Geodesic
Regularization for Optimal Control Problems Associated to Nonlinear Evolution Equations
Journal of convex analysis, Tome 27 (2020) no. 2, pp. 443-485
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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.
Classification : 49K20, 49J20, 47J20, 47J35, 46E40
Mots-clés : Optimal control, regularization, nonlinear evolution equations, compactness, function spaces 
@article{JCA_2020_27_2_JCA_2020_27_2_a3,
     author = {H. Meinlschmidt and C. Meyer and J. Rehberg},
     title = {Regularization for {Optimal} {Control} {Problems} {Associated} to {Nonlinear} {Evolution} {Equations}},
     journal = {Journal of convex analysis},
     pages = {443--485},
     year = {2020},
     volume = {27},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2020_27_2_JCA_2020_27_2_a3/}
}
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H. Meinlschmidt; C. Meyer; J. Rehberg. Regularization for Optimal Control Problems Associated to Nonlinear Evolution Equations. Journal of convex analysis, Tome 27 (2020) no. 2, pp. 443-485. http://geodesic.mathdoc.fr/item/JCA_2020_27_2_JCA_2020_27_2_a3/