Research Article | Open Access

# Stability of Hybrid Stochastic Systems with Time-Delay

**Academic Editor:**G. Ólafsson

#### Abstract

This paper develops some criteria for a kind of hybrid stochastic systems with time-delay, which improve existing results on hybrid systems without considering noises. The improved results show that the presence of noise is quite involved in the stability analysis of hybrid systems. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality (LMI).

#### 1. Introduction

With the development of social production, many practical systems cannot be modeled by linear time-invariant systems. In this case, hybrid systems are employed to model many practical systems. A hybrid system is a dynamical system with continuous dynamics, discrete dynamics, and the interaction between them (see, e.g., [1–3]). Hybrid systems are important from both the practical and theoretical point of view. In fact, hybrid systems naturally represent a wide class of practical systems which are subject to known or unknown abrupt parameter variations and which undergo sudden change of system structures due to the failure of a component. Another practical motivation for studying hybrid dynamical systems originates from the fact that the hybrid control scheme provides an effective approach for controlling highly nonlinear complex dynamical systems and systems with uncertain and/or unknown parameters. From theoretical viewpoints, the interactions between low-level continuous dynamics and high-level discrete logics, mainly governed by the switching and impulsive mechanisms, bring new challenges beyond the conventional system theory. An area of particular interest has been the analysis of stability of hybrid dynamical systems (see, e.g., [4–9]).

Impulsive stabilization of dynamical systems has attracted increasing interests in fields such as population dynamics, automatic control, drug administration, and communication networks (see [4, 5, 8]) and references therein). In many cases, impulsive stabilization may give better performance than continuous stabilization since the former is implemented only at impulsive instants (see [10]) while the latter does so at every moment (see [5]). Even in some cases, only impulsive stabilization can be used for control purpose. For instance, a central bank cannot change its interest rate every day in order to regulate the money supply in a financial market. The fundamental theoretic and systemic method of impulsive dynamical systems have been established in the recent years (see [1, 5]).

On the other hand, many real systems are subject to stochastic effects, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. In the past decade years, the stability and stabilization of stochastic dynamic systems have been intensively investigated. Hence, the theory of hybrid stochastic systems has attracted increasing attention because the relating problems are not only academically challenging, but also of practical importance in many branches of science and engineering (see, e.g., [10, 11]). These systems are adequate mathematical models for many processes and phenomena studied in biology, physics technology, electric power systems, population dynamics, solar-powered systems, and so on (see, e.g., [12]). Consequently, the stability analysis of hybrid stochastic systems has been studied by many researchers (see, e.g., [13–18]). For example, the classical stochastic analysis theory studies stability not only in moment sense but also in almost sure sense (see, e.g., [13]). In paper [14], some sufficient conditions for the existence and global -exponential stability of periodic solution for impulsive stochastic neural networks with delays are given. In paper [15–18], stability of some kinds of stochastic equations is discussed.

However, among the existing results, up to now, very little is known about the stability of hybrid stochastic systems (HSS) with time-delay. In this paper, the stability of HSS with time-delay has been discussed. Some new results of the stability of HSS with time-delay have been attained in this paper. New results can be used to analyze the stability of a kind of stochastic hybrid impulsive and switching neural networks (SHISNN). Therefore, stability analysis of SHISNN can be turned into solving a linear matrix inequality.

This paper is organized as follows. In Section 2, the hybrid stochastic system model and some kinds of stability definition are defined. In Section 3, several criteria for th moment global asymptotical and exponential stability are established. In Section 4, 2th moment global asymptotical and exponential stability for a kind of stochastic hybrid impulsive and switching neural networks is presented.

#### 2. Preliminaries

Throughout this paper, unless otherwise specified, we will employ the following definitions. Let be a complete probability space with filtration satisfying the usual conditions (i.e., it is right continuous and contains all null sets) and let be the expectation operator with respect to the probability space. is -dimensional Brownian defined on . is Euclidean norm of .

Consider the following HSS with time-delay: where is the state variable, is the initial time, is the positive integers, the time sequence satisfies and . Furthermore, we suppose

HSS (1) satisfies conditions (H_{1})–(H_{3}) at any bounded interval : (H_{1}) are continuous functions with for ; (H_{2})there exists nonnegative constant sequence such that ; ;
(H_{3})let be a random variable which is independent of the -algebra generated by ( is a -dimensional normal Brownian motion), and .

Let . And denotes the working time of the subsystem during the interval . In addition, let denote the Lebesgue measure of the set . Then HSS (1) can be written as where and ; furthermore, for all .

Let , and , , where is a given constant and denote the family of all nonnegative functions on that are twice continuously differentiable in and once in *.* If , define an operator associated with HSS (3) from to by
where ,
The purpose of this paper is to develop some stability criteria for systems (1) and (3).

*Definition 1. *(i) th moment asymptotically stable: the trivial of HSS (1) and (3) is th moment asymptotically stable if there exists such that

(ii) th moment exponentially stable: the trivial of HSS (1) and (3) is th moment exponentially stable if there exist positive constants such that

(iii) Asymptotically stable in mean square: the trivial of HSS (1) and (3) is asymptotically stable in mean square if there exists such that

(iv) Exponentially stable in mean square: the trivial of HSS (1) and (3) is th moment exponentially stable if there exist positive constants such that

*Definition 2. * if is differentiable at its right side.

Moreover, in order to finish our results, we will introduce the following lemmas.

Lemma 3. *Let be a nonnegative function defined on the interval and be continuous on the interval . Assume that
**
where and are nonnegative constants satisfying . Then
**
where and satisfies
*

Lemma 4. *If , then
**
for . *

#### 3. Main Results

In this section, we will present some criteria for the stability of systems (3).

Theorem 5. *If there exist switching Lyapunov functions and positive number such that
**
where , and such that
** is a continuous function on . Then
**
implies that the trivial solution of (3) is th moment globally asymptotically stable, and
**
where and are constants, implies that the trivial solution of (3) is th moment globally exponentially stable.*

*Proof. *For any , we denote the solution of (3) by . Let be the switching Lyapunov* function **.* From Ito differential formula, for , we have
Let be small enough such that . Then
Hence, .

It follows from condition (14) that we have
And it follows from Lemma 3 that there exists such that
satisfies the following equation:
Hence, it follows from condition (17) and (18) that we have
Therefore, when ,
It follows from the above discussion that
which implies the conclusions of the theorem. The proof is thus completed.

Using the proof techniques in Theorem 5, we can obtain Corollaries 6–8.

Corollary 6. *Assume that conditions in Theorem 5 hold. Then, the trivial solution of (3) is th moment globally exponentially stable if one of the following conditions is satisfied.**(i)** for all , and there exists a constant such that
**(ii)**, and there exist two positive constants and such that and
*

*Proof. *When , it follows from (29) that
Letting = and noting , we can obtain from Theorem 5 that the trivial solution of (3) is th moment globally exponentially stable.

When , it follows from (30) that

Letting = and noting that , we conclude the proof.

Corollary 7. *In Theorem 5, if there exist switching Lyapunov functions , is a positive-definite matrix and , denote the maximum (minimum) eigenvalue of positive-definite matrices and such that
**
where , and such that
** is a continuous function on . Then
**
implies that the trivial solution of (3) is globally asymptotically stable in mean square, and
**
where and are constants, implies that the trivial solution of (3) is globally exponentially stable in mean square.*

*Proof. *When is a positive-definite matrix, there exists an orthogonal matrix (is an identity matrix) such that
is eigenvalue of positive-definite matrix , and
Here, , and
Hence, we have
Let and = ; in this situation, it is similar to condition (17) in Theorem 5 and ; furthermore, condition (34) is similar to condition (15) in Theorem 5 (, ).

Similar to the proof of Theorem 5, we can easily complete the proof of Corollary 7.

Corollary 8. *In Theorem 5, if in (15) and in (17), then
**
implies that the trivial solution of (3) is asymptotically stable in mean square, and
**
where and are constants, implies that the trivial solution of (3) is exponentially stable in mean square.*

#### 4. Application to a Kind of Stochastic Hybrid Impulsive and Switching Neural Networks (SHISNN)

Similar to paper [9], due to the stochastic distribution, a kind of stochastic hybrid impulsive and switching neural networks (SHISNN) can be described as follows: Similarly, systems (44) can be written as follows:

To simplify, we always suppose that the solution of (45) always exists.

In order to obtain the according result, we give out Lemmas 9 and 10.

Lemma 9. *Let be the real matrix of dimensions and let be a positive matrix; then
*

*Proof. *Since is a positive matrix, there exists an orthogonal matrix such that
Then

Lemma 10. *Let be a symmetric and positive-definite matrix and let be a symmetric matrix; then .*

Theorem 11. *If SHISNN (18) satisfies the following conditions:*(H_{1})*activation functions are bounded; that is, for ,; there exist positive constants such that ;*(H_{2})*activation functions are globally Lipschitzian, for , ; there exist positive constants such that ;*(H_{3})*for , there exist positive constants such that ;*(H_{4})*for SHISNN (45), there exists a constant such that for all ;*(H_{5})*;*(H_{6})*there exist symmetric and positive-definite matrices , positive-definite diagonal matrices and , and positive constant scalars , such that
**where and are defined in Theorem 5, and
**
and is the unique positive root of the equation
**
then implies that the trivial solution of (45) is globally mean square asymptotically stable and , implies that the trivial solution of (45) is globally asymptotically mean square exponentially stable.*

*Proof. *Construct the switching Lyapunov function
Using Ito formula to (52) yields
where
Applying conditions (H_{5}) and Lemmas 9 and 10 to (54) yields
Applying Theorem 5 to (55) yields
satisfies the following equation:
Similar to the proof of Theorem 5,
Therefore, when ,
When , , this implies that the trivial solution of (45) is globally asymptotically stable. When , , this implies that the trivial solution of (45) is globally asymptotically mean square exponentially stable.

It follows from (17) that
The proof is completed.

Via Schur complement, it is easy to show that (49) is equivalent to the LMI

#### 5. Conclusion

The hybrid impulsive and switching stochastic systems model has been discussed based on the existing models of hybrid impulsive and switching systems. Some general criteria for the asymptotic and exponential stability analysis of the new model have been established by using switching Lyapunov functions and stochastic analysis techniques. The new result can be used to analyze mean square global asymptotical and exponential stability for a kind of stochastic hybrid impulsive and switching neural networks. Finally, this paper points out that this kind of stability analysis problem can turn into solving a linear matrix inequality (LMI).

#### Conflict of Interests

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

#### References

- D. Liberzon,
*Switching in Systems and Control*, Birkhauser, Boston, Mass, USA, 2003. View at: Publisher Site - X. Liao and K. W. Wong, “Global exponential stability of hybrid bidirectional associative memory neural networks with discrete delay,”
*Physical Review E*, vol. 67, no. 41, Article ID 042901, 2003. View at: Google Scholar - K. Yuan, J. Cao, and H. X. Li, “Robust stability of switched Cohen-Grossberg neural networks with mixed time-varying delays,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 36, no. 6, pp. 1356–1363, 2006. View at: Publisher Site | Google Scholar - H. Huang, Y. Qu, and H.-X. Li, “Robust stability analysis of switched Hopfield neural networks with time-varying delay under uncertainty,”
*Physics Letters A*, vol. 345, no. 4–6, pp. 345–354, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Z. Sun and S. Ge,
*Switched Linear Systems Control and Assign*, Springer, London, UK, 2005. - M. S. Branicky, V. S. Borkar, and S. K. Mitter, “A unified framework for hybrid control: model and optimal control theory,”
*IEEE Transactions on Automatic Control*, vol. 43, no. 1, pp. 31–45, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Li, Y. Soh, and C. Wen,
*Switched and Impulsive Systems: Analysis,Design and Applications*, Springer, Berlin, Germany, 2005. - Z.-H. Guan, D. J. Hill, and X. Shen, “On hybrid impulsive and switching systems and application to nonlinear control,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 7, pp. 1058–1062, 2005. View at: Publisher Site | Google Scholar - C. Li, G. Feng, and T. Huang, “On hybrid impulsive and switching neural networks,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 38, no. 6, pp. 1549–1560, 2008. View at: Publisher Site | Google Scholar - H. Liu, L. Zhao, Z. Zhang, and Y. Ou, “Stochastic stability of Markovian jumping Hopfield neural networks with constant and distributed delays,”
*Neurocomputing*, vol. 72, no. 16–18, pp. 3669–3674, 2009. View at: Publisher Site | Google Scholar - C. Huang and J. Cao, “On
*p*th moment exponential stability of stochastic Cohen-Grossberg neural networks with time-varying delays,”*Neurocomputing*, vol. 73, no. 4–6, pp. 986–990, 2010. View at: Publisher Site | Google Scholar - Y. Sun and J. Cao, “Stabilization of stochastic delayed neural networks with markovian switching,”
*Asian Journal of Control*, vol. 10, no. 3, pp. 327–340, 2008. View at: Publisher Site | Google Scholar - L. Huang and X. Mao, “On almost sure stability of hybrid stochastic systems with mode-dependent interval delays,”
*IEEE Transactions on Automatic Control*, vol. 55, no. 8, pp. 1946–1952, 2010. View at: Publisher Site | Google Scholar - D. Li, X. Wang, and D. Xu, “Existence and global p-exponential stability of periodic solution for impulsive stochastic neural networks with delays,”
*Nonlinear Analysis: Hybrid Systems*, vol. 6, no. 3, pp. 847–858, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH - L. Pan and J. Cao, “Robust stability for uncertain stochastic neural network with delay and impuses,”
*Neurocomputing*, vol. 94, pp. 102–110, 2012. View at: Publisher Site | Google Scholar - L. Xu, D. He, and Q. Ma, “Impulsive stabilitization of stochastic differential equations with time delays,”
*Mathematical and Computer Modelling*, vol. 57, pp. 997–1004, 2013. View at: Publisher Site | Google Scholar - C. Li, J. Shi, and J. Sun, “Stability of impulsive stochastic differential delay systems and its application to impulsive stochastic neural networks,”
*Nonlinear Analysis: Theory, Methods and Applications*, vol. 74, no. 10, pp. 3099–3111, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH - X. Wu, W. Zhang, and Y. Tang, “
*p*-Moment stability of impulsive stochastic delay differential systems with Markovian switching,”*Communications in Nonlinear Science and Numerical Simulation*, vol. 18, pp. 1870–1879, 2013. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2014 Pu Xing-cheng and Yuan Wei. 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.