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Optimal control of ODEs with state suprema

  • * Corresponding author: D. Wachsmuth

    * Corresponding author: D. Wachsmuth 
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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  • 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.

    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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