Lax Formulae for Efficiently Solving Two Classes of State-Constrained Optimal Control Problems

TL;DR

This paper introduces Lax formulae for efficiently solving two classes of state-constrained optimal control problems, providing a viscosity theory and a polynomial-complexity numerical algorithm applicable to high-dimensional problems.

Contribution

The paper develops novel Lax formulae and a viscosity theory for specific state-constrained optimal control problems, along with a polynomial-time numerical algorithm.

Findings

Lax formulae enable efficient solutions to certain state-constrained control problems.

The numerical algorithm has polynomial complexity in the state dimension.

Examples demonstrate the effectiveness and performance of the proposed methods.

Abstract

This paper presents Lax formulae for solving the following optimal control problems: minimize the maximum (or the minimum) cost over a time horizon, while satisfying a state constraint. We present a viscosity theory, and by applying the theory to the Hamilton-Jacobi (HJ) equations, these Lax formulae are derived. A numerical algorithm for the Lax formulae is presented: under certain conditions, this algorithm's computational complexity is polynomial in the dimension of the state. For each class of optimal control problem, an example demonstrates the use and performance of the Lax formulae.