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

Abstract Approximating the value function $$v^*$$ 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 $$v^*$$ v ∗ is formulated as a collocation-type problem derived from verification conditions for optimality, including the Hamilton–Jacobi–Bellman (HJB) equation, which collectively characterize $$v^*$$ v ∗ . Assuming that $$v^*$$ v ∗ belongs to the chosen RKHS and is either analytic or satisfies two-sided quadratic bounds, we prove that the approximants converge as the collocation points become dense in $$\Omega $$ Ω . In particular, in the analytic case, convergence is global on $$\Omega $$ Ω with respect to the RKHS norm, whereas in the case of two-sided quadratic bounds, convergence holds only locally, in a neighborhood of the origin. Furthermore, we show that a practical numerical realization of the abstract scheme based on an approximate iterative solver reduces to a classical policy-iteration algorithm. Numerical experiments support the effectiveness of the proposed approach.