Volume 27, 2021
Special issue in honor of Enrique Zuazua's 60th birthday
|Number of page(s)||26|
|Published online||20 January 2021|
Sparse optimal control for a semilinear heat equation with mixed control-state constraints – regularity of Lagrange multipliers*,**
Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria,
2 Institut für Mathematik, Technische Universität Berlin, 10623 Berlin, Germany.
*** Corresponding author: firstname.lastname@example.org
Accepted: 30 November 2020
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.
Mathematics Subject Classification: 49K20 / 49N10 / 90C05 / 90C46
Key words: Semilinear heat equation / optimal control / sparse control / mixed control-state constraints / regular Lagrange multipliers
© EDP Sciences, SMAI 2021
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