Discrete time schemes for optimal control problems with monotone controls

Abstract

In this article, we consider the Hamilton–Jacobi–Bellman equation associated with the optimization problem with monotone controls. The problem deals with the infinite horizon case and costs with update coefficients. We study the numerical solution through the discretization in time by finite differences. Without the classical semiconcavity-like assumptions, we prove that the convergence in this problem is of order hγ in contrast with the order hγ/2 valid for general control problems. This difference arises from the simple and precise way the monotone controls can be approximated. We illustrate the result with a simple example.