Dynamic boundary control games with networks of strings

¹ Friedrich-Alexander Universität Erlangen-Nürnberg (FAU), Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany
² RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany

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

Received: 29 March 2017
Accepted: 14 December 2017

Abstract

Consider a star-shaped network of strings. Each string is governed by the wave equation. At each boundary node of the network there is a player that performs Dirichlet boundary control action and in this way influences the system state. At the central node, the states are coupled by algebraic conditions in such a way that the energy is conserved. We consider the corresponding antagonistic game where each player minimizes a certain quadratic objective function that is given by the sum of a control cost and a tracking term for the final state. We prove that under suitable assumptions a unique Nash equilibrium exists and give an explicit representation of the equilibrium strategies.