Impulse and sampled-data optimal control of heat equations, and error estimates

TL;DR

This paper investigates optimal control of heat equations with periodic constraints, introducing impulse and sampled-data approximations, and provides error estimates for these methods compared to the original problem.

Contribution

It develops and analyzes impulse and sampled-data control approximations for heat equations, establishing their uniqueness and deriving precise error bounds.

Findings

Both approximations have unique optimal solutions.

Error estimates quantify the difference between original and approximate controls and states.

The methods provide practical approaches for controlling heat equations with quantifiable accuracy.

Abstract

We consider the optimal control problem of minimizing some quadratic functional over all possible solutions of an internally controlled multi-dimensional heat equation with a periodic terminal state constraint. This problem has a unique optimal solution, which can be characterized by an optimality system derived from the Pontryagin maximum principle. We define two approximations of this optimal control problem. The first one is an impulse approximation, and consists of considering a system of linear heat equations with impulse control. The second one is obtained by the sample-and-hold procedure applied to the control, resulting into a sampled-data approximation of the controlled heat equation. We prove that both problems have a unique optimal solution, and we establish precise error estimates for the optimal controls and optimal states of the initial problem with respect to its impulse and sampled-data approximations.