Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations

TL;DR

This paper advances curse-of-dimensionality-free numerical methods for Hamilton-Jacobi equations, providing a rigorous formulation, practical recommendations, and demonstrating high potential through numerical simulations, while acknowledging current limitations.

Contribution

It extends the Hopf-Lax formula and develops characteristic-based methods to evaluate solutions at different points, avoiding the curse of dimensionality in specific Hamilton-Jacobi problems.

Findings

Numerical simulations show high potential of the proposed techniques.

Methods enable solution evaluation without derivative approximation.

Approaches are currently limited to certain control systems.

Abstract

This paper extends the considerations of the works [1, 2] regarding curse-of-dimensionality-free numerical approaches to solve certain types of Hamilton-Jacobi equations arising in optimal control problems, differential games and elsewhere. A rigorous formulation and justification for the extended Hopf-Lax formula of [2] is provided together with novel theoretical and practical discussions including useful recommendations. By using the method of characteristics, the solutions of some problem classes under convexity/concavity conditions on Hamiltonians (in particular, the solutions of Hamilton-Jacobi-Bellman equations in optimal control problems) are evaluated separately at different initial positions. This allows for the avoidance of the curse of dimensionality, as well as for choosing arbitrary computational regions. The corresponding feedback control strategies are obtained at selected positions without approximating the partial derivatives of the solutions. The results of numerical simulations demonstrate the high potential of the proposed techniques. It is also pointed out that, despite the indicated advantages, the related approaches still have a limited range of applicability, and their extensions to Hamilton-Jacobi-Isaacs equations in zero-sum two-player differential games are currently developed only for sufficiently narrow classes of control systems. That is why further extensions are worth investigating.