Stability and stabilizability of stochastic evolution equations on Hilbert spaces

Abstract

The stochastic evolution equations in Hilbert space with state-and control-dependent noise are dealt with. For linear stochastic evolution equations with statedependent noise, the necessary and sufficient conditions for asymptotic mean-square stability are discussed. For linear and Lure-type non-linear stochastic evolution equations with state-and control-dependent noise, the conditions are derived for which there exists a feedback control law such that the closed-loop system is asymptotically stable in a mean-square sense. Attention is also paid to the case in which there is only state-dependent noise or only control-dependent noise and to the case in which the noise intensity is arbitrarily large.