Abstract

In this work the approximation of non necessarily smooth value functions associated to infinite horizon optimal control problems via sequences of consistent feedback laws based on the verification theorem is addressed. Error bounds of Lp type of approximating smooth feedback laws are derived, depending on either the C1 norm of the value function or its semi-concavity. These error bounds combined with the existence of a Lyapunov type function are used to prove the existence of an approximate optimal sequence of smooth feedback laws. Moreover, we extend this result to the H & ouml;lder continuous case by a diagonalization argument combined with the Moreau envelope. It is foreseen that these error bounds could be applied to study the convergence of synthesis of feedback laws via data driven machine learning methods. Additionally, we provide an example of an infinite horizon optimal control problem for which the value functions are non-differentiable but Lipschitz continuous. We point out that in this example no restrictions on either the controls or the trajectories are assumed. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.