Robust stability and controllability of stochastic differential delay equations with Markovian switching☆
Introduction
Jump system is a hybrid system with state vector that has two components x(t) and r(t). The first one is in general referred to as the state, and the second one is regarded as the mode. In its operation, the jump system will switch from one mode to another in a random way, and the switching between the modes is governed by a Markov process with discrete and finite state space.
The stability and control theory of jump systems has recently received a lot of attention. See, for example, Ji and Chizeck (1990), Mao (1999), Mariton (1990), Pakshin (1997), Pan and Bar-Shalom (1996) and Shaikhet (1996) and the references therein. It is well-known that time-delay cannot be avoided in practice and it often results in instability and poor performance. Besides, it is difficult to obtain the exact value for the delay and we often have to estimate it. Due to the importance of time-delay systems, many researchers have studied the stability of jump systems with time-delay, and the reader is referred to Boukas and Liu (2001), Cao, Sun, and Lam (1998), Jeung, Kim, and Park (1998), Mao, Matasov, and Piunovskiy (2000) and Park (1999), to name a few.
The classical stochastic stability theory deals with not only moment stability but also almost sure stability (cf. Arnold, 1972; Has'minskii, 1981; Kushner, 1967; Mao 1997, Mao 2002; Verriest, 1998). However, to the best of our knowledge, most of the existing results on stochastic differential delay equations with Markovian switching are about the moment stability, while little is known on the almost surely asymptotic stability which is the main topic of the present paper.
The paper is organised as follows. In Section 2, we investigate the asymptotic stability for the stochastic differential delay equations with Markovian switching. In Section 3, the results of Section 2 are then applied to establish a sufficient criterion for the controllability of linear stochastic differential delay equations with Markovian switching. In Section 4, the robustness of stability is discussed.
Section snippets
Asymptotic stability
Throughout this paper, unless otherwise specified, we let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is increasing and right continuous while contains all P-null sets). Let B(t)=(Bt1,…,Btm)T be an m-dimensional Brownian motion defined on the probability space. Let |·| denote the Euclidean norm for vectors or the trace norm for matrices. Let τ>0 and denote the family of all continuous -valued functions on [−
Controllability of linear stochastic differential delay systems
Method by feedback control is one of the most important issues in the control theory (cf. Dragan & Morozan, 2002; Gao & Ahmed, 1987; Lin & Sontag, 1991; Moerder, Halyo, Braussard, & Caglayan, 1989; Verriest, 1998; Willems & Willems, 1976). So far the known results on stabilisation for stochastic differential delay equations with Markovian switching are mainly concerned with the design of feedback controls under which the underlying equations become asymptotically stable in moment, e.g. in mean
Robust stability of linear stochastic differential delay systems
In many practical situations, the system parameters can only be estimated with a certain degree of uncertainty. The robustness of stability is therefore an important issue in the stability theory (cf. Dragan & Morozan, 2002; Mao, 1997; Mao et al., 2000).
Let us now consider the following equation:As before we shall write A(i)=Ai, etc. Assume thatwhere and
Acknowledgements
The authors would like to thank the referee and the associate editor for their very useful comments and suggestions. In particular, it is due to their suggestions that we have been able to weaken the conditions described in (H) while still able to prove the unique global solution under the other hypotheses of Theorem 2.1 (please see Lemma 2.1). The authors would also like to thank the financial supports from the Royal Society (UK) and the London Mathematical Society.
Chenggui Yuan received the BSc degree from Mathematic Department of Central China Normal University, Wuhan, China in 1985; M.Sc. degree from Mathematic Department of Beijing Normal University, Beijing, China in 1988; Ph.D. degree from Research Department of Central South University, Changsha, China in 1994 and was then promoted to Associate Professor at Central South University in 1996. In 2000–2003, as a research student, he studied at the Department of Statistics and Modelling Science,
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Chenggui Yuan received the BSc degree from Mathematic Department of Central China Normal University, Wuhan, China in 1985; M.Sc. degree from Mathematic Department of Beijing Normal University, Beijing, China in 1988; Ph.D. degree from Research Department of Central South University, Changsha, China in 1994 and was then promoted to Associate Professor at Central South University in 1996. In 2000–2003, as a research student, he studied at the Department of Statistics and Modelling Science, University of Strathclyde to get his second Ph.D. He is now a research associate of Department of the Engineering, University of Cambridge.
Xuerong Mao received the Ph.D. degree from Warwick University, England in 1989 and was then SERC (Science and Engineering Research Council, UK) Post-Doctoral Research Fellow 1989–1992. Moving to Scotland, he joined the Department of Statistics and Modelling Science, University of Strathclyde, Glasgow as a lecturer in 1992, was promoted to Reader in 1995, and was made Professor in 1998 which post he still holds. He has authored 3 books and over 100 research papers. His main research interests lie in the field of stochastic analysis. He is a member of the editorial board of Journal of Stochastic Analysis and Applications.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ioannis Paschalidis under the direction of Editor Tamer Başar.
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Partially supported by the Royal Society (UK) and the London Mathematical Society.