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The dissipativity analysis and control problems for a class of nonlinear stochastic impulsive systems (NSISs) are studied. The systems are subject to the nonlinear disturbance, stochastic disturbance, and impulsive effects, which often exist in a wide variety of industrial processes and the sources of instability. Our aim is to analyse the dissipativity and to design the state-feedback controller and impulsive controller based on the dissipativity such that the nonlinear stochastic impulsive systems are stochastic stable and strictly

As we all know that many real-world systems may be disturbed by stochastic factors. Thus, stochastic differential systems appear as a natural description of many observed phenomena of real world. In the past few years, much research effort has paid to the stability analysis and robust control problems for stochastic systems which have been come to play an important role in many fields including population dynamics, macroeconomics, chemical reactor control, communication network, image processes, and mobile robot localization. So far, plenty of significant results also have been published; see, for example, [

Recent years, there are many real-world systems and natural processes which display some kind of dynamic behavior in a style of both continuous and discrete characteristics; we called “impulsive effects,” which exist widely in many evolution process, particularly some biological system such as biological neural networks and bursting rhythm models in pathology as well as optimal control models in economics, frequency-modulated signal processing system, fly object motions, and so on [

On the other hand, since the notation of dissipative dynamical system was introduced by Willems [

The stability and stabilization [

In this paper, we mainly consider the following nonlinear stochastic impulsive systems (NSISs):

The sequence,

In this section, some definitions and lemmas are given.

For systems in (

Before giving the following definition, we firstly give the definition of quadratic energy supply function associated with systems in (

Systems in (

For a given matrix

For any

In this paper, our aim is to develop dissipativity criteria for systems in (

For dissipativity analysis for NSISs, we treat the free systems as follows:

Given a real matrix

Construct a simple Lyapunov function

So

In the following section, we firstly consider the stability of NSISs in (

Applying Dynkin formula and Grownwall-Bellman inequality, together with (

By Lemma

We know that

In summary, we obtain that

Therefore, we get

Then, there exists a constant

Hence, the NSISs in (

Secondly, we consider the

So it follows that (

From (

We are now ready to design the state-feedback controller

Given a real matrix

For NSISs in (

Applying the congruent transformation

When

If

If there exist a scalar

If there exist a scalar

In this section, we will give an example to show the correctness of the derived results and the effectiveness of the proposed methods. Considering NSISs in (

From Figure

The state curves of the uncontrolled NSISs in (

So the controller parameters can be calculated as follows:

The state curves of the NSISs in (

The output curves of the NSISs in (

From Figure

The curves of performance level

The variation curve of

In this paper, the dissipativity analysis and control problems for a class of nonlinear stochastic impulsive systems (NSISs) have been investigated. The systems are subject to the nonlinear disturbance, stochastic disturbance, and impulsive effects, which often exist in a wide variety of industrial processes and the sources of instability. Based on the dissipativity, the state-feedback controller and impulsive controller, such that the nonlinear stochastic impulsive systems are stochastic stable and strictly

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

This work is supported by National Science Foundation of China (NSFC) under Grant nos. 61104127, 51079057, and 61134012, China Postdoctoral Science Foundation with Grant no. 2012M521428, and Hubei Province Key Laboratory of Systems Science in Metallurgical Process (Wuhan University of Science and Technology) under Grant no. Y201101.