Sparse optimal control for the heat equation with mixed control-state constraints

Casas Eduardo; Tröltzsch Fredi

doi:10.3934/mcrf.2020007

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2020, Volume 10, Issue 3: 471-491. Doi: 10.3934/mcrf.2020007

This issue Previous Article Implicit parametrizations and applications in optimization and control Next Article Optimality conditions in variational form for non-linear constrained stochastic control problems

Sparse optimal control for the heat equation with mixed control-state constraints

Casas Eduardo^1, and
Tröltzsch Fredi^2, ,

1.
Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain
2.
Institut für Mathematik, Technische Universität Berlin, D-10623 Berlin, Germany

^* Corresponding author: Fredi Tröltzsch
^* Corresponding author: Fredi Tröltzsch

Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday

Received: February 2018

Revised: December 2018

Early access: December 2019

Published: July 2020

The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P. The second author was supported by the Collaborative Research Center SFB 910, TU Berlin, project B6

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Abstract

A problem of sparse optimal control for the heat equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic tracking type functional is to be minimized that includes a Tikhonov regularization term and the $ L^1 $-norm of the control accounting for the sparsity. Special emphasis is laid on existence and regularity of Lagrange multipliers for the mixed control-state constraints. To this aim, a duality theorem for linear programming problems in Hilbert spaces is proved and applied to the given optimal control problem.

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Mathematics Subject Classification: Primary: 49K20, 49N10; Secondary: 90C05, 90C46.

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References

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