A unified approach for mean square stability of continuous-time Markovian jumping linear systems with additive disturbances

Necessary and sufficient conditions for 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 L₂^m(R⁺), which is the usual scenario for the H_∞ approach. For both cases it is shown that MSS is equivalent to asymptotic wide sense stationarity (AWSS), to the spectrum of an augmented matrix lying in the open left half plane, and to the existence of a solution for a certain Lyapunov equation. Furthermore, it is proved that the Lyapunov equation can be written down in two equivalent forms with each one providing an easier-to-check sufficient condition. It is also shown that MSS is equivalent to the state x(t) belonging to L₂^m whenever the disturbances are in L₂^m-(R ⁺). These results provide, inter alia, a flexible theory, in a unified basis, for MSS of continuous-time Markovian jump linear systems