Sparse optimal control for a semilinear heat equation with mixed control-state constraints - regularity of Lagrange multipliers
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Identificadores
URI: http://hdl.handle.net/10902/20522DOI: 10.1051/cocv/2020084
ISSN: 1292-8119
ISSN: 1262-3377
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2021-01-20Derechos
© EDP Sciences; Société de Mathématiques Appliquées et Industrielles (SMAI). The original publication is available at www.esaim-cocv.org.
Publicado en
ESAIM: Control, optimisation and calculus of variations, 2021, 27, 2
Editorial
EDP Sciences
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Palabras clave
Semilinear heat equation
Optimal control
Sparse control
Mixed control-state constraints
Regular Lagrange multipliers
Resumen/Abstract
An optimal control problem for a semilinear heat equation with distributed control is discussed, where two-sided pointwise box constraints on the control and two-sided pointwise mixed control-state constraints are given. The objective functional is the sum of a standard quadratic tracking type part and a multiple of the L1-norm of the control that accounts for sparsity. Under a certain structural condition on almost active sets of the optimal solution, the existence of integrable Lagrange multipliers is proved for all inequality constraints. For this purpose, a theorem by Yosida and Hewitt is used. It is shown that the structural condition is fulfilled for all sufficiently large sparsity parameters. The sparsity of the optimal control is investigated. Eventually, higher smoothness of Lagrange multipliers is shown up to Hölder regularity.
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