# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021012

## 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

Received  May 2019 Revised  November 2019 Published  March 2021

Fund Project: 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

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.

Citation: Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021012
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Plots for control $u$, state $x$, adjoint $\lambda$ and its time derivative $\mathrm{d}\lambda$
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