On Pareto equilibria for bi-objective diffusive optimal control problems

TL;DR

This paper studies Pareto equilibria in bi-objective optimal control problems involving heat equations and multiplicative controls, introducing numerical methods and demonstrating their effectiveness through experiments.

Contribution

It develops numerical methods combining finite elements and finite differences for bi-objective control problems with heat equations and multiplicative controls.

Findings

Numerical methods successfully compute Pareto equilibria.

Experiments demonstrate the effectiveness of the proposed methods.

Framework handles linear, semilinear, and multiplicative control systems.

Abstract

We investigate Pareto equilibria for bi-objective optimal control problems. Our framework comprises the situation in which an agent acts with a distributed control in a portion of a given domain, and aims to achieve two distinct (possibly conflicting) targets. We analyze systems governed by linear and semilinear heat equations and also systems with multiplicative controls. We develop numerical methods relying on a combination of finite elements and finite differences. We illustrate the computational methods we develop via numerous experiments.