Tailoring Reproducing Kernels for Optimal Control via Policy Iteration

This paper presents a novel approach to formulating the actor-critic method for optimal control by casting policy iteration in reproducing kernel Hilbert spaces (RKHSs -- also known as native spaces). By tailoring the reproducing kernel and RKHS to the dynamics of the nonlinear optimal control problem, we leverage recent advancements in characterizing error bounds from statistical and machine learning theory. These approximations define a general strategy to select the bases of the actor-critic networks, and we formally guarantee for the first time that this basis selection procedure leads to closed-form error bounds for the individual steps of policy iteration. These bounds often have a geometric and computable form, making them potentially useful for a priori or a posteriori evaluation of candidate collections of scattered bases. Numerical studies subsequently provide qualitative evidence of the practical performance achieved for the full recursion using the algorithms and theory developed for the single-step error bounds.