
Journal of Convex Analysis 27 (2020), No. 2, 443485 Copyright Heldermann Verlag 2020 Regularization for Optimal Control Problems Associated to Nonlinear Evolution Equations Hannes Meinlschmidt Johann Radon Institute for Computational and Applied Mathematics, 4040 Linz, Austria hannes.meinlschmidt@ricam.oeaw.ac.at Christian Meyer Technische Universität, Fakultät für Mathematik, 44227 Dortmund, Germany christian2.meyer@tudortmund.de Joachim Rehberg Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany rehberg@wiasberlin.de It is wellknown that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard "calculus of variations" proof for the existence of optimal controls. For timedependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochnertype spaces. In this paper, we propose an abstract function space W_{p}^{1,2}(X;Y) and a suitable regularization or Tychonov term J_{c} for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacletype in time. We establish the necessary properties of W_{p}^{1,2}(X;Y) and J_{c} and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand. Keywords: Optimal control, regularization, nonlinear evolution equations, compactness, function spaces. MSC: 49K20, 49J20, 47J20, 47J35, 46E40. [ Fulltextpdf (298 KB)] for subscribers only. 