Dynamic Programming Principle and Hamilton-Jacobi Equation for Optimal Control Problems with Uncertainty

Abstract

This work establishes that the value function for Mayer’s problem in a Hilbert space is the unique lower semi-continuous solution to the Hamilton-Jacobi-Bellman (HJB) equation under specific conditions. By investigating a parametrized Riemann–Stieltjes problem, we achieve the compactness of its trajectories, which, combined with a characterization of the lower semicontinuity of the associated value function, establishes the existence of optimal controls. Subsequently, utilizing the differential inclusion approach and prior results, we prove the uniqueness of the solution to the HJB equation.