A Second-Order Reachable Sets Computation Scheme via a Cauchy-Type Variational Hamilton-Jacobi-Isaacs Equation

TL;DR

This paper introduces a second-order, successive sweep algorithm for efficiently computing reachable sets in control problems with worst-case disturbances, addressing scalability issues of traditional Eulerian methods.

Contribution

The paper proposes a novel second-order algorithm for reachability analysis based on a Cauchy-type variational Hamilton-Jacobi-Isaacs equation, improving scalability and computational efficiency.

Findings

The algorithm accurately computes reachable sets within a specified time bound.

It demonstrates improved scalability over traditional Eulerian methods.

The approach relies on regularity and continuity assumptions of the HJI PDE.

Abstract

Motivated by the scalability limitations of Eulerian methods for variational Hamilton-Jacobi-Isaacs (HJI) formulations that provide a least restrictive controller in problems that involve state or input constraints under a worst-possible disturbance, we introduce a second-order, successive sweep algorithm for computing the zero sublevel sets of a popular reachability value functional. Under sufficient HJI partial differential equation regularity and continuity assumption throughout the state space, we show that with state feedback control under the worst-possible disturbance, we can compute the state set that are reachable within a prescribed verification time bound.