeno (T–S) fuzzy Markovian jump systems (MJSs) subject to random delays and multiple uncertain transition probabilities. In contrast to existing research, this study presents uncertainty parameters, external disturbance, random delays, and uncertain transition probabilities simultaneously in a unified T–S fuzzy model. Specifically, this study examines multiple Markov chains with partially unknown transition probabilities. These complex imperfections have a substantial adverse impact on system performance and the associated challenge of mixed $H_2/H_\infty$ control remains unresolved. Our innovative contributions are described as follows. The proposed approach utilizes free-weighting matrix technique and Lyapunov–Krasovskii functional to get the $H_2/H_\infty$ controller, which ensures that the stochastic T–S fuzzy systems exhibit stochastic stability and comply with the $H_\infty$ performance index.

Appendices

Appendix 1

Proof

First, define

$$\begin{aligned} P_{im}&=\sum _{s=1}^rh_s(\mu _{k})P_{s,im},\,\,P_{jn}=\sum _{f=1}^rh_f(\mu _{k})P_{f,jn},\\ R_1&=\sum _{s=1}^rh_s(\mu _{k})R_{1s}, \quad R_2=\sum _{s=1}^rh_s(\mu _{k})R_{2s},\\ {\bar{R}}_1&=\sum _{s=1}^rh_s(\mu _{k}){\bar{R}}_{1s}=\sum _{s=1}^rh_s(\mu _{k})(1-{\underline{\pi }}_m)R_{1s},\\ {\bar{P}}_{im}&=\sum _{j=1}^M\sum _{n=0}^N{\hat{\delta }}_{ij}{\hat{\pi }}_{mn}P_{jn}\\&=\sum _{f=1}^r\sum _{j=1}^M\sum _{n=0}^N h_f(\mu _{k}){\hat{\delta }}_{ij}{\hat{\pi }}_{mn}P_{f,jn}\\&=\sum _{f=1}^rh_f(\mu _{k}){\bar{P}}_{f,im}.\end{aligned}$$

Starting from (16) and (17) together with (9) and (11), we can utilize the Schur complement to obtain the following relation

$$\begin{aligned}&\Theta _{im}=\sum _{s=1}^r\sum _{l=1}^r\sum _{f=1}^rh_sh_lh_f(\mu _{k})\Theta _{fsl,im} \\&\quad =\sum _{f=1}^rh_f(\mu _{k})\Bigg (\sum _{s=1}^rh_s^2(\mu _{k})\Theta _{ss,im}+\sum _{s=1}^{r-1}\sum _{l=s+1}^rh_sh_l(\mu _{k}) \\&\qquad \times (\Theta _{sl,im}+\Theta _{ls,im})\Bigg ) <0, \end{aligned}$$

(42)

where

$$\begin{aligned} \Theta _{im}&=\text {sym}\{\bar{\mathbb {G}}^T\mathbb {A}_{i,m}\}+\begin{bmatrix} \tilde{\Upsilon }_{11}&{}{\bar{P}}_{im}&{}R_2&{}0\\ *&{}\tilde{\Upsilon }_{22}&{}0&{}0\\ *&{}*&{}\tilde{\Upsilon }_{33}&{}R_2\\ *&{}*&{}*&{}\tilde{\Upsilon }_{44} \end{bmatrix},\\ \mathbb {A}_{im}&=\begin{bmatrix}{\bar{A}}_{ci}-I&-I&{\bar{A}}_{di}&0\end{bmatrix},\\ \tilde{\Upsilon }_{11}&=-P_{im}+{\bar{P}}_{im}+R_1+(1+{\bar{d}}-{\underline{d}}){\bar{R}}_1-R_2,\\ \tilde{\Upsilon }_{22}&={\bar{P}}_{im}+{\bar{d}}^2R_2,\quad \tilde{\Upsilon }_{33}=-R_1-2R_2,\\ \tilde{\Upsilon }_{44}&={\bar{R}}_1-R_2.\end{aligned}$$

Then, we adopt the following novel Lyapunov–Krasovskii functions with

$$\begin{aligned} V(x_{k})=\sum _{q=1}^4 V_q(x_{k}), \end{aligned}$$

where $\forall y_k=i\in {\mathbb{M}}$ and $\forall d_k=m\in \mathbb {N}$

$$\begin{aligned} V_1(x_{k})&=x_{k}^TP_{im}x_{k},\\ V_2(x_{k})&=\sum _{\tau =k-d_k}^{k-1}x_{\tau }^TR_1x_{\tau },\\ V_3(x_{k})&=\sum _{\theta =-{\bar{d}}+1}^{-{\underline{d}}+1}\sum _{\tau =k+\theta -1}^{k-1}x_{\tau }^T{\bar{R}}_1 x_{\tau },\\ V_4(x_{k})&=\sum _{\theta =-{\bar{d}}+1}^0\sum _{\tau =k+\theta -1}^{k-1}{\bar{d}}\rho _{\tau }^TR_{2}\rho _{\tau },\\ \rho _{\tau }&=x_{\tau +1}-x_{\tau }.\end{aligned}$$

Given $y_k=i,y_{k+1}=j,d_k=m$ and $d_{k+1}=n$, we denote $\textbf{E}[\Delta V_k]$ the expectation of the difference of every term in $V_{q}(x_{k})$ for $q=1,2,\ldots ,4$.

To be specific, define

$$\begin{aligned} \xi _{k} = \begin{bmatrix} x_{k}^T&\rho _{k}^T&x_{k-d_k}^T&x_{k -{\bar{d}}}^T\end{bmatrix}^T. \end{aligned}$$

(43)

For any $\bar{\mathbb {G}}$ associated with the system (11), $\text {let}\, \bar{\mathbb {G}}=.$ $\left. \begin{bmatrix}G_1&G_2&G_3&0\end{bmatrix}\right)$, we can derive

$$\begin{aligned} 2\xi _{k}^T \bar{\mathbb {G}}^T (({\bar{A}}_{ci}-I )x_{k}+{\bar{A}}_{di}x_{k-d_k}-\rho _{k})=0 \end{aligned}$$

(44)

when $v_{k}=0$. Then, we obtain

$$\begin{aligned}&\textbf{E}[\Delta V_1] \\&\quad = \textbf{E}\{(x_{k}+\rho _{k})^T{\bar{P}}_{im}(x_{k}+\rho _{k})\}- \textbf{E}\{x_{k}^TP_{im}x_{k}\} \\&\qquad +2\xi _{k}^T\bar{\mathbb {G}}^T(({\bar{A}}_{ci}-I)x_{k}+{\bar{A}}_{di}x_{k-d_k}-\rho _{k}).\end{aligned}$$

(45)

Combining (6), we obtain

$$\begin{aligned}&\textbf{E}[\Delta V_2] \\&\quad =\pi _{mm}\left( \sum _{\tau =k+1-m}^k-\sum _{\tau =k-m}^{k-1}\right) x_{\tau }^TR_1x_{\tau } \\&\qquad +\sum _{n=0,n\ne m}^N\pi _{mn}\left( \sum _{\tau =k+1-n}^k-\sum _{\tau =k-d_k}^{k-1}\right) x_{\tau }^TR_1x_{\tau } \\&\quad =x_{k}^TR_1x_{k}-x_{k-d_k}^TR_1x_{k-d_k}+\sum _{n=0,n\ne m}^N\pi _{mn} \\&\qquad \times \left( \sum _{\tau =k-{\underline{d}}+1}^{k-1}+\sum _{\tau =k+1-n}^{k-d}-\sum _{l=k-d_k+1}^{k-1}\right) x_{\tau }^TR_1x_{\tau }.\end{aligned}$$

(46)

In fact, we have

$$\begin{aligned} \sum _{\tau =k-{\underline{d}}+1}^{k-1}x_{\tau }^TR_1x_{\tau }&\le \sum _{\tau =k-d_k+1}^{k-1}x_{\tau }^TR_1x_{\tau }. \end{aligned}$$

(47)

$$\begin{aligned} \sum _{\tau =k+1-n}^{k-{\underline{d}}}x_{\tau }^TR_1x_{\tau }&\le \sum _{\tau =k+1-{\bar{d}}}^{k-{\underline{d}}}x_{\tau }^TR_1x_\tau.\end{aligned}$$

(48)

Taking (46)–(48) into account, we obtain that

$$\begin{aligned} \textbf{E}[\Delta V_2]&\le x_{k}^TR_1x_{k}-x_{k-d_k}^TR_1x_{k-d_k} \\&\quad -x_{k -{\bar{d}}}^T{\bar{R}}_1x_{k -{\bar{d}}}+\sum _{\tau =k-{\bar{d}}}^{k-{\underline{d}}}x_{\tau }^T{\bar{R}}_1x_{\tau }.\end{aligned}$$

(49)

Moreover,

$$\begin{aligned} \textbf{E}[\Delta V_3]&=\sum _{\theta =-{\bar{d}}+1}^{-{\underline{d}}+1}\left( \sum _{\tau =k+\theta }^k-\sum _{\tau =k+\theta -1}^{k-1}\right) x_{\tau }^T{\bar{R}}_1x_{\tau } \\&=({\bar{d}}-{\underline{d}}+1)x_{\tau }{\bar{R}}_1x_{\tau }-\sum _{\tau =k-{\bar{d}}}^{k-{\underline{d}}}x_{\tau }^T{\bar{R}}_1x_{\tau }, \end{aligned}$$

(50)

$$\begin{aligned} \textbf{E}[\Delta V_4]&=\sum _{\theta =-{\bar{d}}+1}^0\left( \sum _{\tau =k+\theta }^k-\sum _{\tau =k+\theta -1}^{k-1}\right) {\bar{d}}\rho _{\tau }^TR_2\rho _{\tau } \\&={\bar{d}}^2\rho _{k}^TR_2\rho _{k}-\sum _{\tau =k-{\bar{d}}}^{k-1}{\bar{d}}\rho _{\tau }^TR_2\rho _{\tau }.\end{aligned}$$

(51)

By Jensen’s inequality, one has that

$$\begin{aligned}&-\sum _{\tau =k-{\bar{d}}}^{k-1}{\bar{d}}\rho _{\tau }^TR_2\rho _{\tau } \\&\quad =-\left( \sum _{\tau =k-d_k}^{k-1}+\sum _{\tau =k-{\bar{d}}}^{k-d_k-1}\right) (d_k+{\bar{d}}-d_k)\rho _{\tau }^TR_2\rho _{\tau } \\&\quad \le -\left( \sum _{\tau =k-d_k}^{k-1}\rho _{\tau }\right) ^TR_2\left( \sum _{\tau =k-d_k}^{k-1}\rho _{\tau }\right) \\&\qquad -\left( \sum _{\tau =k-{\bar{d}}}^{k-d_k-1}\rho _{\tau }\right) ^TR_2\left( \sum _{\tau =k-{\bar{d}}}^{k-d_k-1}\rho _{\tau }\right) \\&\quad =-(x_{k}-x_{k-d_k})^TR_2(x_{k}-x_{k-d_k}) \\&\qquad -\left( x_{k-d_k}-x_{k -{\bar{d}}}\right) ^TR_2\left( x_{k-d_k}-x_{k -{\bar{d}}}\right).\end{aligned}$$

(52)

Substituting (52) into (51) and combining (45), (49), and (50), we can infer that

$$\begin{aligned} \textbf{E}[\Delta V]\le \xi _{k}^T\Theta _{im}\xi _{k}.\end{aligned}$$

(53)

According to $\Theta _{im}<0$, we obtain

$$\begin{aligned} \textbf{E}[\Delta V]\le -\mathbb {\delta }_{\min }(-\Theta _{im})\xi _{k}^T\xi _{k}\le -\beta x_{k}^Tx_k, \end{aligned}$$

(54)

where $\beta =inf\{\mathbb {\delta }_{\min }(-\Theta _{im},i\in {\mathbb{M}},m\in \mathbb {N})\}$. Then, for each $T\ge 1$, one has

$$\begin{aligned} \textbf{E}[V_{k+1}(x_{k+1})-V_{0}(x_0)]\le -\beta \sum _{k=0}^T \textbf{E}[x_{k}^Tx_k].\end{aligned}$$

(55)

Moreover,

$$\begin{aligned} \sum _{k=0}^T \textbf{E}[x_{k}^Tx_k]&\le \frac{1}{\beta } \textbf{E}[V_{0}(x_0)-V_T(x_T)] \\&\le \frac{1}{\beta }{} \textbf{E}[V_{0}(x_0)], \quad T>1, \end{aligned}$$

(56)

which indicates that

$$\begin{aligned} \sum _{k=0}^\infty \textbf{E}[x_{k}^Tx_k]\le \frac{1}{\beta }{} \textbf{E}[V_{0}(x_0)]<\infty.\end{aligned}$$

The Definition 1 leads to the conclusion that this situation indicates that (11) is stochastically stable.

Furthermore, one obtains that

$$\begin{aligned} \textbf{E}&[\Delta V]+ \textbf{E}[z_{k}^Tz_k] \\&\le \xi _{k}^T\left\{ \Theta _{im}+\begin{bmatrix}{\bar{C}}_i&0&0&0\end{bmatrix}^T\begin{bmatrix}{\bar{C}}_i&0&0&0\end{bmatrix}\right\} \xi _{k} \\&=\xi _{k}^T\Gamma _{im}\xi _{k}, \end{aligned}$$

(57)

where

$$\begin{aligned} \Gamma _{im}&=\begin{bmatrix}-I&{}\Xi \\ *&{}\Theta _{im}\end{bmatrix},\quad \Xi =\begin{bmatrix}{\bar{C}}_i&0&0&0\end{bmatrix}.\end{aligned}$$

Based on the Schur complement, $\Gamma _{im}<0$ is derived from (16) and

$$\begin{aligned} \textbf{E}[\Delta V]+ \textbf{E}[z_{k}^Tz_k]<0, \end{aligned}$$

(58)

which yields that

$$\begin{aligned} \sum _{k=0}^\infty \textbf{E}[z_k^Tz_k]<-\sum _{k=0}^\infty \textbf{E}[\Delta V], .\end{aligned}$$

Then, in accordance with (54)–(56), we get

$$\begin{aligned} -\sum _{k=0}^\infty \textbf{E}[\Delta V]&\le \beta \sum _{k=0}^\infty \textbf{E}[x_k^Tx_k]\\&\le \textbf{E}[V_0(x_0)] \le V (x_0,y_0,d_0).\end{aligned}$$

Thus, when $k\rightarrow \infty$, it follows that $V_{k}(x_{k})\rightarrow \infty$. Similarly, one has that

$$\begin{aligned} J_2=\sum _{k=0}^{\infty } \textbf{E}[z_{k}^Tz_k\vert \psi _0,y_0,d_0]\le V(x_0,y_0,d_0).\end{aligned}$$

(59)

Therefore, (18) is obtained straightforwardly from (59). $\square$

Appendix 2

Proof

Define $\mathcal {A}=diag\{I,X,\omega _1X,\omega _2X,X\}$, $G_1^{-1}=X$, $G_2^{-1}=\omega _1X$, and $G_3^{-1}=\omega _2X$, where the tuning parameters $\omega _1>0$ and $\omega _2>0$ are known a priori. We also provide the following notations

$$\begin{aligned} {\hat{P}}_{im}&=X^TP_{im}X,\,\hat{{\bar{P}}}_{im}=X^T{\hat{P}}_{im}X,\,{\hat{R}}_2=X^T R_2X,\\ {\hat{R}}_1&=X^TR_1X,\, K_{im}X=Y_{im}.\end{aligned}$$

By pre- and post-multiplying $\mathcal {A}^T$ and $\mathcal {A}$ in (16), we have

$$\begin{aligned} \begin{bmatrix}-I&{}{\tilde{\Xi }}_{ss}\\ *&{} {\tilde{\Theta }}_{fss,im}\end{bmatrix}<0, \end{aligned}$$

(60)

where

$$\begin{aligned} {\tilde{\Theta }}_{fss,im}&\\ =\text {sym}&\left[ {\begin{array}{*{20}{c}} ( A_{s,i}+\Delta A_{ss}-I)X-( B_{s,i}+\Delta B_{ss})Y_{im}\\ ( A_{s,i}+\Delta A_{ss}-I)X-( B_{s,i}+\Delta B_{ss})Y_{im}\\ ( A_{s,i}+\Delta A_{ss}-I)X-( B_{s,i}+\Delta B_{ss})Y_{im}\\ 0 \end{array}}\right. \\&\left. { \begin{array}{*{20}{c}} &{}-\omega _1X&{}\omega _2( A_{ds,i}+\Delta A_{ds,i})X&{}0\\ &{}-\omega _1X&{}\omega _2( A_{ds,i}+\Delta A_{ds,i})X&{}0\\ &{}-\omega _1X&{}\omega _2( A_{ds,i}+\Delta A_{ds,i})X&{}0\\ &{}0&{}0&{}0\\ \end{array} } \right] \\&\quad +\begin{bmatrix} {\bar{\Upsilon }}_{11}&{}\omega _1\hat{\bar{P}}_{im}&{}\omega _2 {\bar{R}}_2&{}0\\ *&{}{\bar{\Upsilon }}_{22}&{}0&{}0\\ *&{}*&{}{\bar{\Upsilon }}_{33}&{}{\hat{R}}_2\\ *&{}*&{}*&{}{\bar{\Upsilon }}_{44} \end{bmatrix},\\ \Delta A_{ss}&=H_{s,i}F_{sk}E_{1s,i},\, \Delta B_{ss}=H_{s,i}F_{sk}E_{3s,i},\\ {\bar{\Upsilon }}_{11}&=-{\hat{P}}_{s,im}+\hat{{\bar{P}}}_{f,im}+{\hat{R}}_{1s}\\&\quad +({\bar{d}}-{\underline{d}}+1)(1-{\underline{\pi }}_m){\hat{R}}_{1s} -{\hat{R}}_{2s},\\ {\bar{\Upsilon }}_{22}&=\omega _1^2\hat{{\bar{P}}}_{f,im}+\omega _1^2{\bar{d}}^2{\hat{R}}_{2s},\\ {\bar{\Upsilon }}_{33}&=-\omega _2^2{\hat{R}}_{1s}-2\omega _2^2{\hat{R}}_{2s},\\ {\bar{\Upsilon }}_{44}&=-(1-\pi _m){\hat{R}}_{1s}-{\hat{R}}_{2s},\\ \hat{{\bar{P}}}_{f,im}&=\sum _{j=1}^M\sum _{n=0}^N{\hat{\delta }}_{ij}\pi _{mn}{\hat{P}}_{f,jn}.\end{aligned}$$

By applying (3) and (5), the above equation becomes

$$\begin{aligned} \bar{{\hat{P}}}_{f,im}&=\sum _{j=1}^M\sum _{n=0}^N{\hat{\delta }}_{ij}{\hat{\pi }}_{mn}{\hat{P}}_{f,jn} \\&=\sum _{j=1}^M\sum _{n=0}^N(\delta _{ij}\pi _{mn}+\delta _{ij}\Delta \pi _{mn} \\&\quad +\Delta \delta _{ij}\pi _{mn}+\Delta \delta _{ij}\Delta \pi _{mn}){\hat{P}}_{f,jn}.\end{aligned}$$

(61)

Under $\vert \Delta \pi _{mm}\vert \le {\bar{\varepsilon }}_1$, it follows from (7) that

$$\begin{aligned} \sum _{j=1}^M&\sum _{n=0}^N\delta _{ij}\Delta \pi _{mn}{\hat{P}}_{f,jn} \\&=\sum _{j=1}^M\delta _{ij}\sum _{n=0,n\ne m}^N \Delta \pi _{mn}({\hat{P}}_{f,jn}-{\hat{P}}_{f,jm}) \\&\le \sum _{j=1}^M\delta _{ij}\Bigg (\frac{1}{4}{\bar{\varepsilon }}_1^2U_1+({\hat{P}}_{f,jm}-{\hat{P}}_{f,j,n\ne m})^TU_1^{-1} \\&\quad \times ({\hat{P}}_{f,jm}-{\hat{P}}_{f,j,n\ne m})\Bigg ).\end{aligned}$$

(62)

Similarly, it generates that

$$\begin{aligned}&\sum _{j=1}^M\sum _{n=0}^N\Delta \delta _{ij}\pi _{mn}{\hat{P}}_{f,jn} \\&\quad \le \sum _{n=0}^N\pi _{mn}\Bigg (\frac{1}{4}{\bar{\varepsilon }}_2^2U_2+({\hat{P}}_{f,in}-{\hat{P}}_{f,j\ne i,n})^T \\&\times U_2^{-1}({\hat{P}}_{f,in}-{\hat{P}}_{f,j\ne i,n})\Bigg ), \end{aligned}$$

(63)

and

$$\begin{aligned}&\sum _{j=1}^M\sum _{n=0}^N\Delta \delta _{ij}\Delta \pi _{mn}{\hat{P}}_{f,jn} \\&\le \sum _{j=0}^M\sum _{n=0}^N\left( \frac{1}{4}{\bar{\varepsilon }}_{ij}^2{\bar{\varepsilon }}_{mn}^2U_3+{\hat{P}}_{f,jn}^TU_3^{-1}{\hat{P}}_{f,jn}\right).\end{aligned}$$

(64)

Using the Schur complement and Lemma 1, together with (61)–(64) and (17), we can easily derive (28) and (29), respectively, from (16). $\square$

Appendix 3

Proof

We generate the same Lyapunov–Krasovskii functions for the system (11) as shown in Appendix 1. From (35) and (36), we can get

$$\begin{aligned} \Phi _{im}&=\sum _{s=1}^r\sum _{l=1}^rh_sh_l(\mu _k)\Phi _{sl,im} \\&=\sum _{s=1}^rh_s^2(\mu _k)\Phi _{ss,im} \\&\quad +\sum _{s=1}^{r-1}\sum _{l=s+1}^{r}h_sh_l(\mu _k)(\Phi _{sl,im}+\Phi _{ls,im}).\end{aligned}$$

(65)

For any nonzero $v_{k}\in L_2[0,\infty )$, define $\zeta _{k}=\begin{bmatrix}\xi _{k}^T&v_{k}^T\end{bmatrix}$. Then,

$$\begin{aligned}&\textbf{E}[\Delta V]+ \textbf{E}[\gamma ^{-1}z_{k}^Tz_k-\gamma v_{k}^Tv_k] \\&\le \zeta _{k}^T\left\{ \text {sym}\Big \{\begin{bmatrix}G_1&G_2&G_3&0&0\end{bmatrix}^T\right. \begin{bmatrix}{\bar{A}}_{ci}-I&-I&{\bar{A}}_{di}&0&{\bar{B}}_v\end{bmatrix}\Big \} \\&\quad +\begin{bmatrix}{\tilde{\Upsilon }}_{11}&{}{\bar{P}}_{im}&{}R_2&{}0&{}0\\ *&{}{\tilde{\Upsilon }}_{22}&{}0&{}0&{}0\\ *&{}*&{}{\tilde{\Upsilon }}_{33}&{}R_2&{}0\\ *&{}*&{}*&{}{\tilde{\Upsilon }}_{44}&{}0\\ *&{}*&{}*&{}*&{}-\gamma I \end{bmatrix} \\&\quad \left. +\begin{bmatrix} {\bar{C}}_{i}&0&0&0&0 \end{bmatrix}^T\gamma ^{-1}\begin{bmatrix} {\bar{C}}_{i}&0&0&0&0 \end{bmatrix} \right\} \zeta _{k} \\&=\zeta _{k}^T\Phi _{im}\zeta _{k}, \end{aligned}$$

(66)

and

$$\begin{aligned} J_T&= \textbf{E}\sum _{k=0}^{T-1}[\gamma ^{-1}z_{k}^Tz_k-\gamma v_{k}^Tv_k].\end{aligned}$$

(67)

For $x_0=0,k=-{\bar{d}},\ldots ,-1$, it follows that $V_{0}(x_0)=V(\psi _0,y_0,d_0)$ and

$$\begin{aligned} \textbf{E}[V_T(x_T)]= \textbf{E}\left[ \sum _{k=0}^{T-1}(V_{k+1}(x_{k+1})-V_{k}(x_{k})\right] >0.\end{aligned}$$

(68)

Accordingly, it implies that

$$\begin{aligned} J_T&\le \textbf{E}\left[ \sum _{k=0}^{T-1}(V_{k+1}(x_{k+1})-V_{k}(x_{k})\right. \\&\quad \left. +\gamma ^{-1}z_{k}^Tz_k-\gamma v_{k}^Tv_k)\right] \\&=\sum _{k=0}^{T-1}\left[ \zeta _{k}^T\Phi _{im}\zeta _{k} \right].\end{aligned}$$

(69)

Given that (35)and (36) guarantees $\Phi _{im}<0$, it concludes

$$\begin{aligned} \lim _{T\rightarrow \infty }\sum _{k=0}^{T-1}[\zeta _{k}^T\Phi _{im}\zeta _{k}]<0.\end{aligned}$$

(70)

For each $v_{k}\in L_2[0,\infty ),z_{k}\in L_2[0,\infty )$, it yields that

$$\begin{aligned} \parallel z_{k}\parallel _2\le \gamma ^2\parallel v_{k}\parallel _2.\end{aligned}$$

(71)

According to Definition 2, the system (11) reaches the given $\gamma >0$ and is stochastically stable. $\square$