# 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

## References

- Chen, C. Explicit solutions and stability properties of homogeneous polynomial dynamical systems via tensor orthogonal decomposition. arXiv
**2021**, arXiv:2107.11438. [Google Scholar] - Liu, X.; Shen, J. Stability theory of hybrid dynamical systems with time delay. IEEE Trans. Autom. Control
**2006**, 51, 620–625. [Google Scholar] [CrossRef] - Haddad, W.M.; L’Afflitto, A. Finite-time stabilization and optimal feedback control. IEEE Trans. Autom. Contro.
**2016**, 61, 1069–1074. [Google Scholar] [CrossRef] - Ahmadi, A.A.; Khadir, B.E. On algebraic proofs of stability for homogeneous vector fields. IEEE Trans. Autom. Control
**2019**, 65, 325–332. [Google Scholar] [CrossRef] [Green Version] - Jungers, R.; Ahmadi, A.A.; Parrilo, P.A.; Roozbehani, M. A characterization of Lyapunov inequalities for stability of switched systems. IEEE Trans. Autom. Control
**2017**, 62, 3062–3067. [Google Scholar] [CrossRef] [Green Version] - Liu, B.; Xu, B.; Zhang, G.; Tong, L. Review of some control theory results on uniform stability of impulsive systems. Mathematics
**2019**, 7, 1186. [Google Scholar] [CrossRef] [Green Version] - Li, H.; Liu, A. Asymptotic stability analysis via indefinite Lyapunov functions and design of nonlinear impulsive control systems. Nonlinear Anal. Hybrid Syst.
**2020**, 38, 100936. [Google Scholar] [CrossRef] - Rao, R.; Lin, Z.; Ai, X.; Wu, J. Synchronization of epidemic systems with Neumann boundary value under delayed impulse. Mathematics
**2022**, 10, 2064. [Google Scholar] [CrossRef] - Li, X.; Li, P. Stability of time-delay systems with impulsive control involving stabilizing delays. Automatica
**2020**, 124, 109336. [Google Scholar] [CrossRef] - Li, X.; Cao, J.; Ho, D.W.C. Impulsive control of nonlinear systems with time-varying delay and applications. IEEE Trans. Cybern.
**2020**, 50, 2661–2673. [Google Scholar] [CrossRef] - Jiang, B.; Lu, J.; Liu, Y. Exponential stability of delayed systems with average-delay impulses. SIAM J. Control Optim.
**2020**, 58, 3763–3784. [Google Scholar] [CrossRef] - Ai, Z.; Chen, C. Asymptotic stability analysis and design of nonlinear impulsive control systems. Nonlinear Anal. Hybrid Syst. Int. Multidiscip. J.
**2017**, 24, 244–252. [Google Scholar] [CrossRef] - Li, G.; Zhang, Y.; Guan, Y.; Li, W. Stability analysis of multi-point boundary conditions for fractional differential equation with non-instantaneous integral impulse. Math. Biosci. Eng.
**2023**, 20, 7020–7041. [Google Scholar] [CrossRef] - Mao, X. Stochastic Differential Equations and Applications; Elsevier: Amsterdam, The Netherlands, 2007. [Google Scholar]
- Calvin, T.; Rostand, N. Impact of financial crisis on economic growth: A stochastic model. Stoch. Qual. Control
**2022**, 37, 45–63. [Google Scholar] - Jin, X.; Li, Y.X. Adaptive fuzzy control of uncertain stochastic nonlinear systems with full state constraints. Inf. Sci.
**2021**, 574, 625–639. [Google Scholar] [CrossRef] - Yu, J.; Yu, S.; Yan, Y. Fixed-time stability of stochastic nonlinear systems and its application into stochastic multi-agent systems. IET Control Theory Appl.
**2021**, 15, 126–135. [Google Scholar] [CrossRef] - Liu, J.; Wu, L.; Wu, C.; Luo, W.; Franquelo, L.G. Event-triggering dissipative control of switched stochastic systems via sliding mode. Automatica
**2019**, 103, 261–273. [Google Scholar] [CrossRef] - Zhu, Q.; Kong, F.; Cai, Z. Special issue “advanced symmetry methods for dynamics, control, optimization and applications”. Symmetry
**2022**, 15, 26. [Google Scholar] [CrossRef] - Cao, W.; Zhu, Q. Razumikhin-type theorem for p th exponential stability of impulsive stochastic functional differential equations based on vector Lyapunov function. Nonlinear Anal. Hybrid Syst.
**2021**, 39, 100983. [Google Scholar] [CrossRef] - Xu, H.; Zhu, Q. New criteria on p th moment exponential stability of stochastic delayed differential systems subject to average-delay impulses. Syst. Control Lett.
**2022**, 164, 105234. [Google Scholar] [CrossRef] - Hu, W.; Zhu, Q. Stability criteria for impulsive stochastic functional differential systems with distributed-delay dependent impulsive effects. IEEE Trans. Syst. Man Cybern. Syst.
**2019**, 51, 2027–2032. [Google Scholar] [CrossRef] - Hu, Z.; Mu, X. Event-triggered impulsive control for nonlinear stochastic systems. IEEE Trans. Cybern.
**2021**, 52, 7805–7813. [Google Scholar] [CrossRef] - Cheng, P.; Deng, F.; Yao, F. Almost sure exponential stability and stochastic stabilization of stochastic differential systems with impulsive effects. Nonlinear Anal. Hybrid Syst.
**2018**, 30, 106–117. [Google Scholar] [CrossRef] - Zhao, Y.; Wang, L. Practical exponential stability of impulsive stochastic food chain system with time-varying delays. Mathematics
**2023**, 11, 147. [Google Scholar] [CrossRef] - He, Z.; Li, C.; Cao, Z.; Li, H. Stability of nonlinear variable-time impulsive differential systems with delayed impulses. Nonlinear Anal. Hybrid Syst.
**2021**, 39, 100970. [Google Scholar] [CrossRef] - Wang, Y.; Lu, J. Some recent results of analysis and control for impulsive systems. Commun. Nonlinear Sci. Numer. Simul.
**2020**, 80, 104862.1–104862.15. [Google Scholar] [CrossRef] - Li, X.; Song, S.; Wu, J. Exponential stability of nonlinear systems with delayed impulses and applications. IEEE Trans. Autom. Control
**2019**, 64, 4024–4034. [Google Scholar] [CrossRef] - Cao, W.; Zhu, Q. Stability of stochastic nonlinear delay systems with delayed impulses. Appl. Math. Comput.
**2022**, 421, 126950. [Google Scholar] [CrossRef] - Ren, W.; Xiong, J. Stability analysis of impulsive stochastic nonlinear systems. IEEE Trans. Autom. Control
**2017**, 62, 4791–4797. [Google Scholar] [CrossRef] - Hu, W.; Zhu, Q.; Karimi, H.R. Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems. IEEE Trans. Autom. Control
**2019**, 64, 5207–5213. [Google Scholar] [CrossRef] - He, W.; Qian, F.; Han, Q.L.; Chen, G. Almost sure stability of nonlinear systems under random and impulsive sequential attacks. IEEE Trans. Autom. Control
**2020**, 65, 3879–3886. [Google Scholar] [CrossRef] - Boyd, S.; El Ghaoui, L.; Feron, E.; Balakrishnan, V. Linear Matrix Inequalities in System and Control Theory; SIAM: Philadelphia, PA, USA, 1994. [Google Scholar]

**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.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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