Journal of Convex Analysis 27 (2020), No. 2, 443--485
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@tu-dortmund.de
Joachim Rehberg
Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany
rehberg@wias-berlin.de
It is well-known 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 time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space Wp1,2(X;Y) and a suitable regularization- or Tychonov term Jc 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 obstacle-type in time. We establish the necessary properties of Wp1,2(X;Y) and Jc 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.
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