
Journal of Convex Analysis 27 (2020), No. 2, 509533 Copyright Heldermann Verlag 2020 On the Metric Regularity of Affine Optimal Control Problems Marc Quincampoix Laboratoire de Mathématiques de Bretagne Atlantique, Unité CNRS UMR6205, Université de Brest, France marc.quincampoix@univbrest.fr Teresa Scarinci Dept. of Statistics and Operations Research, University of Vienna, Austria teresa.scarinci@univie.ac.at Vladimir M. Veliov Inst. of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Austria vladimir.veliov@tuwien.ac.at The paper establishes properties of the type of (strong) metric regularity of the setvalued map associated with the system of necessary optimality conditions for optimal control problems that are affine with respect to the control (shortly, affine problems). It is shown that for such problems it is reasonable to extend the standard notions of metric regularity by involving two metrics in the image space of the map. This is done by introducing (following an earlier paper by the first and the third named author) the concept of (strong) bimetric regularity in a general space setting. LyusternikGravestype theorems are proved for (strongly) bimetrically regular maps, which claim stability of these regularity properties with respect to "appropriately small" perturbations. Based on that, it is proved that in the case of a map associated with affine optimal control problems, the strong bimetric regularity is invariant with respect to linearization. This result is complemented with a sufficient condition for strong bimetric regularity for linearquadratic affine optimal control problems, which applies to the "linearization" of a nonlinear affine problem. Thus the same conditions are also sufficient for strong bimetric regularity in the nonlinear affine problem. Keywords: Optimal control, affine problems, metric regularity, solution stability, perturbed control problems. MSC: 49J30, 49K15, 49K40. [ Fulltextpdf (190 KB)] for subscribers only. 