Journal of Convex Analysis 27 (2020), No. 2, 443--485
Copyright Heldermann Verlag 2020
Regularization for Optimal Control Problems Associated to Nonlinear Evolution Equations
Johann Radon Institute for Computational and Applied Mathematics, 4040 Linz, Austria
Technische Universität, Fakultät für Mathematik, 44227 Dortmund, Germany
Weierstrass Institute for Applied Analysis and Stochastics, 10117 Berlin, Germany
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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