Stochastic optimal control in infinite dimensions with state constraints

Abstract

We consider the state-constrained stochastic optimal control problem in infinitedimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton-Jacobi-Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.