Relaxation Methods for Hyperbolic PDE Mixed-Integer Optimal Control Problems

TL;DR

This paper develops convergence analysis for relaxation methods applied to hyperbolic PDE mixed-integer optimal control problems, with a focus on semilinear systems and traffic flow applications.

Contribution

It extends existing convergence results to hyperbolic PDEs, introducing new a-priori estimates and applying them to flux switching control in conservation laws.

Findings

Convergence of relaxation methods is proven for hyperbolic PDE control problems.

New a-priori estimates for the relaxation gap are established.

Application to traffic flow demonstrates practical effectiveness.

Abstract

We extend the convergence analysis for methods solving PDE-constrained optimal control problems containing both discrete and continuous control decisions based on relaxation and rounding strategies to the class of first order semilinear hyperbolic systems in one space dimension. The results are obtained by novel a-priori estimates for the size of the relaxation gap based on the characteristic flow, fixed-point arguments and particular regularity theory for such mixed-integer control problems. As an application we consider a relaxation model for optimal flux switching control in conservation laws motivated by traffic flow problems.