Computation of Reachable Sets Based on Hamilton-Jacobi-Bellman Equation with Running Cost Function

TL;DR

This paper introduces a new numerical method for computing classical and generalized reachable sets using Hamilton-Jacobi-Bellman equations with running costs, employing recursion and grid interpolation to handle solution discontinuities.

Contribution

The paper presents a novel approach to compute both classical and cost-limited reachable sets via Hamilton-Jacobi-Bellman equations with a new numerical scheme.

Findings

Successfully computes classical reachable sets for various scenarios.

Extends to generalized reachable sets with different cost functions.

Demonstrates the method's validity through illustrative examples.

Abstract

A novel method for computing reachable sets is proposed in this paper. In the proposed method, a Hamilton-Jacobi-Bellman equation with running cost functionis numerically solved and the reachable sets of different time horizons are characterized by a family of non-zero level sets of the solution of the Hamilton-Jacobi-Bellman equation. In addition to the classical reachable set, by setting different running cost functions and terminal conditionsof the Hamilton-Jacobi-Bellman equation, the proposed method allows to compute more generalized reachable sets, which are referred to as cost-limited reachable sets. In order to overcome the difficulty of solving the Hamilton-Jacobi-Bellman equation caused by the discontinuity of the solution, a method based on recursion and grid interpolation is employed. At the end of this paper, some examples are taken to illustrate the validity and generality of the proposed method.