September  2020, 10(3): 471-491. doi: 10.3934/mcrf.2020007

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

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

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

Received  February 2018 Revised  December 2018 Published  September 2020 Early access  December 2019

Fund Project: 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

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.

Citation: Eduardo Casas, Fredi Tröltzsch. Sparse optimal control for the heat equation with mixed control-state constraints. Mathematical Control and Related Fields, 2020, 10 (3) : 471-491. doi: 10.3934/mcrf.2020007
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. CasasR. 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.

show all references

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. CasasR. 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.

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