Fully discrete schemes for monotone optimal control problems

Abstract

In this article, we study an infinite horizon optimal control problem with monotone controls. We analyze the associated Hamilton–Jacobi–Bellman (HJB) variational inequality which characterizes the value function and consider the totally discretized problem using Lagrange elements to approximate the state space $\Omega$. The convergence orders of these approximations are proved, which are in general $(h+\frac{k}{\sqrt{h}})^\gamma$ where $\gamma$ is the Hölder constant of the value function u, h and k are the time and space discretization parameters, respectively. A special election of the relations between h and k allows to obtain a convergence of order $k^{\frac{2}{3}\gamma }$, which is valid without semiconcavity hypotheses over the problem’s data.