Recovery of the optimal control value function in reproducing kernel Hilbert spaces from verification conditions

Approximating the optimal value function $v^*$ for infinite-horizon, nonlinear, autonomous optimal control problems is both challenging and essential for synthesizing real-time optimal feedback. We develop an abstract optimal recovery framework in reproducing kernel Hilbert spaces (RKHS) for reconstructing unknown target functions from mixed equality and inequality functional constraints. Within this framework, the approximation of $v^*$ is cast as a collocation-type problem derived from verification conditions for optimality -- most prominently, the Hamilton-Jacobi-Bellman (HJB) equation -- that uniquely characterizes $v^*$. As the set of collocation points becomes dense in the ambient domain $\Omega$, we establish convergence of the RKHS approximants to $v^*$: globally on $\Omega$ in the RKHS norm when $v^*$ is analytic, and locally (in a neighborhood of the origin) in the RKHS norm when $v^*$ is bounded from above and below by quadratic functions. Furthermore, we show that a practical numerical realization of the abstract scheme reduces to the classical policy iteration algorithm. Numerical experiments support the effectiveness of the proposed approach.