# Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays

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## Abstract

**:**

## 1. Introduction

- (i)
- A DDS theorem for exponential stability of ISDSs is founded by using an appropriate LKF, which can be verified by the feasibility of LMIs.
- (ii)
- When mean-square stability is considered, we propose sufficient conditions for this kind of ISDSs, and the established DDS criterion does not rely on the existence of delays in the diffusion term.

## 2. Preliminaries

**Definition**

**1.**

- (i)
- The set of impulses $\mho =\left\{t\in \left({t}_{0},T\right]\mid t={t}_{k},\right.$ $\left.k\in {\mathbb{N}}^{+}\right\}$ is finite;
- (ii)
- For $t\in \mho $, $x\left(t\right)$ is right-continuous, i.e., $x\left({t}_{k}\right)=x\left({t}_{k}^{+}\right)$. $x\left(t\right)$ is continuous for all non-impulsive times (i.e., $t\in ({t}_{0},T]\setminus \mho $) and ${\mathcal{F}}_{t}-$adapted;
- (iii)
- For any $t\in \left({t}_{0},T\right],\psi \in {\mathcal{L}}_{{\mathcal{F}}_{0}}^{2}\left([-\tau ,0]\right.$, $\left.{\mathbb{R}}^{n}\right)$, the following equation:$$x\left(t\right)=\left\{\begin{array}{c}\psi \left(t\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in [-\tau ,0],\hfill \\ {\displaystyle x\left({t}_{0}\right)+\sum _{i=1}^{m}{\int}_{{t}_{0}}^{t}{A}_{i}x(t-{\tau}_{i})dt+\sum _{i=1}^{m}{\int}_{{t}_{0}}^{t}{B}_{i}x(t-{\delta}_{i})d{w}_{i}\left(t\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t\in ({t}_{0},T]\setminus \mho ,}\hfill \\ {C}_{k}x\left({t}_{k}^{-}\right),\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}t={t}_{k}\in \mho ,\hfill \end{array}\right.$$

**Definition**

**2**

**Definition**

**3**

**([26])**. The trivial solution of (1) is called mean-square exponentially stable if there exist constant $\mathrm{\Gamma}\in {\mathbb{R}}^{+}$, and constant $\gamma \in {\mathbb{R}}^{+}$, independent of the initial value ψ and time t, such that

**Definition**

**4**

**([26])**. For simplicity, let ${x}_{t}=x(t+s)$, $s\in [-\tau ,0]$, $V(t,{x}_{t}):[0,\infty )\times {\mathbb{R}}^{n}\to {\mathbb{R}}^{+}$ is said to belong to the class S if $V(t,{x}_{t})$ satisfies the conditions below:

- (i)
- For every moment ${t}_{k}\in \{{t}_{k},k\in {\mathbb{N}}^{+}\}$, ${lim}_{t\to {t}_{k}^{-}}V(t,{x}_{t})=V({t}_{k}^{-},{x}_{{t}_{k}^{-}})$ and ${lim}_{t\to {t}_{k}^{+}}V(t,{x}_{t})=V({t}_{k}^{+},{x}_{{t}_{k}^{+}})$ exist in ${\mathbb{R}}^{+}$. Moreover, $V({t}_{k}^{+},{x}_{{t}_{k}^{+}})=V({t}_{k},{x}_{{t}_{k}})$;
- (ii)
- For $t\in [{t}_{k-1},{t}_{k})\times {\mathbb{R}}^{n}$, $V(t,{x}_{t})$ is continuously twice differentiable in ${x}_{t}$ and once in t.

**Lemma**

**1**

**([27])**. For matrices $\mathcal{P}={\mathcal{P}}^{T}$, $\mathcal{M}$, and $\mathcal{Q}$ with appropriate dimensions, the following LMI:

## 3. Main Results

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Theorem**

**2.**

**Proof.**

**Remark**

**5.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Xiao, C.; Hou, T.
Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays. *Processes* **2022**, *10*, 1258.
https://doi.org/10.3390/pr10071258

**AMA Style**

Xiao C, Hou T.
Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays. *Processes*. 2022; 10(7):1258.
https://doi.org/10.3390/pr10071258

**Chicago/Turabian Style**

Xiao, Chunjie, and Ting Hou.
2022. "Delay-Dependent Stability of Impulsive Stochastic Systems with Multiple Delays" *Processes* 10, no. 7: 1258.
https://doi.org/10.3390/pr10071258