The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control is Lipschitz in the whole domain. Necessary and sufficient second order conditions are proved with a minimal gap between them.
Mots clés : elliptic control problems, pointwise state constraints, Pontryagin's principle, second order optimality conditions
@article{COCV_2008__14_3_575_0,
author = {Casas, Eduardo},
title = {Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {575--589},
publisher = {EDP-Sciences},
volume = {14},
number = {3},
year = {2008},
doi = {10.1051/cocv:2007063},
mrnumber = {2434067},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv:2007063/}
}
TY - JOUR AU - Casas, Eduardo TI - Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 575 EP - 589 VL - 14 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007063/ DO - 10.1051/cocv:2007063 LA - en ID - COCV_2008__14_3_575_0 ER -
%0 Journal Article %A Casas, Eduardo %T Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 575-589 %V 14 %N 3 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007063/ %R 10.1051/cocv:2007063 %G en %F COCV_2008__14_3_575_0
Casas, Eduardo. Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 3, pp. 575-589. doi : 10.1051/cocv:2007063. http://archive.numdam.org/articles/10.1051/cocv:2007063/
[1] , and , Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201-229. | MR | Zbl
[2] and , Contrôle de systèmes elliptiques semilinéaires comportant des contraintes sur l'état, in Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar 8, H. Brezis and J.-L. Lions Eds., Longman Scientific & Technical, New York (1988) 69-86. | MR | Zbl
[3] , Pontryagin's principle for optimal control problems governed by semilinear elliptic equations, in International Conference on Control and Estimation of Distributed Parameter Systems: Nonlinear Phenomena, F. Kappel and K. Kunisch Eds., Basel, Birkhäuser, Int. Series Num. Analysis. 118 (1994) 97-114. | MR | Zbl
[4] , Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: COCV 8 (2002) 345-374. | Numdam | MR | Zbl
[5] and , Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | MR | Zbl
[6] and , Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67-100. | MR | Zbl
[7] and , Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. App. Math. Optim. 39 (1999) 211-227. | MR | Zbl
[8] and , Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 13 (2002) 406-431. | MR | Zbl
[9] , and , Optimal control problems governed by semilinear elliptic equations with integral control constraints and pointwise state constraints, in International Conference on Control and Estimations of Distributed Parameter Systems, W. Desch, F. Kappel and K. Kunisch Eds., Basel, Birkhäuser, Int. Series Num. Analysis. 126 (1998) 89-102. | MR | Zbl
[10] , and , Second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | MR | Zbl
[11] , A new approach to Lagrange multipliers. Math. Op. Res. 1 (1976) 165-174. | MR | Zbl
[12] , Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985). | MR | Zbl
[13] and , Real and abstract analysis. Springer-Verlag, Berlin-Heidelberg-New York (1965). | MR | Zbl
[14] and and H.F. Weinberger, Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Normale Sup. Pisa 17 (1963) 43-77. | Numdam | MR | Zbl
[15] , Problemas de control óptimo gobernados por ecuaciones semilineales con restricciones de tipo integral sobre el gradiente del estado. Ph.D. thesis, University of Cantabria, Spain (2000).
[16] and , First- and second-order conditions in infinite-dimensional programming problems. Math. Program. 16 (2000) 431-450. | Zbl
[17] and , Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Systems 6 (1979) 98-110. | MR | Zbl
[18] , Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13 (1976) 497-513. | MR | Zbl
[19] and , Sur l'unicité du problème de Cauchy et le prolongement unique pour des équations elliptiques à coefficients non localement bornés. J. Differential Equations 43 (1982) 28-43. | MR | Zbl
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