Error estimates for a multiobjective optimal control of a pointwise tracking problem

TL;DR

This paper develops and analyzes finite element-based scalarization techniques for multiobjective optimal control problems with pointwise tracking, providing error estimates and demonstrating numerical accuracy.

Contribution

It introduces and compares two scalarization methods for multiobjective control, proving a priori error estimates and validating them through numerical experiments.

Findings

Both scalarization methods yield accurate approximations.

Error estimates are rigorously established for the methods.

Numerical results confirm the theoretical error bounds.

Abstract

We analyze a pointwise tracking multiobjective optimal control problem subject to the Poisson problem and bilateral control constraints. To approximate Pareto optimal points and the Pareto front numerically, we consider two different finite element-based scalarization techniques, namely the weighted-sum method and the reference point method, where in both methods many scalar-constrained optimization problems have to be solved. We prove a priori error estimates for both scalarizations. The underlying subproblems of either method are solved with a Barzilai-Borwein gradient method. Numerical experiments illustrate the accuracy of the proposed method.