Be greedy and learn: efficient and certified algorithms for parametrized optimal control problems

¹ Institute for Analysis and Numerics, Mathematics Münster, University of Münster, Einsteinstrasse 62, 48149 Münster, Germany
² Department of Electrical Engineering and Computing, University of Dubrovnik, Ćira Carića 4, 20 000 Dubrovnik, Croatia
³ Dipartimento di Matematica, University of Genova, Via Dodecaneso 35, 16146 Genoa, Italy

^* Corresponding author: hendrik.kleikamp@uni-muenster.de

Received: 7 August 2023
Accepted: 28 September 2024

Abstract

We consider parametrized linear-quadratic optimal control problems and provide their online-efficient solutions by combining greedy reduced basis methods and machine learning algorithms. To this end, we first extend the greedy control algorithm, which builds a reduced basis for the manifold of optimal final time adjoint states, to the setting where the objective functional consists of a penalty term measuring the deviation from a desired state and a term describing the control energy. Afterwards, we apply machine learning surrogates to accelerate the online evaluation of the reduced model. The error estimates proven for the greedy procedure are further transferred to the machine learning models and thus allow for efficient a posteriori error certification. We discuss the computational costs of all considered methods in detail and show by means of two numerical examples the tremendous potential of the proposed methodology.