Stability and robust stabilization to linear stochastic systems described by differential equations with Markovian jumping and multiplicative white noise

In this paper we consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping. Our aim is to provide a mathematical background in order to give unified approach for a large class of problems associated to linear controlled systems subjected both to multiplicative white noise perturbations and Markovian jumping. First we prove an Ito type formula. Our result extends the result of Ref. [24], to the case when the stochastic process x(t) has not all moments bounded. Necessary and sufficient conditions assuring the exponential stability in mean square for the zero solution of a linear stochastic system with multiplicative white noise and Markovian jumping are provided. Some estimates for solutions of affine stochastic systems are derived, and necessary and sufficient conditions assuring the stochastic stabilizability and stochastic delectability are given. A stochastic version of Bounded Real Lemma is proved and several aspects of the problem of robust stabilization by state feedback for a class of linear systems with multiplicative white noise and Markovian jumping are investigated.