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。