Optimal control of ODEs with state suprema

Tobias Geiger; Daniel Wachsmuth; Gerd Wachsmuth

doi:10.3934/mcrf.2021012

Article Contents

2021, Volume 11, Issue 3: 555-578. Doi: 10.3934/mcrf.2021012

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Optimal control of ODEs with state suprema

1.
Institute of Mathematics, University of Würzburg, Emil-Fischer-Str. 40, 97074 Würzburg, Germany
2.
Brandenburgische Technische Universität Cottbus-Senftenberg, Institute of Mathematics, 03046 Cottbus, Germany

^* Corresponding author: D. Wachsmuth
^* Corresponding author: D. Wachsmuth

Received: April 30, 2019

Revised: October 31, 2019

Early access: March 2021

Published: September 2021

The second and third author were supported by DFG grants within the Priority Program SPP 1962 (Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization), which is gratefully acknowledged

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Abstract

We consider the optimal control of a differential equation that involves the suprema of the state over some part of the history. In many applications, this non-smooth functional dependence is crucial for the successful modeling of real-world phenomena. We prove the existence of solutions and show that related problems may not possess optimal controls. Due to the non-smoothness in the state equation, we cannot obtain optimality conditions via standard theory. Therefore, we regularize the problem via a LogIntExp functional which generalizes the well-known LogSumExp. By passing to the limit with the regularization, we obtain an optimality system for the original problem. The theory is illustrated by some numerical experiments.

Keywords:

Functional differential equations,
differential equations with state suprema,
optimality conditions,
maximum principle,
LogIntExp.

Mathematics Subject Classification: Primary: 49K21; Secondary: 34K35, 49J21.

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Figure 1. Plots for control $ u $, state $ x $, adjoint $ \lambda $ and its time derivative $ \mathrm{d}\lambda $

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