Contraction of Riccati flows applied to the convergence analysis of a max-plus curse of dimensionality free method

TL;DR

This paper improves the convergence analysis of a max-plus method for solving Hamilton-Jacobi-Bellman equations by applying Riccati flow contraction, leading to exponential error decay and better practical performance.

Contribution

It introduces a new contraction-based analysis of Riccati flows that refines error bounds for the max-plus curse-of-dimensionality free method.

Findings

Error decays exponentially with iterations under certain conditions

Improved estimates for execution time and pruning precision

Applicable to a broad class of Hamilton-Jacobi-Bellman problems

Abstract

Max-plus based methods have been recently explored for solution of first-order Hamilton-Jacobi-Bellman equations by several authors. In particular, McEneaney's curse-of-dimensionality free method applies to the equations where the Hamiltonian takes the form of a (pointwise) maximum of linear/quadratic forms. In previous works of McEneaney and Kluberg, the approximation error of the method was shown to be $O(1/(N\tau))+O(\sqrt{\tau})$ where $\tau$ is the time discretization step and $N$ is the number of iterations. Here we use a recently established contraction result for the indefinite Riccati flow in Thompson's metric to show that under different technical assumptions, still covering an important class of problems, the error is only of order $O(e^{-\alpha N\tau})+O(\tau)$. This also allows us to obtain improved estimates of the execution time and to tune the precision of the pruning procedure, which in practice is a critical element of the method.