In the present article, we consider a von Kárman equation with long memory. The goal is to study a quadratic cost minimax optimal control problems for the control system governed by the equation. First, we show that the solution map is continuous under a weak assumption on the data. Then, we formulate the minimax optimal control problem. We show the first and twice Fréchet differentiabilities of the nonlinear solution map from a bilinear input term to the weak solution of the equation. With the Fréchet differentiabilities of the control to solution mapping, we prove the uniqueness and existence of an optimal pair and find its necessary optimality condition.

中文翻译：

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具有长记忆的vonKárman系统的双线性Minimax最优控制问题

在本文中，我们考虑具有长记忆的vonKárman方程。目的是研究方程式控制系统的二次成本最小极大最优控制问题。首先，我们证明了在弱假设下对数据的求解图是连续的。然后，我们制定了极大极小最优控制问题。我们显示了从双线性输入项到方程的弱解的非线性解图的第一和两次Fréchet微分。利用控件到解映射的Fréchet可微性，我们证明了最优对的唯一性和存在性，并找到了其必要的最优性条件。