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Optimal control of ODEs with state suprema
Mathematical Control and Related Fields ( IF 1.0 ) Pub Date : 2021-03-04 , DOI: 10.3934/mcrf.2021012
Tobias Geiger , Daniel Wachsmuth , Gerd Wachsmuth

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.

中文翻译:

具有状态至上的 ODE 的最优控制

我们考虑微分方程的最优控制,该微分方程涉及历史的某些部分的状态至上。在许多应用中,这种非平滑函数依赖性对于成功建模现实世界现象至关重要。我们证明了解决方案的存在,并表明相关问题可能不具有最优控制。由于状态方程的非光滑性,我们无法通过标准理论获得最优条件。因此,我们通过泛化著名的 LogSumExp 的 LogIntExp 函数对问题进行正则化。通过正则化传递到极限,我们获得了原始问题的最优系统。该理论是通过一些数值实验来说明的。
更新日期:2021-03-04
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