# Riccati Equations and Delay-Dependent BIBO Stabilization of Stochastic Systems with Mixed Delays and Nonlinear Perturbations

- Xia Zhou
^{1}Email author and - Shouming Zhong
^{1}

**2010**:494607

https://doi.org/10.1155/2010/494607

© Xia Zhou and Shouming Zhong. 2010

**Received: **21 August 2010

**Accepted: **9 December 2010

**Published: **20 December 2010

## Abstract

The mean square BIBO stability is investigated for stochastic control systems with mixed delays and nonlinear perturbations. The system with mixed delays is transformed, then a class of suitable Lyapunov functionals is selected, and some novel delay-dependent BIBO stabilization in mean square criteria for stochastic control systems with mixed delays and nonlinear perturbations are obtained by applying the technique of analyzing controller and the method of existing a positive definite solution to an auxiliary algebraic Riccati matrix equation. A numerical example is given to illustrate the validity of the main results.

## Keywords

## 1. Introduction

In recent years, Bounded-Input Bounded-Output (BIBO) stabilization has been investigated by many researchers in order to track out the reference input signal in real world, see [1–6] and some references therein. On the other hand, because of the finite switching speed, memory effects, and so on, time delays are unavoidable in technology and nature, commonly exist in various mechanical, chemical engineering, physical, biological, and economic systems. They can make the concerned control system be of poor performance and instable, which cause the hardware implementation of the control system to become difficult. It is necessary to introduce the distributed delay in control systems, which can describe mathematical modeling of many biological phenomena, for instance, in prey-predator systems, see [7–9]. And so, BIBO stabilization analysis for mixed delays and nonlinear systems is of great significance.

In [10, 11], the sufficient condition for BIBO stabilization of the control system with no delays was proposed by the Bihari-type inequality. In [12, 13], employing the parameters technique and the Gronwall inequality investigated the BIBO stability of the system without distributed time delays. In [14–16], based on Riccati-equations, by constructing appropriate Lyapunov functions, some BIBO stabilization criteria for a class of delayed control systems with nonlinear perturbations were established. In [17], the BIBO stabilization problem of a class of piecewise switched linear system was further investigated. However, up to now, these previous results have been assumed to be in deterministic systems, including continuous time deterministic systems and discrete time deterministic systems, but seldom in stochastic systems (see [18], Fu and Liao get several mean square BIBO stabilization criteria in terms of Razumikhin technique and comparison principle). In practice, stochastic control systems are more applicable to problems that are environmentally noisly in nature or related to biological realities. Thus, the BIBO stabilization analysis problems for stochastic case are necessary.

Up to now, to the best of authors knowledge, the method of existence of a positive definite solution to an auxiliary algebraic Riccati matrix equation is only used to deal with the BIBO stabilization for deterministic differential equations [14–17], not for stochastic differential equations.

Motivated by the above discussions, the main aim of this paper is to study the BIBO stabilization in mean square for the stochastic control system with mixed delays and nonlinear perturbations. Based on the technique of analyzing controller and transforming of the system, various suitable Lyapunov functionals are selected, different Riccati matrix equations are established, and some sufficient conditions guaranteeing BIBO stabilization in mean square are obtained. Finally, a numerical example is provided to demonstrate the effectiveness of the derived results.

## 2. Problem Formulation and Preliminaries

where is known positive constant.

where , , are the feedback gain matrices, , , are the reference inputs.

To derive our main results, we need to introduce the following definitions and lemmas.

Definition 2.1 (see [18]).

*A vector function*
*is said to be an element of*
*if*
*, where*
*denotes the Euclid norm in*
*or the norm of a matrix.*

Definition 2.2 (see [18]).

*The nonlinear stochastic control system ( 2.2 ) is said to be BIBO stabilized in mean square if one can construct a controller ( 2.5 ) such that the output*

*satisfies*

Definition 2.3 (*L*-operator).

Lemma 2.4 (see [19]).

Lemma 2.5 (see [20]).

## 3. BIBO Stabilization for Nonlinear Stochastic Systems

Theorem 3.1.

Proof.

By Definition 2.2, the nonlinear stochastic control system (3.1) with the control law (2.3) is said to be BIBO stabilized in mean square. This completes the proof.

we can get the following result.

Theorem 3.2.

Proof.

Let , . The rest of the proof is essentially as that of Theorem 3.1, and hence is omitted. This completes the proof.

Remark 3.3.

using the same process of Theorem 3.1, we can get the corresponding BIBO stability in mean square results. Here we omitted it.

Theorem 3.4.

Proof.

Let , . The following proof runs as that of Theorem 3.1., and hence is omitted. This completes the proof.

Remark 3.5.

The systems (2.1), (3.1), and (3.23) are also asymptotically stable in mean square when all the conditions in Theorems 3.1–3.4 are satisfied, if , , in (2.3).

Remark 3.6.

In [18], the authors studied the BIBO stabilization of stochastic delayed systems with uncertainty in terms of Razumikhin technique and comparison principle. In the present paper, we are the first to introduce a new way in the study of BIBO stabilization for stochastic delayed systems by using algebraic Riccati matrix equation, which makes the stability conditions be more feasible.

Remark 3.7.

In [18], researchers investigate the BIBO stabilization of stochastic delayed systems with uncertainty without distributed time delays and nonlinear perturbations. In [15], the authors discussed the BIBO stabilization of mixed time-delayed systems with nonlinear perturbations, but the environmental noise is not taken into account in the models. Therefore, compared with ([14–18]), the systems reported in this paper are more general.

Remark 3.8.

The criteria given in Theorem 3.1–3.4 are delay dependent with respect to delays. Generally speaking, the delay-dependent stability criterion is less conservative than delay-independent stability when the time delay is small.

## 4. Example

In this section, a numerical example will be presented to show the effectiveness of the main results derived in this paper.

Example 4.1.

## 5. Conclusions

The problem of delay-dependent BIBO stabilization in mean square for the stochastic control systems with mixed delays and nonlinear perturbations was investigated. A suitable class of Lyapunov functional combined with the descriptor model transformation and decomposition technique of controller were constructed to derive some novel mean square BIBO stability criteria. This paper was the first to successfully introduce the method of Riccati matrix equation to stochastic BIBO stabilization. A numerical example was given to illustrate the validity of the main results.

## Declarations

### Acknowledgments

The authors would like to thank the editor and the reviewers for their detailed comments and valuable suggestions which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China (Grant no. 60736029) and the National Basic Research Program of China (2010CB732501).

## Authors’ Affiliations

## References

- Kotsios S, Feely O:
**A BIBO stability theorem for a two-dimensional feedback discrete system with discontinuities.***Journal of the Franklin Institute B*1998,**335**(3):533-537. 10.1016/S0016-0032(96)00134-2MathSciNetView ArticleGoogle Scholar - Bose T, Chen MQ:
**BIBO stability of the discrete bilinear system.***Digital Signal Processing*1995,**5**(3):160-166. 10.1006/dspr.1995.1016View ArticleGoogle Scholar - Partington JR, Bonnet C:
**and BIBO stabilization of delay systems of neutral type.***Systems and Control Letters*2004,**52**(3-4):283-288. 10.1016/j.sysconle.2003.09.014MathSciNetView ArticleMATHGoogle Scholar - You KH, Lee EB:
**BIBO stability integral**(**-gain) for second-order systems with numerator dynamics.***Automatica*2000,**36**(11):1693-1699. 10.1016/S0005-1098(00)00075-3MathSciNetView ArticleMATHGoogle Scholar - Tomerlin AT, Edmonson WW:
**BIBO stability of D-dimensional filters.***Multidimensional Systems and Signal Processing*2002,**13**(3):333-340. 10.1023/A:1015864514815MathSciNetView ArticleMATHGoogle Scholar - Wang W, Zou Y:
**The stabilizability and connections between internal and BIBO stability of 2-D singular systems.***Multidimensional Systems and Signal Processing*2004,**15**(1):37-50. 10.1023/B:MULT.0000003930.53696.1bMathSciNetView ArticleMATHGoogle Scholar - Ma Z-P, Huo H-F, Liu C-Y:
**Stability and Hopf bifurcation analysis on a predator-prey model with discrete and distributed delays.***Nonlinear Analysis: Real World Applications*2009,**10**(2):1160-1172. 10.1016/j.nonrwa.2007.12.006MathSciNetView ArticleMATHGoogle Scholar - Lin G, Yuan R:
**Periodic solution for a predator-prey system with distributed delay.***Mathematical and Computer Modelling*2005,**42**(9-10):959-966. 10.1016/j.mcm.2005.05.015MathSciNetView ArticleGoogle Scholar - Guo H, Chen L:
**The effects of impulsive harvest on a predator-prey system with distributed time delay.***Communications in Nonlinear Science and Numerical Simulation*2009,**14**(5):2301-2309. 10.1016/j.cnsns.2008.05.010MathSciNetView ArticleMATHGoogle Scholar - Xu D, Zhong S, Li M:
**BIBO stabilization of large-scale systems.***Control Theory & Applications*1995,**12**(6):758-763.MathSciNetGoogle Scholar - Xu DY, Zhong SM:
**BIBO stabilization of multivariable feedback systems.***Journal of University of Electronic Science and Technology of China*1995,**24**(1):90-96.MathSciNetGoogle Scholar - Zhong SM, Huang YQ:
**BIBO stabilization of nonlinear system with time-delay.***Journal of University of Electronic Science and Technology of China*2000,**32**(6):655-657.MathSciNetGoogle Scholar - Cao KC, Zhong SM, Liu BS:
**BIBO and robust stabilization for system with time-delay and nonlinear perturbations.***Journal of University of Electronic Science and Technology of China*2003,**32**(6):787-789.MATHGoogle Scholar - Li P, Zhong S-M:
**BIBO stabilization of time-delayed system with nonlinear perturbation.***Applied Mathematics and Computation*2008,**195**(1):264-269. 10.1016/j.amc.2007.04.081MathSciNetView ArticleMATHGoogle Scholar - Li P, Zhong S-M:
**BIBO stabilization for system with multiple mixed delays and nonlinear perturbations.***Applied Mathematics and Computation*2008,**196**(1):207-213. 10.1016/j.amc.2007.05.046MathSciNetView ArticleMATHGoogle Scholar - Li P, Zhong S-M:
**BIBO stabilization of piecewise switched linear systems with delays and nonlinear perturbations.***Applied Mathematics and Computation*2009,**213**(2):405-410. 10.1016/j.amc.2009.03.029MathSciNetView ArticleMATHGoogle Scholar - Li P, Zhong S-M, Cui J-Z:
**Delay-dependent robust BIBO stabilization of uncertain system via LMI approach.***Chaos, Solitons & Fractals*2009,**40**(2):1021-1028. 10.1016/j.chaos.2007.08.059MathSciNetView ArticleMATHGoogle Scholar - Fu Y, Liao X:
**BIBO stabilization of stochastic delay systems with uncertainty.***IEEE Transactions on Automatic Control*2003,**48**(1):133-138. 10.1109/TAC.2002.806666MathSciNetView ArticleGoogle Scholar - Gu K:
**An integral inequality in the stability problem of time-delay systems.***Proceedings of the IEEE Conference on Decision and Control*2000,**3:**2805-2810.Google Scholar - Wang Z, Shu H, Liu Y, Ho DWC, Liu X:
**Robust stability analysis of generalized neural networks with discrete and distributed time delays.***Chaos, Solitons & Fractals*2006,**30**(4):886-896. 10.1016/j.chaos.2005.08.166MathSciNetView ArticleMATHGoogle Scholar - Kolmanovskii V, Myshkis A:
*Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications*.*Volume 463*. Kluwer Academic, Dordrecht, The Netherlands; 1999:xvi+648.View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.