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Critical cones for sufficient second order conditions in PDE constrained optimization

Abstract: In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed, and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

 Autoría: Casas E., Mateos M.,

 Fuente: SIAM Journal on Optimization, 2020, 30(1), 585-603 - (CORRIGENDUM), 2022, 32 (1), 319--320

 Editorial: Society for Industrial and Applied Mathematics

 Fecha de publicación: 01/02/2020

 Nº de páginas: 19

 Tipo de publicación: Artículo de Revista

 DOI: 10.1137/19M1258244

 ISSN: 1052-6234,1095-7189

 Proyecto español: MTM2017-83185-P