Optimal control of semilinear stochastic evolution equations

Abstract

The aim of this paper is to initiate a semigroup theory-based approach to characterization of optimal Markov controls for controlled semilinear stochastic evolution equations. (It may be recalled that Markov controls are those that depend only on the current state at each time.) For finite dimensional controlled stochastic differential equations with a nondegenerate diffusion matrix, this task is traditionally achieved through the Hamilton-Jacobi-Bellman equation of dynamic programming associated with the problem and an accompanying verification theorem. The latter states that an optimal Markov control can be explicitly obtained by the pointwise minimization of a Hamiltonian derivable from the solution of the HJB equation. Moreover, any optimal Markov control is obtainable in this manner.