Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions

Abstract

We study a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of “slow” and “fast” components. We use the theory of viscosity solutions in Hilbert spaces to show that as the speed of the fast component goes to infinity, the value function of the optimal control problem converges to the viscosity solution of a reduced effective equation. We consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. The results of this paper generalize to the infinite dimensional case the finite dimensional results of Alvarez and Bardi [SIAM J. Control Optim. 40 (2001/02) 1159–1188] and complement the results in Hilbert spaces obtained recently in Guatteri and Tessitore [To appear in: Appl. Math. Optim. (2019) https://doi.org/10.1007/s00245-019-09577-y].