Mean Square Exponential Stability for some Stochastic Linear Discrete Time Systems

Abstract

The problem of the mean square exponential stability for a class of discrete-time linear stochastic systems subject to independent random perturbations and 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 if the system is affected by Markovian jumping the property of exponential stability is independent of the initial distribution of the Markov chain. The definition expressed in terms of exponential stability of the evolution generated by a sequence of linear positive operators, allows us to characterize the mean square exponential stability based on the existence of some quadratic Lyapunov functions. 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. One of the aim of this paper is to show that in the general case of discretetime time-varying linear stochastic systems subject to an homogeneous or an inhomogeneous Markov chain, exponential stability mean square defined in terms of state space trajectories of the systems cannot be always characterized via quadratic Lyapunov functions. 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.