The Focusing Energy-Critical Wave Equation

Kenig, Carlos E.

doi:10.1007/978-3-319-52742-0_7

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2020

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“…Of course, this is all for the defocusing case since the mentioned works establish global existence for initial data of arbitrary size. Extensions of the mentioned results and especially the focusing case are in the focus of current research; we mention exemplarily [11,13,14,21,25].…”

Section: Contextmentioning

confidence: 95%

Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints

Kunisch

Meinlschmidt

2020

Journal de Mathématiques Pures et Appliquées

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This paper is concerned with an optimal control problem subject to the H 1 -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) L 2 (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 L 1 (0, T; L 2 (Ω)), which also promotes sparsity in time of an optimal control, is used in the objective functional.

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