Input-to-state stability of hybrid stochastic systems with unbounded delays and impulsive effects

Abstract

In this paper, we investigate input-to-state stability (ISS) problem for a general model of hybrid stochastic systems with unbounded delays and impulsive effects, in which stochastic disturbances involve white noise and Markov chain. Firstly, we establish a novel differential inequality with unbounded delays and variable inputs, which extends and improves some existing results. Then, by using the obtained inequality, the sufficient conditions are derived to determine ISS properties on impulsive robustness and impulsive stabilization for the impulsive stochastic systems with unbounded delays. The results not only show that the ISS properties still remain under certain impulsive perturbations for some continuous stable systems, but also indicate that an unstable system can be successfully stabilized to be input-to-state stable by impulses even if the corresponding continuous system is unstable. The obtained criteria are applied to study the ISS problems for impulsive stochastic neural networks and the systems of energy-storing electrical circuit. Finally, two numerical examples and their simulations are given to illustrate the validity of theoretical results.

Appendices

Appendix A: Proof of Theorem 4.1

Proof

Let $V_i(x(t),r)=\zeta ^p_i(r)|x_i(t)|^p$, $V(x(t),r)=(V_1(x(t),r)$, $V_2(x(t),r),\cdots ,V_n(x(t),r))^T$, where $x(t)=(x_1(t),x_2(t),\cdots ,x_n(t))^T$ is the solution of system (1) with the initial condition (2). We have

$$\begin{aligned}&\frac{\partial V_i(x(t),r)}{\partial x_i}=p\zeta ^p_i(r)|x_i(t)|^{p-1}sgn(x_i)\\&\qquad \qquad =p\zeta ^p_i(r)|x_i(t)|^{p-2}x_i(t),\\&\frac{\partial ^2 V_i(x(t),r)}{\partial x^2_i}=p(p-1)\zeta ^p_i(r)|x_i(t)|^{p-2}, \end{aligned}$$

where $sgn(\cdot )$ is the sign function. Then, by generalized $It{\hat{o}}$ formula, we have

$$\begin{aligned} {\mathscr {L}}V_i(x(t),r)= & {} p\zeta ^p_i(r)|x_i(t)|^{p-2}x_i(t)\\&\bigg [-\alpha _i(t,r)x_i(t)+\sum _{j=1}^na_{ij}(t,r)f_j(x_j(t)) \\&+\sum _{j=1}^nb_{ij}(t,r)g_j(x_j(t-\tau _{ij}(t))\\&+u_i(t)\bigg ]+\frac{1}{2}p(p-1)\zeta ^p_i(r)|x_i(t)|^{p-2} \\&\sum _{j=1} ^nh_{ij}^2(x_j(t),x_j(t-\tau _{ij}(t),r)\\&+\sum _{l=1}^m\gamma _{rl}\zeta ^p_i(l)|x_i(t)|^p\\\le & {} -p\alpha _i(t,r)\zeta ^p_i(r)|x_i(t)|^p\\&+p\sum _{j=1}^n\zeta ^p_i(r)|x_i(t)|^{p-1}|a_{ij}(t,r)||f_j(x_j(t))| \\&+p\sum _{j=1}^n\zeta ^p_i(r)|x_i(t)|^{p-1}\\&|b_{ij}(t,r)||g_j(x_j(t-\tau _{ij}(t))| \\&+\frac{1}{2}p(p-1)\sum _{j=1}^n\zeta ^p_i(r)|x_i(t)|^{p-2}\\&h^2_{ij}(x_j(t), x_j(t-\tau _{ij}(t))\\&+p\zeta ^p_i(r)|x_i(t)|^{p-1}u_i(t)\\&+\sum _{l=1}^m\gamma _{rl} \zeta ^p_i(l)|x_i(t)|^p\\\le & {} -p\alpha _i(t,r)\zeta ^p_i(r)|x_i(t)|^p\\&+p\sum _{j=1}^n|a_{ij}(t,r)\\&|M_j \zeta ^p_i(r)|x_i(t)|^{p-1} |x_j(t)| \\&+p\sum _{j=1}^n|b_{ij}(t,r)|L_j\zeta ^p_i(r)|x_i(t)|^{p-1}\\&|x_j(t-\tau _{ij}(t))|\\&+\frac{1}{2}p(p-1)\zeta ^p_i(r) \\&|x_i(t)|^{p-2}\sum _{j=1}^n(\mu _{ij}(r)x_j^2(t)\\&+\nu _{ij}(r)x_j^2(t-\tau _{ij}(t))) +(p-1)\zeta ^p_i(r)|x_i(t)|^{p} \\&+\zeta ^p_i(r)|u_i(t)|^p+\sum _{l=1}^m\gamma _{rl}\zeta ^p_i(l)|x_i(t)|^p\\\le & {} -p\alpha _i(t,r)\zeta ^p_i(r)|x_i(t)|^p+p\frac{\zeta _i(r)}{\zeta _j(r)}\\&\sum _{j=1}^n|a_{ij}(t,r)|M_j \zeta ^{p-1}_i(r)|x_i(t)|^{p-1}\\&\zeta _j(r) |x_j(t)|+p\sum _{j=1}^n|b_{ij}(t,r)|L_j\\&\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )}\zeta ^{p-1}_i(r)|x_i(t)|^{p-1} \\&\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )|x_j(t-\tau _{ij}(t))|\\&+\frac{1}{2}p(p-1)\\&\frac{\zeta ^{2}_i(r)}{\zeta ^{2}_j(r)}\zeta ^{p-2}_i(r)|x_i(t)|^{p-2} \\&\sum _{j=1}^n\mu _{ij}(r)\zeta ^{2}_j(r)|x_j(t)|^2\\+&\frac{1}{2}p(p-1)\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )}\\&\zeta ^{p-2}_i(r)|x_i(t)|^{p-2} \\&\sum _{j=1}^n \nu _{ij}(r) \min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )|x_j(t-\tau _{ij}(t))|^2\\&+(p-1)\zeta ^p_i(r)|x_i(t)|^{p} \\&+\zeta ^p_i(r)|u_i(t)|^p\\&+\sum _{l=1}^m \frac{\zeta ^p_i(l)}{\zeta ^p_i(r)} \gamma _{rl}\zeta ^p_i(r) |x_i(t)|^p\\\le & {} \sum _{j=1}^n\Big [{\hat{P}}_{ij}(t)V^{\frac{1}{p}}_{j}(x(t),r) V^{\frac{p-1}{p}}_{i}(x(t),r)\\&+ {\hat{Q}}_{ij}(t)\min \limits _{\iota \in {\mathbb {M}}}V^{\frac{1}{p}}_{j}(x(t-\tau _{ij}(t)),\iota )\\&V^{\frac{p-1}{p}}_{i}(x(t),r)\\&+{\hat{S}}_{ij}(t)V^{\frac{p-2}{p}}_{j}(x(t),r) V^{\frac{2}{p}}_{i}(x(t),r) +{\hat{T}}_{ij}(t)\\&\min \limits _{\iota \in {\mathbb {M}}}V^{\frac{2}{p}}_{j}(x(t-\tau _{ij}(t)),\iota )\Big ]\\&V^{\frac{p-2}{p}}_{i} (x(t),r)\\&+\max \limits _{r\in {\mathbb {M}}}\{\zeta ^p_i(r)|u_i(t)|^p\}. \end{aligned}$$

On the other hand, when $t=t_k,$ from ($A_3$) and ($A_5$) , we get

$$\begin{aligned} V_i(x(t_k^+),l)= & {} \zeta ^p_i({r}) |x_i(t_k^-)+I_{ik}(x(t_k^-))|^p\\\le & {} \zeta ^p_i({r}) [\sum _{j=1}^n d_{ijk}|x_j(t_k^{-})|]^p\\\le & {} \frac{\max \zeta ^p_i({r})}{\min \limits _{\iota \in {\mathbb {M}}} \zeta ^p_j({\iota })} (\sum _{m=1}^n d_{imk })^{p-1}\sum _{j=1}^n d_{ijk} \zeta ^p_j({v}) |x_j(t_k^{-})|^p\\\le & {} \sum _{j=1}^n {\hat{D}}_{ij} V_j(x(t_k^-),l). \end{aligned}$$

It follows from Theorem 3.1 that

$$\begin{aligned} E|x_i(t)|^p\le {\underline{\eta }}^{-1} E\Vert \phi \Vert ^p \frac{\sigma }{\omega (t)}+ \frac{e^\rho \overline{\eta } \sup \limits _{0\le s\le t}|u_i(s)|^p}{{\underline{\eta }}\delta }, t\ge 0, \end{aligned}$$

where ${\underline{\eta }}=\min \limits _{r\in {\mathbb {M}},1\le i\le n} \zeta ^p_i(r), \overline{\eta }=\max \limits _{r\in {\mathbb {M}},1\le i\le n} \zeta ^p_i(r)$. We get

$$\begin{aligned} E|x(t)|^p\le n {\underline{\eta }}^{-1} E\Vert \phi \Vert ^p \frac{\sigma }{\omega (t)}+ \frac{e^\rho \overline{\eta }\Vert u\Vert _\infty ^p}{{\underline{\eta }}\delta }. \end{aligned}$$

This finishes the proof of Theorem 4.1. $\square$

Appendix B: Proof of Theorem 4.2

Proof

Let $W_i(x(t),r)=\zeta ^p_i(r)|x_i(t)|^p$. According to Theorem 4.1, we have

$$\begin{aligned} {\mathscr {L}} W_i(x(t),r)&\le -p\alpha _i(t,r)\zeta ^p_i(r)|x_i(t)|^p\\&+p\sum _{j=1}^n\frac{\zeta _i(r)}{\zeta _j(r)}|a_{ij}(t,r)|M_j \zeta ^{p-1}_i(r)\\&|x_i(t)|^{p-1}\zeta _j(r) |x_j(t)|\\&+p\sum _{j=1}^n|b_{ij}(t,r)|L_j\\&\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )}\zeta ^{p-1}_i(r) \\&|x_i(t)|^{p-1}\\&\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )|x_j (t-\tau _{ij}(t))|\\&+\sum _{j=1}^n\frac{1}{2}p(p-1)\\&\frac{\zeta ^{2}_i(r)}{\zeta ^{2}_j(r)}\zeta ^{p-2}_i(r)|x_i(t)|^{p-2} \mu _{ij}(r)\zeta ^{2}_j(r)x_j^2(t)\\&+ \frac{1}{2}p(p-1)\sum _{j=1}^n\\&\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )}\zeta ^{p-2}_i(r)|x_i(t)|^{p-2} \nu _{ij}(r)\\&\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )x_j^2(t-\tau _{ij}(t))\\&+(p-1)\zeta ^p_i(r)|x_i(t)|^{p}+\zeta ^p_i(r)|u_i(t)|^p\\&+\sum _{l=1}^m\frac{\zeta ^p_i(l)}{\zeta ^p_i(r)}\gamma _{rl}\zeta ^p_i(r) |x_i(t)|^p\\&\le -p\alpha _i(t,r)\zeta ^p_i(r)|x_i(t)|^p\\&+(p-1)\sum _{j=1}^n\frac{\zeta _i(r)}{\zeta _j(r)}|a_{ij}(t,r)| M_j\zeta ^p_i(r)|x_i(t)|^p\\&+\sum _{j=1}^n\frac{\zeta _i(r)}{\zeta _j(r)}|a_{ij}(t,r)| M_j \zeta ^p_j(r)|x_j(t)|^p\\&+(p-1)\sum _{j=1}^n|b_{ij}(t,r)|L_j\\&\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )} \zeta ^p_i(r)|x_i(t)|^p\\&+\sum _{j=1}^n|b_{ij}(t,r)| L_j\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )}\\&\min \limits _{\iota \in {\mathbb {M}}}\zeta ^p_j(\iota )|x_j(t-\tau _{ij}(t))|^p\\&+\frac{(p-1)(p-2)}{2}\sum _{j=1}^n\frac{\zeta ^2_i(r)}{\zeta ^2_j(r)}\mu _{ij}(r) \zeta ^p_i(r)|x_i(t)|^p\\&+(p-1)\sum _{j=1}^n\\&\frac{\zeta ^2_i(r)}{\zeta ^2_j(r)} \mu _{ij}(r)\zeta ^p_j(r)|x_j(t)|^p\\&+\frac{(p-1)(p-2)}{2} \sum _{j=1}^n\nu _{ij}(r)\\&\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )} \zeta ^p_i|x_i(t)|^p\\&+(p-1)\sum _{j=1}^n\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )} \nu _{ij}(r)\min \limits _{\iota \in {\mathbb {M}}}\zeta ^p_j(\iota )|x_j(t-\tau _{ij}(t))|^p\\&+(p-1)\zeta ^p_i(r)|x_i(t)|^{p} +\zeta ^p_i(r)|u_i(t)|^p\\&+\sum _{l=1}^m \frac{\zeta ^p_i(l)}{\zeta ^p_i(r)}\gamma _{rl}\zeta ^p_i(r)|x_i(t)|^p\\&=-[p\alpha _i(t,r)-\sum _{j=1}^n(p-1)\\&(\frac{\zeta _i(r)}{\zeta _j(r)}|a_{ij}(t,r)| M_j+|b_{ij}(t,r)|L_j\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )})\\&-\frac{(p-1)(p-2)}{2}\sum _{j=1}^n\\&(\frac{\zeta ^2_i(r)}{\zeta ^2_j(r)}\mu _{ij}(r) +\nu _{ij}(r)\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )})\\&-a_{ii}(t,r)M_i-(p-1)\mu _{ii} \\&-(p-1)-\sum _{l=1}^m\frac{\zeta ^p_i(l)}{\zeta ^p_i(r)}\gamma _{rl}]\zeta ^p_i(r)|x_i(t)|^p\\&+\sum _{i\ne j,j=1}^n\Big (|a_{ij}(t,r)|L_j +(p-1)\frac{\zeta ^2_i(r)}{\zeta ^2_j(r)} \mu _{ij}(r)\Big )\\&\zeta ^p_j(r)|x_j(t)|^{p}\\&+\sum _{j=1}^n\Big (|b_{ij}(t,r)| L_j\\&\frac{\zeta _i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta _j(\iota )}+(p-1)\\&\frac{\zeta ^2_i(r)}{\min \limits _{\iota \in {\mathbb {M}}}\zeta ^2_j(\iota )} \nu _{ij}(r)\Big )\\&\min \limits _{\iota \in {\mathbb {M}}}\zeta ^p_j(\iota )|x_j(t-\tau _{ij}(t))|^p +\zeta ^p_i(r)|u_i(t)|^p\\&\le \sum _{j=1}^n {\tilde{P}}_{ij}(t) W_j(x(t),r) \\&+\sum _{j=1}^n {\tilde{Q}}_{ij}(t) \min \limits _{\iota \in {\mathbb {M}}}W_j(x(t-\tau _{ij}(t)),\iota )\\&+\max \limits _{\iota \in {\mathbb {M}}}\zeta ^p_i(\iota )|u_i(t)|^p, \end{aligned}$$

where ${\tilde{P}}_{ij}(t), {\tilde{Q}}_{ij}(t)$ are defined by (20) and $U_i(t)=\max \limits _{r\in {\mathbb {M}}}\{\zeta ^p_i(r)|u_i(t)|^p\}$. On the other hand, when $t=t_k$, we get

$$\begin{aligned} W_i(x(t_k),l)=\zeta ^p_i({r}) |x_i(t_k)|^p= & {} \zeta ^p_i({r}) |x_i(t_k^-)+I_{ik}(x(t_k^-))|^p\\= & {} \zeta ^p_i({r}) |x_i(t_k^-)+d_{iik}x(t_k^-)|^p\\\le & {} \zeta ^p_i({r}) \gamma |x_i(t_k^{-})|^p\\\le & {} \gamma V_i(x(t_k^-), l). \end{aligned}$$

It follows from Theorem 3.2 that

$$\begin{aligned}&E|x_i(t)|^p \le K\left[ \frac{e^{-\mu (t)}}{\min \limits _{1\le i\le n}z_i}\frac{E\Vert \phi \Vert ^p}{\gamma }\right. \\&\left. \quad +\frac{\max \limits _{\iota \in {\mathbb {M}}}\zeta ^p_i(\iota ) \Vert u\Vert _\infty }{\gamma \inf \limits _{ t\ge 0}-({\tilde{P}}_{ii}(t)z_i+\frac{\sum _{i\ne j,j=1}^n{\tilde{P}}_{ij}(t)+\sum _{j=1}^n{\tilde{Q}}_{ij}(t)}{\gamma }z_j +\frac{\ln \gamma }{\rho }z_i)}\right] , \end{aligned}$$

where $t\ge 0, K=\frac{z_i\max \limits _{r\in {\mathbb {M}},1\le i\le n} \zeta ^p_i(r)}{\min \limits _{r\in {\mathbb {M}},1\le i\le n} \zeta ^p_i(r)}$. Therefore, we have

$$\begin{aligned}&E|x(t)|^p \le \frac{nKe^{-\mu (t)}}{\min \limits _{1\le i\le n}z_i}\frac{E\Vert \phi \Vert ^p}{\gamma }\\&\quad +\frac{nK\max \limits _{\iota \in {\mathbb {M}}}\zeta ^p_i(\iota ) \Vert u\Vert _\infty }{\gamma \inf \limits _{ t\ge 0}-({\tilde{P}}_{ii}(t)z_i+\frac{\sum _{i\ne j,j=1}^n{\tilde{P}}_{ij}(t)+\sum _{j=1}^n{\tilde{Q}}_{ij}(t)}{\gamma }z_j +\frac{\ln \gamma }{\rho }z_i)}, \end{aligned}$$

where $t\ge 0$. The proof is completed. $\square$