Dragan, Vasile (Romanian Acad.), Morozan, Toader (Romanian Acad.)

Mean Square Exponetial Stability for Discrete-Time Time-Varying Linear Systems with Markovian Switching

Scheduled for presentation during the Regular Session "Stochastic systems" (FrP08), Friday, July 28, 2006, 17:25−17:50, Room I

17th International Symposium on Mathematical Theory of Networks and Systems, July 24-28, 2006, Kyoto, Japan

This information is tentative and subject to change. Compiled on April 19, 2024

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to Markovian switching is investigated. Four different definitions of the concept of exponential stability in mean square are introduced and it is shown that they are not always equivalent. One definition of the concept of mean square exponential stability is done in terms of the exponential stability of the evolution defined by a sequence of linear positive operators on an ordered Hilbert space. The other three definitions are given in terms of different types of exponential behavior of the trajectories of the considered system. In our approach the Markov chain is not prefixed. The only available information about the Markov chain is the sequence of probability transition matrices and the set of its states. In this way one obtains that in the case of the systems affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain.

Unlike the continuous time framework, for the discrete time linear stochastic systems with Markovian jumping two types of Lyapunov operators are introduced. Therefore in the case of discrete-time linear stochastic systems subject to Markovian perturbations one obtains characterizations of the mean square exponential stability which do not have an analogous in the continuous time.

The results developed in this paper may be used to derive some procedures for designing stabilizing controllers for the considered class of discrete-time linear stochastic systems.