State Feedback H∞-Control for Discrete-Time Infinite-Dimensional Stochastic Bilinear Systems

In this paper we consider the class of infinite-dimensional discrete-time linear systems with multiplicative random disturbances (i.e. with states multiplied by a random sequence), also known as stochastic bilinear systems. We obtain necessary and sufficient conditions, in terms of an algebraic Riccati-like operator equation, for existence of a state-feedback controller that stabilizes the system and ensures that the influence of the additive disturbance on the output is smaller than some prespecified bound. In a deterministic framework this problem is equivalent to the H1-control problem in a state-space formulation. Our results, when specialized to the case with no multiplicative random disturbance, reduces to the ones known for the deterministic case. Due to the intrinsic probabilistic nature of the stochastic bilinear model (the multiplicative noise acting on the state of the system makes it a stochastic process, regardless the additive disturbance) a probabilistic framework for th..