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Implicit parametrizations and applications in optimization and control
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 |
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.
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. |
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. 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. |
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