The patchy Method for the Infinite Horizon Hamilton-Jacobi-Bellman Equation and its Accuracy

TL;DR

This paper presents a modified patchy method for solving the stationary Hamilton-Jacobi-Bellman equation, producing polynomial approximations of optimal cost and control with an error bound under certain smoothness assumptions.

Contribution

It introduces a new modification to the patchy method with an error analysis, enhancing the accuracy of solutions for the HJB equation.

Findings

Provides an error bound for the numerical method

Produces polynomial approximations of optimal cost and control

Validates the method under smooth Lyapunov function assumption

Abstract

We introduce a modification to the patchy method of Navasca and Krener for solving the stationary Hamilton Jacobi Bellman equation. The numerical solution that we generate is a set of polynomials that approximate the optimal cost and optimal control on a partition of the state space. We derive an error bound for our numerical method under the assumption that the optimal cost is a smooth strict Lyupanov function. The error bound is valid when the number of subsets in the partition is not too large.