Critical cones for sufficient second order conditions in PDE constrained optimization
Identificadores
URI: http://hdl.handle.net/10902/18552DOI: 10.1137/19M1258244
ISSN: 1052-6234
ISSN: 1095-7189
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2020-02Derechos
© Society for Industrial and Applied Mathematics
Publicado en
SIAM Journal on Optimization, 2020, 30(1), 585-603 - (CORRIGENDUM), 2022, 32(1), 319-320
Editorial
Society for Industrial and Applied Mathematics
Palabras clave
Optimal control
Semilinear partial differential equation
Optimality conditions
Sparse controls
Resumen/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.
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