Relaxation Methods for Mixed-Integer Optimal Control of Partial Differential Equations

TL;DR

This paper develops relaxation techniques for mixed-integer optimal control problems governed by partial differential equations, providing conditions for approximation accuracy and introducing a constructive numerical method.

Contribution

It extends relaxation methods from ODEs to PDEs, offering theoretical guarantees and a numerical approach for mixed-integer control of distributed parameter systems.

Findings

Relaxation techniques can approximate integer controls in PDE systems with arbitrary precision.

The method is constructive and suitable for numerical implementation.

Numerical experiments validate the theoretical results.

Abstract

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.