Approximation of the value function for optimal control problems on stratified domains

TL;DR

This paper introduces a semi-Lagrangian approximation scheme for optimal control problems on stratified domains, proving its convergence and providing a software tool for numerical solutions in complex scenarios.

Contribution

It presents a novel approximation method for value functions on stratified domains and demonstrates its convergence, along with a software implementation for practical use.

Findings

The scheme converges to the true value function.

The software handles complex stratified domain problems.

Numerical examples illustrate phenomena absent in continuous frameworks.

Abstract

In optimal control problems defined on stratified domains, the dynamics and the running cost may have discontinuities on a finite union of submanifolds of RN. In [8, 5], the corresponding value function is characterized as the unique viscosity solution of a discontinuous Hamilton-Jacobi equation satisfying additional viscosity conditions on the submanifolds. In this paper, we consider a semi-Lagrangian approximation scheme for the previous problem. Relying on a classical stability argument in viscosity solution theory, we prove the convergence of the scheme to the value function. We also present HJSD, a free software we developed for the numerical solution of control problems on stratified domains in two and three dimensions, showing, in various examples, the particular phenomena that can arise with respect to the classical continuous framework.