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NECESSARY CONDITIONS OF FIRST-ORDER FOR AN OPTIMAL BOUNDARY CONTROL PROBLEM FOR VISCOUS DAMAGE PROCESSES IN 2D

机译：一维粘性损伤过程最优边界控制问题的一阶必要条件

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Controlling the growth of material damage is an important engineering task with plenty of real world applications. In this paper we approach this topic from the mathematical point of view by investigating an optimal boundary control problem for a damage phase-field model for viscoelastic media. We consider non-homogeneous Neumann data for the displacement field which describe external boundary forces and act as control variables. The underlying hyberbolic-parabolic PDE system for the state variables exhibit highly nonlinear terms which emerge in context with damage processes. The cost functional is of tracking type, and constraints for the control variable are prescribed. Based on recent results from [M.H. Farshbaf Shaker and C. Heinemann, Math. Models Methods Appl. Sci. 25 (2015) 2749-2793], where global-in-time well-posedness of strong solutions to the lower level problem and existence of optimal controls of the upper level problem have been established, we show in this contribution differentiability of the control-to-state mapping, well-posedness of the linearization and existence of solutions of the adjoint state system. Due to the highly nonlinear nature of the state system which has by our knowledge not been considered for optimal control problems in the literature, we present a very weak formulation and estimation techniques of the associated adjoint system. For mathematical reasons the analysis is restricted here to the two-dimensional case. We conclude our results with first-order necessary optimality conditions in terms of a variational inequality together with PDEs for the state and adjoint state system.
机译：在大量实际应用中，控制材料损伤的增长是一项重要的工程任务。在本文中，我们通过研究粘弹性介质损伤相场模型的最佳边界控制问题，从数学的角度探讨了该主题。我们考虑位移场的非均匀诺伊曼数据，该数据描述了外部边界力并充当控制变量。用于状态变量的基本双曲线-抛物线PDE系统显示出高度非线性的项，这些项在破坏过程中出现。成本函数是跟踪类型的，并且规定了控制变量的约束条件。根据[M.H. Farshbaf Shaker和C.Heinemann，数学。模型方法应用科学25（2015）2749-2793]，其中已经建立了针对下层问题的强大解决方案的全局及时适时性和上层问题的最优控制的存在，我们在此贡献中显示了控制的可区分性-状态映射，线性化的适定性以及伴随状态系统的解的存在性。由于状态系统的高度非线性性质（据我们所知，文献中尚未将其视为最佳控制问题），因此我们提出了相关联的伴随系统的非常弱的公式化和估算技术。出于数学原因，此处的分析仅限于二维情况。我们用一阶必要最优条件（根据变分不等式）以及状态和伴随状态系统的PDE得出结论。

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