A functional analytic approach to infinite dimensional stochastic linear systems

Abstract

In this paper, we study infinite dimensional stochastic systems having both unbounded control and observation operators. First of all, using a semigroup approach, we give another take of the well-posedness of such systems treated in [SIAM J. Control Optim., 53 (2015), pp. 3457--3482]. We further prove a result on the exact controllability of such systems. Second, we propose a new variation of constants formula for mild solutions of perturbed abstract stochastic Cauchy problems using the concept of Yosida extensions of admissible operators. Third, we prove the well-posedness of perturbed boundary control systems. Fourth, we apply this result to a general class of stochastic systems with delays in the state, control, and observation parts. Finally, we study admissible observation operators and exact observability for semilinear stochastic systems.