Robust stabilization of systems under singular estimates with respect to stochastic perturbations

The present paper is devoted to the robust stabilization problem for linear systems with stochastic uncertainties. We consider systems on Hilbert spaces where the free dynamics operator $ A $ and the control operator $ B $ yield a singular estimate for $ S(t)B $ where $ S(t) $ is the semigroup generated by $ A $, which is not necessary analytic. Necessary and sufficient conditions for the existence of suboptimal controllers are established. Using these conditions, we show that the supreme achievable stability radius can be characterized via an infinite dimensional Riccati equation which satisfies an operator inequality. An important example is provided for illustrating the theory.