Infinite Horizon Optimal Control Problems for a Class of Semilinear Parabolic Equations

TL;DR

This paper studies infinite horizon optimal control problems for semilinear parabolic equations, deriving first order optimality conditions that reveal sparsity in the optimal controls over time, using an approximation approach.

Contribution

It introduces a novel method to derive optimality conditions without relying on a control-to-state mapping, enabling analysis of sparsity in infinite horizon problems.

Findings

Optimal controls exhibit temporal sparsity.

First order optimality conditions are established.

Approximation by finite horizon problems is effective.

Abstract

Infinite horizon open loop optimal control problems for semilinear parabolic equations are investigated. The controls are subject to a cost-functional which promotes sparsity in time. The focus is put on deriving first order optimality conditions. This is achieved without relying on a well-defined control-to-state mapping in a neighborhood of minimizers. The technique of proof is based on the approximation of the original problem by a family of finite horizon problems. The optimality conditions allow to deduce sparsity properties of the optimal controls in time.