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Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints
Journal de Mathématiques Pures et Appliquées ( IF 2.3 ) Pub Date : 2020-03-16 , DOI: 10.1016/j.matpur.2020.03.006
Karl Kunisch , Hannes Meinlschmidt

This paper is concerned with an optimal control problem subject to the H1-critical defocusing semilinear wave equation on a smooth and bounded domain in three spatial dimensions. Due to the criticality of the nonlinearity in the wave equation, unique solutions to the PDE obeying energy bounds are only obtained in special function spaces related to Strichartz estimates and the nonlinearity. The optimal control problem is complemented by pointwise-in-time constraints of Trust-Region type u(t)L2(Ω)ω(t). We prove existence of globally optimal solutions to the optimal control problem and give optimality conditions of both first- and second order necessary as well as second order sufficient type. A nonsmooth regularization term for the natural control space L1(0,T;L2(Ω)), which also promotes sparsity in time of an optimal control, is used in the objective functional.



中文翻译:

具有空间集成控制约束的3D能量临界半线性波动方程的最优控制

本文关注的是一个最优控制问题。 H1个临界散焦半线性波动方程在三个空间维上的光滑有界域上。由于波动方程中非线性的重要性,仅在与Strichartz估计和非线性有关的特殊函数空间中才能获得PDE服从能界的唯一解。最佳控制问题由Trust-Region类型的时间点约束补充üŤ大号2ΩωŤ。我们证明了最优控制问题的全局最优解的存在,并给出了一阶和二阶必要以及二阶充分类型的最优性条件。自然控制空间的非平滑正则项大号1个0Ť;大号2Ω在目标功能中使用了,它还可以促进最优控制的时间稀疏性。

更新日期:2020-03-16
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