A class of stochastic optimal control problems in Hilbert spaces: BSDEs and optimal control laws, state constraints, conditioned processes

We consider a nonlinear controlled stochastic evolution equation in a Hilbert space, with a Wiener process affecting the control, assuming Lipschitz conditions on the coefficients. We take a cost functional quadratic in the control term, but otherwise with general coefficients that may even take infinite values. Under a mild finiteness condition, and after appropriate formulation, we prove existence and uniqueness of the optimal control. We construct the optimal feedback law by means of an associated backward stochastic differential equation. In this Hilbert space setting we are able to treat some state constraints and in some cases to recover conditioned processes as optimal trajectories of appropriate optimal control problems. Applications to optimal control of stochastic partial differential equations are also given.