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

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 L₂ᵐ(Ω, F, P), which is the usual scenario for the H∞ approach. The Markov process is assumed to take values in an infinite countable set 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 S finite) asymptotic wide sense stationarity (AWSS). It is also shown that SS is equivalent to the state $x(t)$ belonging to L₂ⁿ(Ω, F, P) whenever the disturbances are in L₂ᵐ(Ω, F, P). For the case in which 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.