Regularity and stability of feedback relaxed controls

This paper proposes a relaxed control regularization with general exploration rewards to design robust feedback controls for multidimensional continuous-time stochastic exit time problems. We establish that the regularized control problem admits a Hölder continuous optimal feedback control and demonstrate that both the value function and the feedback control of the regularized control problem are Lipschitz stable with respect to parameter perturbations. Moreover, we show that a precomputed feedback relaxed control gives a robust performance in a perturbed system and derive a first-order sensitivity equation for both the value function and optimal feedback relaxed control. These stability results provide a theoretical justification for recent reinforcement learning heuristics, including that an exploration reward in the optimization objective leads to more robust decision-making. We finally prove first-order monotone convergence of the value functions for relaxed control problems with vanishing exploration parameters, which subsequently enables us to construct the pure exploitation strategy of the original control problem based on the feedback relaxed controls