# 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

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topBergounioux, 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

top- 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.
- M. Bergounioux and F. Tröltzsch, Optimal Control of Linear Bottleneck Problems. ESAIM: Cont. Optim. Cal. Var.3 (1998) 235-250.
- E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim.35 (1997) 1297-1327.
- J.P. Raymond, Non Linear Boundary Control of Semilinear Parabolic Problems with Pointwise State Constraints. Discrete and Continuous Dynamical Systems3 (1997) 341-370.
- J.P. Raymond and H. Zidani, Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations. Appl. Math. Optim., to appear
- F. Tröltzsch, Optimality conditions for parabolic control problems and applications, Teubner Texte zur Mathematik, Teubner, Leipzig (1984).
- J. Zowe and S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim.5 (1979) 49-62.

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