Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type

Maïtine Bergounioux; Fredi Tröltzsch

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 4, page 595-608
  • ISSN: 1292-8119

Abstract

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We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming problems.

How to cite

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Bergounioux, Maïtine, and Tröltzsch, Fredi. "Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type." ESAIM: Control, Optimisation and Calculus of Variations 4 (2010): 595-608. <http://eudml.org/doc/197341>.

@article{Bergounioux2010,
abstract = { We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming problems. },
author = {Bergounioux, Maïtine, Tröltzsch, Fredi},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = { Parabolic equation; optimal control; pointwise state-constraint; bottleneck problem.; pointwise state constraints; semilinear parabolic equations; constraints of bottleneck; Lagrange multipliers; continuous linear programming},
language = {eng},
month = {3},
pages = {595-608},
publisher = {EDP Sciences},
title = {Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type},
url = {http://eudml.org/doc/197341},
volume = {4},
year = {2010},
}

TY - JOUR
AU - Bergounioux, Maïtine
AU - Tröltzsch, Fredi
TI - Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 4
SP - 595
EP - 608
AB - We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming problems.
LA - eng
KW - Parabolic equation; optimal control; pointwise state-constraint; bottleneck problem.; pointwise state constraints; semilinear parabolic equations; constraints of bottleneck; Lagrange multipliers; continuous linear programming
UR - http://eudml.org/doc/197341
ER -

References

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  1. M. Bergounioux and F. Tröltzsch, Optimality Conditions and Generalized Bang-Bang Principle for a State Constrained Semilinear Parabolic Problem. Num. Funct. Anal. Opt.15 (1996) 517-537.  Zbl0858.49021
  2. M. Bergounioux and F. Tröltzsch, Optimal Control of Linear Bottleneck Problems. ESAIM: Cont. Optim. Cal. Var.3 (1998) 235-250.  Zbl0904.49020
  3. E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim.35 (1997) 1297-1327.  Zbl0893.49017
  4. J.P. Raymond, Non Linear Boundary Control of Semilinear Parabolic Problems with Pointwise State Constraints. Discrete and Continuous Dynamical Systems3 (1997) 341-370.  Zbl0953.49026
  5. J.P. Raymond and H. Zidani, Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations. Appl. Math. Optim., to appear  Zbl0922.49013
  6. F. Tröltzsch, Optimality conditions for parabolic control problems and applications, Teubner Texte zur Mathematik, Teubner, Leipzig (1984).  Zbl0587.49021
  7. J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim.5 (1979) 49-62.  Zbl0401.90104

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