Iterative Method using the Generalized Hopf Formula: Avoiding Spatial Discretization for Computing Solutions of Hamilton-Jacobi Equations for Nonlinear Systems

Abstract

Hamilton-Jacobi (HJ) analysis provides globally optimal solutions for multiple player game problems in a variety of fields, including robotics, control, logistics, manufacturing, and finance. Despite its importance, computation complexity limits its use. Motivated by recent work in using the Hopf formula for a grid-free solution to Hamilton-Jacobi equations for linear systems, this paper proposes an iterative method that provides a sub-optimal solution and corresponding control law for the solution of Hamilton-Jacobi partial differential equations for nonlinear systems. This allows efficient computation of Hamilton-Jacobi solutions in high dimensions, for a broader class of systems than has been treated in prior work, and also provides a conservative solution that guarantees goal-reaching in goal-reaching problems or safety in collision avoidance problems. We name our method the Iterative Hopf Method. We demonstrate our method in two examples: goal-reaching and collision avoidance problems with a three-dimensional vehicle model to analyze performance of the iterative Hopf method by comparing with the level set method for computing a convergent solution to Hamilton-Jacobi equations, and a goal-reaching problem with a seven-dimensional vehicle whose computation for the solution is considered to be intractable.