Feedback stabilization of some nonlinear stochastic systems on Hilbert space

Abstract

has been considered by Ichikawa [lo], where A generates a strongly continuous semigroup T(t), t 2 0, on a real separable Hilbert space X, f and G are nonlinear and satisfy Lipschitz condition together with f(0) = 0 and G(0) = 0. That is, f and G are bounded perturbations. But in many practical situations, f and G might be bounded or unbounded, even uncertain; that is, f, G E 6, where 6 is a family of nonlinear unbounded operators. The problem of stabilization is to design a state feedback control law that assures exponential stability of the zero state uniformly with respect to perturbations f, G E 6. This stabilization problem for perturbed (uncertain) linear systems in finite dimensions has received considerable attention in current literature [3, 13 and references thereof]. We have considered similar problems for linear evolution equations in infinite dimensional spaces in our earlier work [ 14, 151. In this paper, we show that our approach can be extended to both deterministic and stochastic semilinear evolution equations based on perturbation theory of semigroups and some results of Ichikawa. Furthermore, in addition to exponential stability in the mean square sense for stochastic systems, stability of sample paths is also obtained. To our knowledge, this problem, involving infinite dimensional systems with unbounded and nonlinear perturbations, has not been considered in the literature before.