Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems

TL;DR

This paper introduces a penalty method for numerically solving discrete Hamilton-Jacobi-Bellman equations with control and obstacle constraints, providing error estimates and iterative solution techniques.

Contribution

The paper develops a novel penalty approach with convergence analysis and demonstrates its effectiveness through numerical experiments.

Findings

Penalisation error estimates for a class of penalty terms.

Newton's method variations achieve global convergence.

Numerical results show the method's competitiveness.

Abstract

In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.