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Necessary and sufficient optimality conditionsfor elliptic control problemswith finitely many pointwise state constraints

Published online by Cambridge University Press:  21 December 2007

Eduardo Casas
Eduardo Casas*
Affiliation:
Dpt. Matemática Aplicada y Ciencias de la Computación, E.T.S.I.I y T., Universidad de Cantabria, Av. Los Castros s/n 39005 Santander, Spain; eduardo.casas@unican.es
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Abstract

The goal of this paper is to prove the first and second orderoptimality conditions for some control problems governed bysemilinear elliptic equations with pointwise control constraintsand finitely many equality and inequality pointwise stateconstraints. To carry out the analysis we formulate a regularityassumption which is equivalent to the first order optimalityconditions. Though the presence of pointwise state constraintsleads to a discontinuous adjoint state, we prove that the optimalcontrol is Lipschitz in the whole domain. Necessary and sufficientsecond order conditions are proved with a minimal gap betweenthem.

Keywords

Elliptic control problemspointwise state constraintsPontryagin's principlesecond order optimality conditions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 3 , July 2008 , pp. 575 - 589
Copyright
© EDP Sciences, SMAI, 2007

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References

Arada, N., Casas, E. and Tröltzsch, F., Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. 23 (2002) 201–229. CrossRef
J.F. Bonnans and E. Casas, 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.
E. Casas, 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.
Casas, E., Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints. ESAIM: COCV 8 (2002) 345–374. CrossRef
Casas, E. and Mateos, M., Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431–1454. CrossRef
Casas, E. and Mateos, M., Uniform convergence of the FEM. Applications to state constrained control problems. Comp. Appl. Math. 21 (2002) 67–100.
Casas, E. and Tröltzsch, F., Second order necessary optimality conditions for some state-constrained control problems of semilinear elliptic equations. App. Math. Optim. 39 (1999) 211–227. CrossRef
Casas, E. and Tröltzsch, F., Second order necessary and sufficient optimality conditions for optimization problems and applications to control theory. SIAM J. Optim. 13 (2002) 406–431. CrossRef
E. Casas, J.P. Raymond and H. Zidani, 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.
Casas, E., Tröltzsch, F. and Unger, A., Second order sufficient optimality conditions for some state constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369–1391. CrossRef
Clarke, F.H., A new approach to Lagrange multipliers. Math. Op. Res. 1 (1976) 165–174. CrossRef
P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston-London-Melbourne (1985).
E. Hewitt and K. Stromberg, Real and abstract analysis. Springer-Verlag, Berlin-Heidelberg-New York (1965).
Littman, W. and Stampacchia, G. and Weinberger, H.F., Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Normale Sup. Pisa 17 (1963) 43–77.
M. Mateos, 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).
Maurer, H. and Zowe, J., First- and second-order conditions in infinite-dimensional programming problems. Math. Program. 16 (2000) 431–450.
Raymond, J.-P. and Tröltzsch, F., Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dynam. Systems 6 (1979) 98–110.
Robinson, S.M., Stability theory for systems of inequalities, Part II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13 (1976) 497–513. CrossRef
Saut, J.C. and Scheurer, B., 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. CrossRef