Research Article | Open Access

Jian Wang, Cuixia Li, "Stabilization of a Class of Stochastic Systems with Time Delays", *Journal of Applied Mathematics*, vol. 2014, Article ID 274681, 6 pages, 2014. https://doi.org/10.1155/2014/274681

# Stabilization of a Class of Stochastic Systems with Time Delays

**Academic Editor:**Shi-Liang Wu

#### Abstract

The problem of exponential stability is investigated for a class of stochastic time-delay systems. By using the decomposition technique and Lyapunov stability theory, two improved exponential stability criteria are derived. Finally, a numerical example is given to illustrate the effectiveness and the benefit of the proposed method.

#### 1. Introduction

In fact, time delay constantly occurs in the real world, which results in instability of systems. Thus, the stability problem for time-delay systems has been studied for many years [1–8]. On the other hand, stochastic modelling has come to play an important role in many fields of science or industry. Stability analysis for stochastic systems has become increasingly meaningful. A number of results have appeared in the literature [9–19]. For instance, in [11], the author provided the criteria for the stability of a class of stochastic systems by Lyapunov theory.

It’s worth noting that, the grey systems can be established when parameters are evaluated by grey numbers (see [20]). Until now, there have been a few papers tackling the stability of the systems; some important and innovative results are obtained [20–23]. In [23], the authors provided the delay-dependent criteria for exponential robust stability in the forms of nonlinear matrix inequalities and linear matrix inequalities.

In this paper, we deal with the exponential stability for the time-delay grey stochastic systems. By using the method of [21–23] and Lyapunov stability theory, two improved criteria of mean-square exponential stability are proposed. At last, a numerical example is given to verify the criteria.

*Notations*. denotes the dimensional Euclidean space, the superscript “” denotes matrix transposition, and the notation , where and are symmetric matrices, means that is positive semidefinite. Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all -null sets). Let be an -dimensional Brownian motion defined on the probability space. Let and be the family of continuous functions from to . Let be the family of all -measurable -valued random variables such that .

#### 2. Problem Formulation

Consider

*Definition 1. *If there is at least a grey matrix among matrices , and of system (1), then (1) is called grey stochastic time-delay system.

Hence
where and are grey matrices and , .

Clearly, if matrices and are replaced by the deterministic matrices and , the grey system (2) becomes system (1). The equations
are said to be the continuous matrix-covered sets of and .

*Definition 2 (see [20]). *System (2) is said to be robustly exponentially stable in the mean square, if, for all and arbitrary matrices , , there exist scalars and , such that
In addition, the following assumptions are made on the system (2). (H1). (H2)Supposing there exist scalars , , such that, for , the inequality holds, .

Before giving the main results, we first present Lemmas 3 and 4, which are important for the proof of main theorems.

Lemma 3 (see [20]). *For arbitrary whitened matrix , it follows that*(i)(ii)(iii)*,**where , , , .*

Lemma 4 (see [18]). *Let be a real matrix of appropriate dimensions; for any vectors , one has .*

#### 3. Proof of the Main Theorem

In this section, we discuss the exponential stability for system (2); two improved criteria for robust exponential stability in mean square are proposed.

Theorem 5. *System (2) is exponentially robustly stable in mean square. If there exist positive scalars , , and , such that
**
here
**
Then, for all , the following inequality holds:
**
where is the unique positive solution of the following equation:
*

*Proof. *First, define Lyapunov-Krasovskii functional as follows:
Then, we have
Using Lemmas 3 and 4, we derive
By assumption (H2), we can obtain
Substituting (11)–(12) into (10), we see that
Using Itô’s differential formula and integrating both sides, we obtain
Moreover, we have
Combining (14) with (15) and noting the definitions of , , we see that
Furthermore,
Since , and .

When (5) holds, (8) must have a unique solution .

Hence, we have
This completes the proof of Theorem 5.

By a similar approach in [22], whitened system of (2) can be written as Since then, we can introduce Combining (19) and (21) together, whitened system of (2) can be rewritten as

Lemma 6. *For all and , the following inequality holds:
**
where
*

*Proof. *First, by the definition of , we can derive
Clearly,
By assumption (H2), we see that
By (25)–(27) and noting the definitions of , , we can obtain
By (28) and integrating both sides, we have
Moreover, we can obtain
Substituting (30) into (29) and noting the definitions of , , and , (23) holds. The proof of Lemma 6 is completed.

By (22) and Lemma 6, another criterion for system (2) will be given.

Theorem 7. *System (2) is exponentially robustly stable in mean square. If there exist positive scalars , , and , such that
**
here
**
Then, for all , the following inequality holds:
**
where is the unique positive solution of the following equation:
*

*Proof. *Similar to the proof process of Theorem 5, we can derive
Using Itô’s differential formula and integrating and by the definition of , we obtain
If , then .

Because , and .

If (31) holds, (34) must have a unique solution .

Therefore, we can easily get
or equivalently (33) holds; the proof of Theorem 7 is completed.

#### 4. Examples

Consider a stochastic time-delay system where

Respectively, , , , and are the lower bound and upper bound matrices of and .

In addition, Clearly, By using the method of [21], we can obtain that or , which indicates that the system (38) is exponentially stable in mean square.

#### 5. Conclusion

In this paper, we have investigated a class of grey stochastic systems with time delay; by constructing a suitable Lyapunov-Krasovskii functional combined with Itô’s differential formula, two improved exponential stability criteria are derived. The criteria obtained in this paper are so conveniently verified that the results in this paper should be proved to be very useful in applications.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The research is supported by the National Natural Science Foundation of China (no. 11301009) and the Natural Science Foundation of Henan Province (no. 0511013800).

#### References

- S. Arik, “Global robust stability analysis of neural networks with discrete time delays,”
*Chaos, Solitons and Fractals*, vol. 26, no. 5, pp. 1407–1414, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. K. Boukas, Z. K. Liu, and G. X. Liu, “Delay-dependent robust stability and ${H}_{\infty}$ control of jump linear systems with time-delay,”
*International Journal of Control*, vol. 74, no. 4, pp. 329–340, 2001. View at: Publisher Site | Google Scholar | MathSciNet - Y. S. Lee, Y. S. Moon, W. H. Kwon, and P. G. Park, “Delay-dependent robust H
_{∞}control for uncertain systems with a state-delay,”*Automatica*, vol. 40, no. 1, pp. 65–72, 2004. View at: Publisher Site | Google Scholar | MathSciNet - J. H. Park, “Robust stabilization for dynamic systems with multiple time-varying delays and nonlinear uncertainties,”
*Journal of Optimization Theory and Applications*, vol. 108, no. 1, pp. 155–174, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Park, “A delay-dependent stability criterion for systems with uncertain time-invariant delays,”
*IEEE Transactions on Automatic Control*, vol. 44, no. 4, pp. 876–877, 1999. View at: Publisher Site | Google Scholar | MathSciNet - E. K. Boukas and Z. K. Liu, “Robust stability and stabilizability of Markov jump linear uncertain systems with mode-dependent time delays,”
*Journal of Optimization Theory and Applications*, vol. 109, no. 3, pp. 587–600, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - W. Xiong and J. Liang, “Novel stability criteria for neutral systems with multiple time delays,”
*Chaos, Solitons and Fractals*, vol. 32, no. 5, pp. 1735–1741, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Cao and J. Wang, “Global exponential stability and periodicity of recurrent neural networks with time delays,”
*IEEE Transactions on Circuits and Systems I: Regular Papers*, vol. 52, no. 5, pp. 920–931, 2005. View at: Publisher Site | Google Scholar | MathSciNet - A. V. Skorokhod,
*Asymptotic Methods in the Theory of Stochastic Differential Equations*, American Mathematical Society, Providence, RI, USA, 1989. View at: MathSciNet - L. Huang and X. Mao, “Robust delayed-state-feedback stabilization of uncertain stochastic systems,”
*Automatica*, vol. 45, no. 5, pp. 1332–1339, 2009. View at: Publisher Site | Google Scholar | MathSciNet - E. I. Verriest, “Stochastic stability of a class of distributed delay systems,” in
*Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05)*, pp. 5048–5053, Seville, Spain, December 2005. View at: Publisher Site | Google Scholar - T. Shaikhet, “Stability of stochastic hereditarty systems with Markov switching,”
*Theory of Stochastic Processes*, vol. 18, no. 2, pp. 180–184, 1996. View at: Google Scholar - S. Xu, J. Lam, X. Mao, and Y. Zou, “A new LMI condition for delay-dependent robust stability of stochastic time-delay systems,”
*Asian Journal of Control*, vol. 7, no. 4, pp. 419–423, 2005. View at: Publisher Site | Google Scholar | MathSciNet - S. Xu, J. Lam, and D. W. C. Ho, “Novel global robust stability criteria for interval neural networks with multiple time-varying delays,”
*Physics Letters, Section A: General, Atomic and Solid State Physics*, vol. 342, no. 4, pp. 322–330, 2005. View at: Publisher Site | Google Scholar - H. Yan, X. Huang, H. Zhang, and M. Wang, “Delay-dependent robust stability criteria of uncertain stochastic systems with time-varying delay,”
*Chaos, Solitons and Fractals*, vol. 40, no. 4, pp. 1668–1679, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - X. Mao, N. Koroleva, and A. Rodkina, “Robust stability of uncertain stochastic differential delay equations,”
*Systems & Control Letters*, vol. 35, no. 5, pp. 325–336, 1998. View at: Publisher Site | Google Scholar | MathSciNet - L. Wan and J. Sun, “Mean square exponential stability of stochastic delayed Hopfield neural networks,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 343, no. 4, pp. 306–318, 2005. View at: Publisher Site | Google Scholar - A. Friedman,
*Stochastic Differential Equations and their Applications*, Academic Press, New York, NY, USA, 1976. View at: MathSciNet - Z. Wang and H. Qiao, “Robust filtering for bilinear uncertain stochastic discrete-time systems,”
*IEEE Transactions on Signal Processing*, vol. 50, no. 3, pp. 560–567, 2002. View at: Publisher Site | Google Scholar | MathSciNet - C. H. Su and J. J. Li, “Research on stability of grey neutral stochastic linear delay systems,”
*Acta Analysis Functionalis Applicata*, vol. 12, no. 4, pp. 328–334, 2010. View at: Google Scholar | MathSciNet - C. H. Su and S. F. Liu, “Robust stability of grey stochastic nonlinear systems with distributed delays,”
*Mathematics in Practice and Theory*, vol. 38, no. 22, pp. 218–223, 2008. View at: Google Scholar | MathSciNet - C. H. Su and S. F. Liu, “Mean-square Exponential Robust Stability for a Class of Grey Stochastic Syst em s with Distr ib uted Delays,”
*Journal of Xinyang Normal University*, vol. 23, no. 4, pp. 501–505, 2010. View at: Google Scholar - C. H. Su and S. F. Liu, “Exponential robust stability of grey neutral stochastic systems with distributed delays,”
*Chinese Journal of Engineering Mathematics*, vol. 27, no. 3, pp. 403–414, 2010. View at: Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Jian Wang and Cuixia Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.