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The mean square BIBO stabilization is investigated for the stochastic control systems with time delays and nonlinear perturbations. A class of suitable Lyapunov functional is constructed, combined with the descriptor model transformation and the decomposition technique of coefficient matrix; thus some novel delay-dependent mean square BIBO stabilization conditions are derived. These conditions are expressed in the forms of linear matrix inequalities (LMIs), whose feasibility can be easily checked by using Matlab LMI Toolbox. Finally, three numerical examples are given to demonstrate that the derived conditions are effective and much less conservative than those given in the literature.

Because of the finite switching speed, memory effects, and so on, time delay is unavoidable in technology and nature, which commonly exists in various mechanical, chemical engineering, physical, biological, and economic systems. It can make the concerned control systems become of poor performance and unstable, which leads to the difficulty of hardware implementation of the control system. Thus, the stability of time-delay systems has been widely investigated. See [

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 [

Motivated by the previous discussions, this paper mainly aims to study the BIBO stabilization in mean square for the stochastic control systems with time delays and nonlinear perturbations. Based on the descriptor model transformation and the decomposition technique of coefficient matrix, some sufficient conditions guaranteeing BIBO stabilization in mean square are obtained. Finally, three numerical examples provided to demonstrate the derived conditions are valid and much less conservative than those given in the literature.

The notations are quite standard. Throughout this letter,

Consider the stochastic control system described by the following equation:

To obtain the control law described by (

At the end of this section, let us introduce some important definitions and lemmas which will be used in the sequel.

A vector function

The nonlinear stochastic control system (

Let Lyapunov functionals

For any constant symmetric matrix

Let

To derive delay-dependent mean square stabilization conditions, which include the information of the time delay

Letting

let

For the mean square BIBO stabilization of the system described by (

For any given positive constants

We define a Lyapunov functional

By Lemmas

Thus, according to Theorem 1.3 in page 331 of [

For any given positive integer

We define a Lyapunov functional

If the stochastic term disappears, the control system (

We have the following stabilization results.

For any given positive constants

For any given positive integer

In this section, Example

As a simple application of Theorem

Let us consider the delayed control system (

Let us decompose matrix

For the convenience of comparison, let us consider a delayed control system (

Now we use Corollary

The problem of the mean square BIBO stabilization for the stochastic control systems with delays and nonlinear perturbations is investigated. A class of suitable Lyapunov functional combined with the descriptor model transformation and the decomposition technique of coefficient matrix is constructed to derive some novel delay-dependent BIBO stabilization criteria. Numerical examples have shown that the derived conditions are valid and improvements over the existing results are significant.

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. The work of Xia Zhou is supported by the National Natural Science Foundation of China (no. 11226140), the Anhui Provincial Colleges and Universities Natural Science Foundation (no. KJ2013Z267), and Fuyang Teachers College Natural Science Foundation (no. 2012FSKJ08). The work of Shouming Zhong is supported by National Basic Research Program of China (no. 2010CB732501).