Turnpike properties of optimal boundary control problems with random linear hyperbolic systems

¹ Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematic, Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur), Cauerstr. 11, 91058 Erlangen, Germany
² Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Templergraben 55, 52062 Aachen, Germany

^* Corresponding author: martin.gugat@fau.de

Received: 14 February 2022
Accepted: 25 June 2023

Abstract

In many applications, in systems that are governed by linear hyperbolic partial differential equations some of the problem parameters are uncertain. If information about the probability distribution of the parametric uncertainty, distribution is available, the uncertain state of the system can be described using an intrinsic formulation through a polynomial chaos expansion. This allows to obtain solutions for optimal boundary control problems with random parameters. We show that similar to the deterministic case, a turnpike result holds in the sense that for large time horizons the optimal states for dynamic optimal control problems on a substantial part of the time interval approaches the optimal states for the corresponding uncertain static optimal control problem. We show turnpike results both for the full uncertain system as well as for a generalized polynomial chaos approximation.