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

Kunisch K, Meinlschmidt H (2020)

Publication Type: Journal article

Publication year: 2020

Journal

Journal De Mathematiques Pures Et Appliquees Elsevier

Book Volume: 138

Pages Range: 46-87

DOI: 10.1016/j.matpur.2020.03.006

Abstract

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.

Authors with CRIS profile

Hannes Meinlschmidt

Involved external institutions

Karl-Franzens-Universität Graz AT Austria (AT) Johann Radon Institut für Angewandte Mathematik / Johann Radon Institute for Computational and Applied Mathematics (RICAM) AT Austria (AT)

How to cite

APA:

Kunisch, K., & Meinlschmidt, H. (2020). Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints. Journal De Mathematiques Pures Et Appliquees, 138, 46-87. https://doi.org/10.1016/j.matpur.2020.03.006

MLA:

Kunisch, Karl, and Hannes Meinlschmidt. "Optimal control of an energy-critical semilinear wave equation in 3D with spatially integrated control constraints." Journal De Mathematiques Pures Et Appliquees 138 (2020): 46-87.

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