Elsevier

Automatica

Volume 46, Issue 6, June 2010, Pages 1068-1073
Automatica

Brief paper
Complete controllability of impulsive stochastic integro-differential systems

https://doi.org/10.1016/j.automatica.2010.03.002Get rights and content

Abstract

This paper is concerned with the controllability of impulsive stochastic integro-differential systems. Sufficient conditions of complete controllability for impulsive stochastic integro-differential systems are obtained by using Schaefer’s fixed point theorem. A numerical example is provided to show the effectiveness of the proposed results.

Introduction

It is well known that the concept of controllability plays an important role in control theory and engineering. Controllability has been studied extensively in the fields of finite-dimensional nonlinear systems, infinite-dimensional systems (see e.g., Bemporad et al., 2000, Li and Rao, 2003, Mahmudov, 2003).

Impulsive systems arise naturally in various fields, such as mechanical systems and biological systems, economics, etc. (see Lakshmikantham, Bainov, & Simeonov, 1989, and the references therein). Impulsive dynamical systems exhibit the continuous evolutions of the states typically described by ordinary differential equations coupled with instantaneous state jumps or impulses. And the presence of impulses implies that the trajectories of the system do not necessarily preserve the basic properties of the non-impulsive dynamical systems. To this end the theory of impulsive differential systems has emerged as an important area of investigation in applied sciences. In the last few years many papers have been published about the controllability of impulsive differential systems. Guan, Qian, and Yu (2002) considered the controllability and observability for a class of time-varying impulsive control systems;Li, Wang, and Zhang (2006) investigated the controllability of the first-order impulsive functional differential systems in Banach space; Xie and Wang (2004) studied the controllability of switched impulsive control systems; Liu and Marquez (2008) discussed the controllability and observability problem for a class of controlled switching impulsive systems; in Sakthivel, Mahmudov, and Kim (2009), sufficient conditions were formulated for the exact controllability of second-order nonlinear impulsive control differential systems.

On the other hand noise is ubiquitous. Systems, both natural and artificial ones, often possess various structures subject to stochastically abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, sudden environment changes, etc. More details on stochastic differential equations can be found in the books of Mao (1997) and Oksendal (2003). For linear stochastic system the controllability problem of the form dx(t)=[Ax(t)+Bu(t)]dt+g(t)dw(t),t[0,T],x(0)=x0, has been studied by several authors (e.g., Mahmudov, 2001a, Mahmudov, 2001b). Here A,B are both n×n matrices, and g():[0,T]Rn×l for n,lN. For nonlinear stochastic systems there are also many results on the control theory, including Mahmudov (2003), Mahmudov and Zorlu (2003), Wang, Ho, Liu, and Liu (2009) and Niu, Ho, and Wang (2007) dealt with the problem of sliding mode control for a class of nonlinear uncertain stochastic systems with Markovian switching. Dong and Sun (2008) gave a detailed discussion on hybrid control for a class of nonlinear stochastic Markovian switching systems.

For the controllability problem there are different methods for various types of nonlinear systems. And the most common methods for controllability of stochastic systems as we know are: Picard type iteration (e.g. Balachandran, Karthikeyan, & Kim, 2007), contraction mapping principle (e.g., Mahmudov & Zorlu, 2003), and Lyapunov approach (e.g. Zhao, 2008). However, the complete controllability problem of impulsive stochastic integro-differential systems has not been investigated yet, to the best of our knowledge, although Karthikeyan and Balachandran (2009) and Sakthivel, Mahmudov, and Lee (2009) respectively investigated the controllability of impulsive stochastic control systems by using contraction mapping principle, and Subalakshmi and Balachandran (2009) studied the approximate controllability of nonlinear stochastic impulsive systems in Hilbert spaces by using Nussbaum’s fixed point theorem. Based on Schaefer’s fixed point theorem, the proposed work in this paper on the complete controllability of integro-differential systems with both noise perturbations and impulsive effects is new in the literature. This problem is important and challenging in both theory and practice, which has motivated us for this study.

In this paper our main aim is to show the complete controllability of the impulsive stochastic integro-differential systems of the form dx(t)=[Ax(t)+F(t,x(t),0tf1(t,s,x(s))ds,0tf2(t,s,x(s))dw(s))]dt+Bu(t)dt+G(t,x(t),0tg1(t,s,x(s))ds,0tg2(t,s,x(s))dw(s))dw(t),t[0,T],tτk,k=1,2,,m,Δx(t)=Ik(x(t)),t=τk,k=1,2,,m,x(0)=x0 under some basic assumptions via Schaefer’s fixed point theorem. Here A and B are both n×n matrices; and the functions in the equation are: F:[0,T]×Rn×Rn×RnRn,G:[0,T]×Rn×Rn×RnRn×l,f1,g1:[0,T]×[0,T]×RnRn,f2,g2:[0,T]×[0,T]×RnRn×l;Δx(t) denotes the jump of x at t, i.e. Δx(t)=x(t+)x(t)=x(t)x(t);IkC(Rn,Rn); The initial value x0 is a 0-measurable random variable with Ex02<; u(t) is a feedback control and w isl-dimensional Wiener process; and t is the filtration generated by B(s),0st.

The system (2) is in a very general form and it covers many possible models with various definitions of f1,f2,g1,g2. We would like to mention that Balachandran and Karthikeyan (2008) obtained the controllability results of system (2) when Ik=0. The controllability problem with f2=g1=g2=Ik=0 was studied by Sakthivel, Kim, and Mahmudov (2006) and Sakthivel et al. (2009) studied the case f1=f2=g1=g2=0. The system (2) with f1=f2=g1=g2=Ik=0 was investigated by Mahmudov and Zorlu (2003). However, all the papers listed above obtained the controllability results by using the contraction mapping principle which seems to be restrictive in some cases. In Section 4 an example will illustrate it.

The paper is organized as follows. In Section 2, some basic notations and preliminary facts are recalled. Some lemmas and the results are given in Section 3. And an example in Section 4 is discussed to illustrate the efficiency of the results. Finally, conclusions are given in Section 5.

Section snippets

Preliminaries

In this section, we introduce notations, definitions and preliminary facts which are used throughout the paper.

Let {Ω,,P} be a complete probability space with a filtration {t}t0 satisfying the usual conditions (i.e. right continuous and 0 containing all P-null sets). E() is the expectation with respect to the measure P. 2t([0,T]×Ω,Rn) is the Banach space of all square integrable and t-adapted processes x(t) mapping [0,T]×Rn into Rn equipped with the norm x2=supt[0,T]Ex(t)2.

For the

Main results

In this section we discuss the controllability of the stochastic impulsive integro-differential systems (2). For the study of this problem we hence introduce the following hypotheses.

(H1) The linear control system (1) is completely controllable in [0,T].

By (H1) there exists a positive constant l1, such that for t[0,T] (Mahmudov, 2001a) Φ(t)2l1. (H2)(F̃,G̃) are t-adapted with respect to t. And for every positive constant k1, there exists function q()L1(R+,R+) such that for every t[0,T],

A numerical example

Consider the following two-dimensional impulsive integro-differential stochastic system in the form of (2), where A=(1111),B=(1001),F=(F1,F2)T and F1=1+0tsinx1ds+0tcosx1dw1+|x1|,F2=1+0tsinx2ds+0tcosx2dw1+|x2|G=(gij)2×2, with g12=g21=0, g11=et|x1+1|+|0tarctanx1ds|+|0tsinx1dw|, g22=et|x2+1|+|0tarctanx2ds|+|0tsinx2dw|, Ik=diag(1+e212(k+1),1+e212(k+1)) for x=(x1,x2) with the initial value x0 and final point xTR2. For this system the controllability matrix is Ψst=12(1e2(ts)001

Conclusions

The controllability of impulsive stochastic integro-differential systems has been investigated in this paper. By defining an operator on a proper function space, the controllability problem has been transformed into the existence of solution to the operator equation. Based on Schaefer’s fixed point theorem, sufficient conditions of complete controllability for the system have been obtained. To illustrate the obtained results, a numerical example has been analyzed.

Acknowledgements

The authors would like to thank the editor and the reviewers for their constructive comments and suggestions which improved the quality of the paper.

Lijuan Shen received her B.S. degree from Henan Normal University in Xinxiang, in 2003, and her M.S. degree from Dalian University of Technology in Dalian, in 2006. Since 2006 she has been employed at Luoyang Normal University, and now is pursuing her Ph.D. study at Department of Mathematics, Tongji University in Shanghai. Her research interests include existence and control analysis of stochastic impulsive differential systems.

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    Lijuan Shen received her B.S. degree from Henan Normal University in Xinxiang, in 2003, and her M.S. degree from Dalian University of Technology in Dalian, in 2006. Since 2006 she has been employed at Luoyang Normal University, and now is pursuing her Ph.D. study at Department of Mathematics, Tongji University in Shanghai. Her research interests include existence and control analysis of stochastic impulsive differential systems.

    Junping Shi studied for B.S. degree in mathematics in Nankai University, Tianjin, China from 1990 to 1993, and he obtained Ph.D degree in mathematics from Brigham Young University, Provo, Utah, USA in 1998. After 2 years in Tulane University, New Orleans, Louisiana, USA as visiting assistant professor, he has been a faculty member in College of William and Mary, Williamsburg, Virginia, USA since 2000, and currently he holds the position of Distinguished Associate Professor of Mathematics. Since 2002, he is also a special invited professor, and Longjiang professorship from 2006 to 2009, in Harbin Normal University, Harbin, China. He is the author or co-author of over 60 scholarly publications. His main research interests include partial differential equations, nonlinear analysis, and mathematical biology. He is an associate editor for the journals Applicable Analysis and Journal of Mathematical Analysis and Applications.

    Jitao Sun was born in Jiangsu, China, in 1963. He received the B.Sc. degree in Mathematics from Nanjing University, China, in 1983, and Ph.D. degree in Control Theory and Control Engineering from the South China University of Technology, China, in 2002, respectively.

    He was affiliated with Anhui University of Technology from July 1983 to September 1997. From September 1997 to April 2000, he was with Shanghai Tiedao University. In April 2000, he joined the Department of Applied Mathematics, Tongji University, Shanghai, China. From March 2004 to June 2004, he was a Senior Research Assistant in the Centre for Chaos Control and Synchronization, City University of Hong Kong, China. From February 2005 to May 2005, he was a Research Fellow in the Department of Applied Mathematics, City University of Hong Kong, China. From July 2005 to September 2005, he was a Visiting Professor in the Faculty of Informatics and Communication, Central Queensland University, Australia. From February 2006 to October 2006, August 2007 to October 2007, and April 2008 to June 2008, he was a Research Fellow in the Department of Electrical & Computer Engineering, National University of Singapore, Singapore, respectively. From November 2009 to May 2010, he was a Visiting Scholar in the Department of Mathematics, College of William & Mary, USA. He is currently a Professor at the Tongji University. Prior to this, he was a Professor at Anhui University of Technology and Shanghai Tiedao University from 1995 to 2000, respectively. He is the author or coauthor of more than 100 journals papers. His recent research interests include impulsive control, robust control, time delay systems, and hybrid systems.

    Prof. Sun is the Member of Technical Committee on Nonlinear Circuits and Systems, Part of the IEEE Circuits and Systems Society, and reviewer of Mathematical Reviews on AMS.

    This work is supported by the NNSF of China under grant 60874027, and NSF of Education Department of Henan Province (2008B110010). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen.

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