# Stability analysis for highly nonlinear switched stochastic systems with time-varying delays

*Complex Eng Syst*2022;2:17.

## Abstract

In this paper, we examine the stability of highly nonlinear switched stochastic systems (SSSs) with time-varying delays, where the switching time instants are deterministic rather than stochastic. Herein, the boundedness of the global solution is first proven for highly nonlinear SSSs via the average dwell time (ADT) method and multiple Lyapunov function (MLF) approach. Then, the stability criteria for *q*th moment exponential stability and almost surely exponential stability are presented. The main difficulty lies in the presence of switching and time-varying delay terms, which prevents the validation of existing methods. New inequality techniques have been developed to counteract the effects of switching signals and time-varying delays. Finally, an example is provided to verify the effectiveness of the results.

## Keywords

*,*deterministic switching signal

*,*time-varying delays

*,*average dwell time

*,*multiple Lyapunov function

## 1. INTRODUCTION

Switched systems are important dynamic systems. The idea of switching has been widely applied in various fields, such as aircraft attitude control ^{[1]}, ecological dynamics ^{[2]}, and financial markets ^{[3]}. With the increasing complexity of system architectures, dynamical analysis of switched systems has attracted significant academic interest. A switched system consists of a family of continuous-time dynamics, discrete-time dynamics, and switching rules between subsystems. According to the switching signal features, switched systems are divided into two categories, {*namely*}, deterministic switched systems and randomly switched systems. Many researchers have focused on stabilization and stability analyses of various switched systems. For example, in ^{[4]}, a series of results on stochastic differential equations (SDEs) with Markovian switching was obtained. In particular, the authors have provided some useful stability criteria. In ^{[5]}, the authors studied the input-to-state stability of time-varying switched systems by employing the ADT method coupled with the MLF approach. The authors of ^{[6]} investigated the stability of switched stochastic delay neural networks with all unstable subsystems based on discretized Lyapunov-Krasovskii functions (DLKFs). In ^{[7]}, a novel Lyapunov function was designed to ensure a non-weighted ^{[8]}, a hidden Markov model was proposed to study the finite region ^{[9-11]} and references therein.

The linear growth condition (LGC) is crucial for ensuring the existence of a global solution for a stochastic system. However, many stochastic systems do not satisfy LGC. Hence, the solution of a stochastic system may explode in a finite time. Recently, the stability of stochastic systems without LGC has drawn considerable attention. For instance, the authors of ^{[12]} investigated the stability and boundedness of nonlinear hybrid stochastic differential delay equations without LGC based on a Lyapunov function approach. By introducing a polynomial growth condition (PGC), ^{[13]} discussed the stabilization problem of highly nonlinear hybrid SDEs. The input-to-state practically exponential stability in the sense of mean square was introduced in ^{[14]}. Sufficient conditions for stability have been obtained. Additionally, other meaningful results were reported in ^{[15]} and ^{[16]}.

Time-delay is an important factor that affects dynamical performances of stochastic systems. By constructing a suitable Lyapunov function, the authors of ^{[12]} studied the stability and boundedness of highly nonlinear hybrid stochastic systems with a time delay. The authors of ^{[17]} used the ADT method to study the stability problem of SSSs, where the switching signals are deterministic. Based on the stability criteria for stochastic time-delay systems, the authors of ^{[18]} introduced a suitable Lyapunov-Krasovskii (L-K) functional, and discussed the global probabilistic asymptotic stability of the closed-loop system. In ^{[19]}, the Razumikhin approach was presented to study the exponential stability of a class of impulsive stochastic delay differential systems. Using the piecewise dynamic gain method, the authors of ^{[20]} studied the global uniform ultimate boundedness of switched linear time-delay systems. Motivated by the aforementioned literature, the stability of highly nonlinear SSSs with time-varying delays is studied in this paper. Figure 1 shows the framework of this paper.

The challenges of this article lie in the following two parts: (1) The time delay studied here is merely a Borel measurable function of time

The main advantages of this paper are as follows:

The remainder of this paper is organized as follows. An introduction of the model and important assumptions are given in Section 2. The existence of a unique global solution and stability analysis are presented in Sections 3. In Section 4, a simulation example is presented to validate our theoretical results. Finally, Section 5 concludes the paper.

*Note*: In this paper,

## 2. PRELIMINARIES

Model descriptions and assumptions are introduced in this section. In this study, we analyzed the following highly nonlinear SSS with time-varying delays:

with the initial value:

where ^{[13]}, one of the merits of this paper is that the time delay

**Assumption 1.*** The time-varying delay *

*where *

**Remark 1*** Assumption 1 reveals that the time delay in SSS (1) is merely a Borel measurable function of time *

^{[21-25]}), the time delay

The following lemma provides a useful inequality to obtain the stability of the SSS (1) with time-varying delays, and its proof can be found in ^{[16]}.

**Lemma 1**^{[16]} Let

The conditions for the existence and uniqueness of global solution are the local Lipschitz condition (LLC) and the LGC (see, e.g., ^{[4, 7, 20, 26]}). In this paper, the highly nonlinear SSS (1) generally does not require the LGC. Consequently, we must impose the PGC on it.

**Assumption 2.*** (LLC & PGC) For any real number *

*for all *

*where *

**Assumption 3*** Assume that there are two functions *

*where *

*Moreover, assume that there exists a constant *

**Remark 2*** The system studied in this research has the property of high nonlinearity. In other words, the LGC is removed from the SSS (1), which makes the considered system more general. Without the LGC, the solution of a stochastic system may explode in a finite time. To ensure the existence of a global solution, a PGC (i.e., condition (6)) is imposed on the SSS (1) (see, e.g., ^{[13, 27, 28]}). Therefore, the system (1) we studied obeys the LLC (i.e., condition (5)) and the PGC. By combining the MLF approach and ADT method, we then prove the existence and uniqueness of the global solution*.

Before presenting the main results, the definition of ADT is revisited.

**Definition 1**^{[28]} For a switching signal

*where *.

## 3. MAIN RESULTS

In this section, we prove the existence of a unique global solution for a highly nonlinear SSS (1) by using the ADT and MLF approaches. Then, both the

**Theorem 1*** Under Assumptions 1-3, if there exists a constant *

*Then, for any initial data (2), there exists a unique global solution *

**Proof.** We divide the whole proof into two steps. In step 1, for all

**Step 1.** For all

where

Clearly,

By Lemma 1, we have

and

Hence,

where

is a finite constant. Applying (10) and (11) from Assumption 3, we can deduce that

Recalling the condition (7), we can get

This implies

We observe that

Letting

**Step 2.** This section proves the existence of a unique global solution for SSS (1). Let

Clearly,

where

For

Combining (18) and (19), it implies that

For

By mathematical induction, for

It follows from (8) and (21) that

Because

Similar to the proof stated in Part 1, we can derive

where

is finite. Then,

Recalling condition (7), we obtain

This implies

Letting

Using Definition 1, we have that for

where

Therefore, for all

This means that the unique solution

The proof is completed.

**Remark 3*** To deal with the time-varying delay $$ \delta_t $$ , some new inequalities (e.g., see (15) and (16) for details) are constructed in the proof for Theorem 3.1. Compared with the results reported in existing studies ^{[21-25]}, the time delay $$ \delta_t $$ in this paper is merely a Borel-measurable function, which invalidates these existing methods. By virtue of Lemma 1, a more general form of time delay can be imposed on system (1)*.

We now refer to the equation (25) in the proof of Theorem 1. The following theorem provides sufficient conditions for the

**Theorem 2*** Under the same conditions as those considered in Theorem 1, the solution of system (1) with the initial value (2) is *

**Proof**. Applying (25) yields

Recalling condition (7), we have

Hence, from (12), we observe that

where

**Remark 4*** The difficulty of the proof is that the time delay *.

^{[13, 28]}. This means that the existing results

^{[13, 28]}cannot be applied to SSS (1). By selecting a suitable form of MLF, the existence and uniqueness of the global solution are initially proven via an inequality scaling technique (i.e., Lemma 1). Subsequently, the

The following theorem demonstrates that a stronger result can be obtained under proper conditions.

**Theorem 3*** Let Assumptions 1-3 hold. If *

**Proof.** Let ^{[26]}, we obtain

From condition (6), we have

where

Similarly, we also have

From (28), it follows that

where

By the Doob martingale inequality, it follows that

From the well-known Borel-Cantelli lemma^{[4]}, it follows that for almost all

Therefore, for almost all

Then, we can obtain

which is the required assertion in (29). Thus, the proof is completed.

So far, we can conclude that under Assumptions 1-3, system (1) is not only

**Remark 5*** In general, for a stochastic nonlinear system, the *

**Remark 6*** In this paper, the highly nonlinear SSSs with time-varying delays are considered, in which the switching signal is deterministic and differs from those considered in ^{[13, 16, 29-32]}. In the current study on stochastic systems with Markovian switching ^{[13, 16, 29-32]}, *

## 4. NUMERICAL EXAMPLE

In this section, a numerical example is presented to validate the derived results. Consider the following highly nonlinear SSS with a time-varying delay:

where the time-varying delay

In addition, we set

and

Then, we obtain

which means that the condition (9) holds with

## 5. CONCLUSIONS

In this paper, the existence of a unique global solution for a highly nonlinear SSS with a deterministic switching signal is examined by using the ADT method coupled with the MLF approach. The stability criteria of ^{[7, 20, 33-36]}, two further research directions have emerged: (1) Solving the problem of stability for highly nonlinear SSSs with impulsive effects under asynchronous switching, and (2) designing a control input function to stabilize a highly nonlinear SSS with a time-varying delay.

## DECLARATIONS

### Authors' contributions

Made substantial contributions to supervision, writing, review, editing and methodology: Wang H

Performed writing-original draft, software, validation and visualization: Sun J

### Availability of data and materials

Not applicable.

### Financial support and sponsorship

This work was jointly supported by the National Natural Science Foundation of China (62003170), and the Natural Science Foundation of Jiangsu Province (BK20190770).

### Conflicts of interest

All authors declared there are no conflicts of interest.

### Ethical approval and consent to participate

Not applicable.

### Consent for publication

Not applicable.

### Copyright

© The Author(s) 2022.

## REFERENCES

1. Zhang R, Quan Q, Cai KY. Attitude control of a quadrotor aircraft subject to a class of time-varying disturbances. *IET Control Theory & Applications* 2011;5:1140-6.

2. Yoshida T, Jones LE, Ellner SP, Fussmann GF, Hairston NG Jr. Rapid evolution drives ecological dynamics in a predator-prey system. *Nature* 2003;424:303-6.

3. Preis T, Schneider JJ, Stanley HE. Switching processes in financial markets. *Proc Natl Acad Sci U S A* 2011;108:7674-8.

4. Mao XR, Yuan CG. Stochastic Differential Equations with Markovian Switching. London: Imperial College Press, 2006.

5. Wu XT, Tang Y, Cao JD. Input-to-state stability of time-varying switched systems with time delay. *IEEE Trans Automat Contr* 2019;64:2537-44.

6. Xiao HN, Zhu QX, Karimi HR. Stability of stochastic delay switched neural networks with all unstable subsystems: a multiple discretized Lyapunov-Krasovskii functionals method. *Inf Sci* 2022;582:302-15.

7. Yuan S, Zhang LX, Schutter BD, Baldi S. A novel Lyapunov function for a non-weighted *Automatica* 2018;87:310-17.

8. Cheng P, He SP, Luan XL, Liu F. Finite-region asynchronous *Automatica* 2021;129:109590.

9. Wang B, Zhu QX. Stability analysis of Markov switched stochastic differential equations with both stable and unstable subsystems. *Systems & Control Letters* 2017;105:55-61.

10. Liberzon D. Finite data-rate feedback stabilization of switched and hybrid inear systems. *Automatica* 2014;50:409.

11. https://doi.org/10.1007/978-0-85729-256-8 [Last accessed on 22 Dec 2022] ]]>.

12. Hu LJ, Mao XR, Shen Y. Stability and boundedness of nonlinear hybrid stochastic differential delay equations. *Syst & Contr Let* 2013;62:178.

13. Li XY, Mao XR. Stabilisation of highly nonlinear hybrid stochastic differential delay equations by delay feedback control. *Automatica* 2020;112:108657.

14. Zhu QX. Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. *IEEE Trans Automat Contr* 2019;64:3764-71.

15. Zhu QX, Song SY, Shi P. Effect of noise on the solutions of non-linear delay systems. *IET Control Theory & Appl* 2018;12:1822-9.

16. Dong HL, Mao XR. Advances in stabilization of highly nonlinear hybrid delay systems. *Automatica* 2022;136:110086.

17. Zhao Y, Wu XT, Gao JD. Stability of highly non-linear switched stochastic systems. *IET Control Theory & Applications* 2019;13:1940-44.

18. Liu L, Yin S, Zhang LX, Yin XY, Yan HC. Improved results on asymptotic stabilization for stochastic nonlinear time-delay systems with application to a chemical reactor system. *IEEE Transactions on Sysems, Man, and Cybernetics: Systems* 2016;47:195-204.

19. Hu W, Zhu QX, Karimi HR. Some improved razumikhin stability criteria for impulsive stochastic delay differential systems. *IEEE Trans Automat Contr* 2019;64:5207-13.

20. Yuan S, Zhang LX, Baldi S. Adaptive stabilization of impulsive switched linear time-delay systems: a piecewise dynamic gain approach. *Automatica* 2019;103:322-29.

21. Lian J, Feng Z. Passivity analysis and synthesis for a class of discrete-time switched stochastic systems with time-varying delay. *Asian J Control* 2013;15:501-11.

22. Cong S, Yin LP. Exponential stability conditions for switched linear stochastic systems with time-varying delay. *IET Control Theory & Applications* 2012;6:2453-59.

23. Yue D, Han QL. Delay-dependent exponential stability of stochastic systems with time-varying delay, nonlinearity, and markovian switching. *IEEE Trans Automat Contr* 2005;50:217-22.

24. Zeng HB, Liu XG, Wang W. A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems. *Applied Mathematics and Computation* 2019;354:1-8.

25. Chen HB, Shi P, Lim CC, Hu P. Exponential stability for neutral stochastic markov systems with time-varying delay and its applications. *IEEE Trans Cybernetics* 2015;46:1350-62.

26. Mao XR. Stochastic differential equations and applications. Elsevier; 2007.

27. Zhao Y, Zhu QX. Stability of highly nonlinear neutral stochastic delay systems with non-random switching signals. *Syst & Contr Let* 2022;165:105261.

28. Zhao Y, Zhu QX. Stabilization by delay feedback control for highly nonlinear switched stochastic systems with time delays. *Int J Robust Nonlinear Control* 2021;31:3070-89.

29. Fei WY, Hu LJ, Mao XR, Shen MX. Structured robust stability and boundedness of nonlinear hybrid delay systems. *SIAM J Control Optim* 2018;56:2662-89.

30. Shen MX, Fei C, Fei WY, Mao XR. Stabilisation by delay feedback control for highly nonlinear neutral stochastic differential equations. *Syst & Contr Let* 2020;137:104645.

31. Shen MX, Fei WY, Mao XR, Deng SN. Exponential stability of highly nonlinear neutral pantograph stochastic differential equations. *Asian J Control* 2020;22:436-48.

32. Song RL, Wang B, Zhu QX. Delay-dependent stability of nonlinear hybrid neutral stochastic differential equations with multiple delays. *Int J Robust Nonlinear Control* 2021;31:250-67.

33. Kang Y, Zhai DH, Liu GP, Zhao YB, Zhao P. Stability analysis of a class of hybrid stochastic retarded systems under asynchronous switching. *IEEE Trans on Automat Contr* 2014;59:1511-23.

34. Shen MX, Fei C, Fei WY, Mao XR, Mei CH. Delay-dependent stability of highly nonlinear neutral stochastic functional differential equations. *Intl J Robust & Nonlinear* 2022;32:9957-76.

35. Ren MF, Zhang QC, Zhang JH. An introductory survey of probability density function control. *Syst Sci & Contr Eng* 2019;7:158-70.

## Cite This Article

## How to Cite

Sun, J.; Wang H. Stability analysis for highly nonlinear switched stochastic systems with time-varying delays. *Complex. Eng. Syst.* **2022**, *2*, 17. http://dx.doi.org/10.20517/ces.2022.48

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