# Stability Analysis for a Class of Stochastic Differential Equations with Impulses

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Remark**

**1.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5**

**Lemma**

**1**

**Lemma**

**2**

**Lemma**

**3**

- (1)
- ${P}_{2}<0,{P}_{1}-{P}_{3}{P}_{2}^{-1}{P}_{3}^{\top}<0$,
- (2)
- $\left[\begin{array}{cc}{P}_{1}& {P}_{3}\\ {P}_{3}^{\top}& {P}_{2}\end{array}\right]<0$.

## 3. Main Results

**Theorem**

**1.**

- (1)
- $$RE+{E}^{\top}R+{F}^{\top}RF-\eta R<0,$$

- (2)
- $$\underset{j\to \infty}{lim}\left\{\prod _{k=1}^{{l}_{j}}{\left({\mu}_{k}\right)}^{2}\right\}=0.$$

**Proof.**

## 4. Numerical Simulations

**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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**Figure 1.**State trajectories of system (17) without stabilizing impulses.

**Figure 2.**State trajectories of system (17) with stabilizing impulses.

**Figure 3.**State trajectories of system (18) without stabilizing impulses.

**Figure 4.**State trajectories of system (18) with stabilizing impulses.

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**MDPI and ACS Style**

Xia, M.; Liu, L.; Fang, J.; Zhang, Y.
Stability Analysis for a Class of Stochastic Differential Equations with Impulses. *Mathematics* **2023**, *11*, 1541.
https://doi.org/10.3390/math11061541

**AMA Style**

Xia M, Liu L, Fang J, Zhang Y.
Stability Analysis for a Class of Stochastic Differential Equations with Impulses. *Mathematics*. 2023; 11(6):1541.
https://doi.org/10.3390/math11061541

**Chicago/Turabian Style**

Xia, Mingli, Linna Liu, Jianyin Fang, and Yicheng Zhang.
2023. "Stability Analysis for a Class of Stochastic Differential Equations with Impulses" *Mathematics* 11, no. 6: 1541.
https://doi.org/10.3390/math11061541