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

Casas Eduardo; Tröltzsch Fredi

doi:10.3934/mcrf.2020007

Article Contents

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: January 31, 2018

Revised: November 30, 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

[1]	J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer Series in Operations Research, Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1394-9.
[2]	E. Casas, R. Herzog and G. Wachsmuth, Optimality conditions and error analysis of semilinear elliptic control problems with $L^1$ cost functional, SIAM J. Optim., 22 (2012), 795-820. doi: 10.1137/110834366.
[3]	R. C. Grinold, Continuous programming. I. Linear objectives, J. Math. Anal. Appl., 28 (1969), 32-51. doi: 10.1016/0022-247X(69)90106-1.
[4]	R. C. Grinold, Symmetric duality for continuous linear programs, SIAM J. Appl. Math., 18 (1970), 84-97. doi: 10.1137/0118011.
[5]	J. Jahn, Vector Optimization. Theory, Applications, and Extensions, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-24828-6.
[6]	W. Krabs, Zur Dualitätstheorie bei linearen Optimierungsproblemen in halbgeordneten Vektorräumen, Math. Z., 121 (1971), 320-328. doi: 10.1007/BF01109978.
[7]	O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I., 1968.
[8]	J.-L. Lions, Contrôle Optimal de Systèmes Gouvernès par des Équations aux Dérivées Partielles, Avant Propos de P. Lelong Dunod, Paris, Gauthier-Villars, Paris, 1968.
[9]	D. G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., New York-London-Sydney, 1969.
[10]	A. Rösch and F. Tröltzsch, On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints, SIAM J. Control and Optimization, 46 (2007), 1098-1115. doi: 10.1137/060671565.
[11]	F. Tröltzsch, Existenz- und Dualitätsaussagen für lineare Optimierungsaufgaben in reflexiven Banach-Räumen., Math. Operationsforschung und Statistik, 6 (1975), 901-912. doi: 10.1080/02331887508801268.
[12]	F. Tröltzsch, A minimum principle and a generalized bang-bang-principle for a distributed optimal control problem with constraints on the control and the state, Z. Angew. Math. Mech., 59 (1979), 737-739. doi: 10.1002/zamm.19790591208.
[13]	W. F. Tyndall, A duality theorem for a class of continuous linear programming problems, J. Soc. Indust. Appl. Math., 13 (1965), 644-666. doi: 10.1137/0113043.