A Unified Approach for Stochastic and Mean Square Stability of Continuous-Time Linear Systems with Markovian Jumping Parameters and Additive Disturbances

Abstract

Necessary and sufficient conditions for stochastic stability (SS) and mean square stability (MSS) of continuous-time linear systems subject to Markovian jumps in the parameters and additive disturbances are established. We consider two scenarios regarding the additive disturbances: one in which the system is driven by a Wiener process, and one characterized by functions in ${L_2^m}{(\Omega,{\cal F}, {\mathbb{P}})}$, which is the usual scenario for the $H_{\infty}$ approach. The Markov process is assumed to take values in an infinite countable set $\mathcal{S}$. It is shown that SS is equivalent to the spectrum of an augmented matrix lying in the open left half plane, to the existence of a solution for a certain Lyapunov equation, and implies (is equivalent for $\mathcal{S}$ finite) asymptotic wide sense stationarity (AWSS). It is also shown that SS is equivalent to the state $x(t)$ belonging to ${L_2^n}{(\Omega,{\cal F}, {\mathbb{P}})}$ whenever the disturbances are in ${L_2^m}{(\Omega,{\cal F}, {\mathbb{P}})}$. For the case in which $\mathcal{S}$ is finite, SS and MSS are equivalent, and the Lyapunov equation can be written down in two equivalent forms with each one providing an easier-to-check sufficient condition.