K. Deckelnick and M. Hinze, Convergence of a finite element approximation to a state constraint elliptic control problem, 2006.

I. Ekeland and R. Temam, Analyse convexe etprobì emes variationnels, 1974.

H. O. Fattorini, Infinite dimensional optimization and control theory, 1998.
DOI : 10.1017/CBO9780511574795

P. Grisvard, Elliptic problems in nonsmooth domains, 1985.
DOI : 10.1137/1.9781611972030

E. B. Lee and L. Marcus, Foundations of optimal control theory, 1967.

X. Li and J. Yong, Optimal control theory for infinite dimensional systems, 1995.
DOI : 10.1007/978-1-4612-4260-4

J. L. Lions, Contrôle optimal de systèmes gouvernés par des equations aux dérivées partielles, 1968.

P. A. Raviart and J. M. Thomas, IntroductionàIntroductionà l'analyse numérique des equations aux dérivées partielles, 1983.

T. Roubí?ek, Relaxation in optimization theory and variational calculus, 1997.
DOI : 10.1515/9783110811919

A. H. Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates, Mathematics of Computation, vol.67, issue.223, pp.877-899, 1998.
DOI : 10.1090/S0025-5718-98-00959-4

S. S. Sritharan, Optimal control of viscous flow, 1998.
DOI : 10.1137/1.9781611971415

G. Stampacchia, Le probl??me de Dirichlet pour les ??quations elliptiques du second ordre ?? coefficients discontinus, Annales de l???institut Fourier, vol.15, issue.1, pp.189-258, 1965.
DOI : 10.5802/aif.204

F. Tröltzsch, Optimale steuerung partieller differetialgleichungen-theorie, verfahren und anwendungen, 2005.

J. Warga, Optimal control of differential and functional equations, 1972.

L. C. Young, Lectures on the calculus of variations and optimal control theory, 1969.

R. 1. Arada, E. Casas, and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, vol.23, issue.2, pp.201-229, 2002.
DOI : 10.1023/A:1020576801966

V. Arnautu and P. Neittaanmäki, Discretization estimates for an elliptic control problem, Numer. Funct. Anal. and Optimiz, vol.19, pp.431-464, 1998.

M. Bergounioux, K. Ito, and K. Kunisch, Primal-Dual Strategy for Constrained Optimal Control Problems, SIAM Journal on Control and Optimization, vol.37, issue.4, pp.1176-1194, 1999.
DOI : 10.1137/S0363012997328609

URL : https://hal.archives-ouvertes.fr/hal-00023011

M. Bergounioux and K. Kunisch, Primal-dual strategy for state-constrained optimal control problems, Computational Optimization and Applications, vol.22, issue.2, pp.193-224, 2002.
DOI : 10.1023/A:1015489608037

URL : https://hal.archives-ouvertes.fr/hal-00022023

J. Bonnans and H. Zidani, Optimal Control Problems with Partially Polyhedric Constraints, SIAM Journal on Control and Optimization, vol.37, issue.6, pp.1726-1741, 1999.
DOI : 10.1137/S0363012998333724

URL : https://hal.archives-ouvertes.fr/inria-00073340

E. Casas and M. Mateos, Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints, SIAM Journal on Control and Optimization, vol.40, issue.5, pp.1431-1454, 2002.
DOI : 10.1137/S0363012900382011

E. Casas and M. Mateos, Uniform convergence of the FEM, applications to state constrained control problems, Comp. Appl. Math, vol.21, pp.67-100, 2002.

E. Casas and F. Tröltzsch, Error estimates for linear-quadratic elliptic controls problems, " in Analysis and Optimization of Differential Systems, pp.89-100, 2003.

A. L. Dontchev and W. W. Hager, The Euler approximation in state constrained optimal control, Mathematics of Computation, vol.70, issue.233, pp.173-203, 2000.
DOI : 10.1090/S0025-5718-00-01184-4

A. L. Dontchev, W. W. Hager, and V. M. Velì-ov, Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control, SIAM Journal on Numerical Analysis, vol.38, issue.1, pp.202-226, 2000.
DOI : 10.1137/S0036142999351765

F. Falk, Approximation of a class of optimal control problems with order of convergence estimates, Journal of Mathematical Analysis and Applications, vol.44, issue.1, pp.28-47, 1973.
DOI : 10.1016/0022-247X(73)90022-X

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Analyse num??rique, vol.13, issue.4, pp.313-328, 1979.
DOI : 10.1051/m2an/1979130403131

P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985.
DOI : 10.1137/1.9781611972030

W. W. Hager, Multiplier Methods for Nonlinear Optimal Control, SIAM Journal on Numerical Analysis, vol.27, issue.4, pp.1061-1080, 1990.
DOI : 10.1137/0727063

W. W. Hager, Numerical Analysis in Optimal Control, Optimal Control of Complex Structures. International Series of Numerical Mathematics, pp.83-93, 2001.
DOI : 10.1007/978-3-0348-8148-7_7

M. Heinkenschloss and F. Tröltzsch, Analysis of the Lagrange-SQP-Newton method for the control of a phase field equation, Control and Cybernetics, vol.28, pp.178-211, 1999.

M. Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, Computational Optimization and Applications, vol.30, issue.1
DOI : 10.1007/s10589-005-4559-5

URL : https://hal.archives-ouvertes.fr/hal-01395598

D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bulletin of the American Mathematical Society, vol.4, issue.2, pp.203-207, 1981.
DOI : 10.1090/S0273-0979-1981-14884-9

D. Jerison and C. Kenig, The Inhomogeneous Dirichlet Problem in Lipschitz Domains, Journal of Functional Analysis, vol.130, issue.1, pp.161-219, 1995.
DOI : 10.1006/jfan.1995.1067

C. Kelley and E. Sachs, Approximate quasi-Newton methods, Mathematical Programming, pp.41-70, 1990.
DOI : 10.1007/BF01582251

C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS, vol.83, 1994.
DOI : 10.1090/cbms/083

K. Kunisch and E. Sachs, Reduced SQP Methods for Parameter Identification Problems, SIAM Journal on Numerical Analysis, vol.29, issue.6, pp.1793-1820, 1992.
DOI : 10.1137/0729100

K. Kunisch and A. Rösch, Primal-Dual Active Set Strategy for a General Class of Constrained Optimal Control Problems, SIAM Journal on Optimization, vol.13, issue.2, pp.321-334, 2002.
DOI : 10.1137/S1052623499358008

K. Malanowski, Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems, Applied Mathematics & Optimization, vol.117, issue.1, pp.69-95, 1981.
DOI : 10.1007/BF01447752

K. Malanowski, C. Büskens, and H. Maurer, Convergence of approximations to nonlinear control problems, Mathematical Programming with Data Perturbation, pp.253-284, 1997.

C. Meyer and A. Rösch, Superconvergence Properties of Optimal Control Problems, SIAM Journal on Control and Optimization, vol.43, issue.3
DOI : 10.1137/S0363012903431608

F. Tröltzsch, An SQP method for the optimal control of a nonlinear heat equation, Control and Cybernetics, vol.23, pp.267-288, 1994.

A. Unger, Hinreichende Optimalitätsbedingungen 2. Ordnung und Konvergenz des SQP-Verfahrens für semilineare elliptische Randsteuerprobleme, 1997.

J. Bonnans and H. Zidani, Optimal Control Problems with Partially Polyhedric Constraints, SIAM Journal on Control and Optimization, vol.37, issue.6, pp.1726-1741, 1999.
DOI : 10.1137/S0363012998333724

URL : https://hal.archives-ouvertes.fr/inria-00073340

E. Casas, Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints, ESAIM: Control, Optimisation and Calculus of Variations, vol.8, pp.345-374, 2002.
DOI : 10.1051/cocv:2002049

E. Casas and M. Mateos, Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints, SIAM Journal on Control and Optimization, vol.40, issue.5, pp.1431-1454, 2002.
DOI : 10.1137/S0363012900382011

E. Casas and M. Mateos, Uniform convergence of the FEM. Applications to state constrained control problems, Comput. Appl. Math, vol.21, pp.67-100, 2002.

E. Casas, M. Mateos, and F. Tröltzsch, Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems, Computational Optimization and Applications, vol.23, issue.2
DOI : 10.1007/s10589-005-2180-2

E. Casas and J. Raymond, Error Estimates for the Numerical Approximation of Dirichlet Boundary Control for Semilinear Elliptic Equations, SIAM Journal on Control and Optimization, vol.45, issue.5
DOI : 10.1137/050626600

E. Casas and F. Tröltzsch, Second-Order Necessary and Sufficient Optimality Conditions for Optimization Problems and Applications to Control Theory, SIAM Journal on Optimization, vol.13, issue.2, pp.406-431, 2002.
DOI : 10.1137/S1052623400367698

P. Ciarlet, The Finite Element Method for Elliptic Problems (North-Holland, 1978.

A. Dontchev and W. Hager, The Euler approximation in state constrained optimal control, Mathematics of Computation, vol.70, issue.233, pp.173-203, 2000.
DOI : 10.1090/S0025-5718-00-01184-4

A. Dontchev and W. Hager, Second-Order Runge--Kutta Approximations in Control Constrained Optimal Control, SIAM Journal on Numerical Analysis, vol.38, issue.1, pp.202-226, 2000.
DOI : 10.1137/S0036142999351765

R. Falk, Approximation of a class of optimal control problems with order of convergence estimates, Journal of Mathematical Analysis and Applications, vol.44, issue.1, pp.44-72, 1973.
DOI : 10.1016/0022-247X(73)90022-X

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO Numer, Anal, vol.13, pp.313-328, 1979.

P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985.
DOI : 10.1137/1.9781611972030

W. Hager, Multiplier Methods for Nonlinear Optimal Control, SIAM Journal on Numerical Analysis, vol.27, issue.4, pp.1061-1080, 1990.
DOI : 10.1137/0727063

W. Hager, Numerical Analysis in Optimal Control, International Series of Numerical Mathematics, vol.139, pp.83-93, 2001.
DOI : 10.1007/978-3-0348-8148-7_7

G. Knowles, Finite Element Approximation of Parabolic Time Optimal Control Problems, SIAM Journal on Control and Optimization, vol.20, issue.3, pp.414-427, 1982.
DOI : 10.1137/0320032

I. Lasiecka, Boundary control of parabolic systems: Finite-element approximation, Applied Mathematics & Optimization, vol.28, issue.1, pp.287-333, 1980.
DOI : 10.1007/BF01442882

I. Lasiecka, Ritz???Galerkin Approximation of the Time Optimal Boundary Control Problem for Parabolic Systems with Dirichlet Boundary Conditions, SIAM Journal on Control and Optimization, vol.22, issue.3, pp.477-500, 1984.
DOI : 10.1137/0322029

K. Malanowski, C. Büskens, and H. Maurer, Convergence of approximations to nonlinear control problems, in: Mathematical Programming with Data Perturbation, pp.253-284, 1997.

R. Mcknight and W. Bosarge, The Ritz???Galerkin Procedure for Parabolic Control Problems, SIAM Journal on Control, vol.11, issue.3, pp.510-524, 1973.
DOI : 10.1137/0311040

P. Raviart and J. Thomas, Introduction à l'analyse numérique des equations aux dérivées partielles, 1983.

J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints, Discrete Contin, Dynam. Systems, vol.6, pp.431-450, 2000.

D. Tiba and F. Tröltzsch, Error estimates for the discretization of state constrained convex control problems, Numerical Functional Analysis and Optimization, vol.45, issue.9-10, pp.1005-1028, 1996.
DOI : 10.1007/BF01189480

F. Tröltzsch, Semidiscrete finite element approximation of parabolic boundary control problemsconvergence of switching points, in: Optimal Control of Partial Differential Equations II, International Series of Numerical Mathematics, vol.78, pp.219-232, 1987.

F. Tröltzsch, Approximation of nonlinear parabolic boundary problems by the Fourier methodconvergence of optimal controls, pp.83-98, 1991.

F. Tröltzsch, On Convergence of Semidiscrete Ritz-Galerkin Schemes Applied to the Boundary Control of Parabolic Equations with Non-Linear Boundary Condition, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift f??r Angewandte Mathematik und Mechanik, vol.17, issue.7, pp.72-291, 1992.
DOI : 10.1002/zamm.19920720712

F. Tröltzsch, Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems?Strong convergence of optimal controls, Applied Mathematics & Optimization, vol.72, issue.3, pp.309-329, 1994.
DOI : 10.1007/BF01189480

N. Arada, E. Casas, and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, vol.23, issue.2, pp.201-229, 2002.
DOI : 10.1023/A:1020576801966

S. C. Brenner and L. R. Scott, The mathematical theory of finite element methods, of Texts in Applied Mathematics, 1994.

E. Casas, Using piecewise linear functions in the numerical approximation of semilinear elliptic control problems, Advances in Computational Mathematics, vol.29, issue.1-3, 2005.
DOI : 10.1007/s10444-004-4142-0

E. Casas and M. Mateos, Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints, SIAM Journal on Control and Optimization, vol.40, issue.5, pp.1431-1454, 2002.
DOI : 10.1137/S0363012900382011

E. Casas, M. Mateos, and F. Tröltzsch, Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems, Computational Optimization and Applications, vol.23, issue.2, pp.31-193, 2005.
DOI : 10.1007/s10589-005-2180-2

E. Casas and J. Raymond, The stability in W s,p (?) spaces of the L 2 -projections on some convex sets of finite element function spaces

P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985.
DOI : 10.1137/1.9781611972030

M. Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, Computational Optimization and Applications, vol.30, issue.1, pp.45-61, 2005.
DOI : 10.1007/s10589-005-4559-5

URL : https://hal.archives-ouvertes.fr/hal-01395598

D. Jerison and C. Kenig, The Neumann problem on Lipschitz domains, Bulletin of the American Mathematical Society, vol.4, issue.2, pp.203-207, 1981.
DOI : 10.1090/S0273-0979-1981-14884-9

C. Meyer and A. Rösch, Superconvergence Properties of Optimal Control Problems, SIAM Journal on Control and Optimization, vol.43, issue.3, pp.970-985, 2005.
DOI : 10.1137/S0363012903431608

N. Arada, E. Casas, and F. Tröltzsch, Error estimates for the numerical approximation of a semilinear elliptic control problem, Computational Optimization and Applications, vol.23, issue.2, pp.201-229, 2002.
DOI : 10.1023/A:1020576801966

N. Arada and J. Raymond, Dirichlet Boundary Control of Semilinear Parabolic Equations Part 1: Problems with No State Constraints, Applied Mathematics and Optimization, vol.45, issue.2, pp.125-143, 2002.
DOI : 10.1007/s00245-001-0035-5

M. Berggren, Approximations of Very Weak Solutions to Boundary-Value Problems, SIAM Journal on Numerical Analysis, vol.42, issue.2, pp.860-877, 2004.
DOI : 10.1137/S0036142903382048

J. Bonnans and H. Zidani, Optimal Control Problems with Partially Polyhedric Constraints, SIAM Journal on Control and Optimization, vol.37, issue.6, pp.1726-1741, 1999.
DOI : 10.1137/S0363012998333724

URL : https://hal.archives-ouvertes.fr/inria-00073340

J. Bramble, J. Pasciak, and A. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math, Comp, vol.47, pp.103-134, 1986.

S. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, 1994.

E. Casas, Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints, ESAIM: Control, Optimisation and Calculus of Variations, vol.8, pp.345-374, 2002.
DOI : 10.1051/cocv:2002049

E. Casas and M. Mateos, Second Order Optimality Conditions for Semilinear Elliptic Control Problems with Finitely Many State Constraints, Uniform convergence of the FEM. Applications to state constrained control problems, pp.1431-1454, 2002.
DOI : 10.1137/S0363012900382011

E. Casas, M. Mateos, and F. Tröltzsch, Error Estimates for the Numerical Approximation of Boundary Semilinear Elliptic Control Problems, Computational Optimization and Applications, vol.23, issue.2, pp.31-193, 2005.
DOI : 10.1007/s10589-005-2180-2

E. Casas and J. Raymond, The stability in W s,p (?) spaces of the L 2 -projections on some convex sets of finite element function spaces

P. Ciarlet and P. Raviart, Maximum principle and uniform convergence for the finite element method, Computer Methods in Applied Mechanics and Engineering, vol.2, issue.1, pp.17-31, 1973.
DOI : 10.1016/0045-7825(73)90019-4

M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integral Equations and Operator Theory, vol.11, issue.4, pp.227-261, 1992.
DOI : 10.1007/BF01204238

A. Dontchev, W. Hager, and J. , The Euler approximation in state constrained optimal control, Second-order Runge-Kutta approximations in constrained optimal control, pp.173-203, 2000.
DOI : 10.1090/S0025-5718-00-01184-4

J. J. Douglas, T. Dupont, and L. Wahlbin, The stability inL q of theL 2-projection into finite element function spaces, Numerische Mathematik, vol.46, issue.3, pp.193-197, 1975.
DOI : 10.1007/BF01400302

R. Falk, Approximation of a class of optimal control problems with order of convergence estimates, Journal of Mathematical Analysis and Applications, vol.44, issue.1, pp.44-72, 1973.
DOI : 10.1016/0022-247X(73)90022-X

D. French and J. King, Approximation of an elliptic control problem by the finite element method, Numerical Functional Analysis and Optimization, vol.104, issue.3-4, pp.299-314, 1991.
DOI : 10.1137/0717002

T. Geveci, On the approximation of the solution of an optimal control problem governed by an elliptic equation, RAIRO. Analyse num??rique, vol.13, issue.4, pp.313-328, 1979.
DOI : 10.1051/m2an/1979130403131

P. Grisvard, Elliptic Problems in Nonsmooth Domains, 1985.
DOI : 10.1137/1.9781611972030

M. D. Gunzburger, L. S. Hou, and T. P. Svobodny, Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls, ESAIM: Mathematical Modelling and Numerical Analysis, vol.25, issue.6, pp.711-748, 1991.
DOI : 10.1051/m2an/1991250607111

W. Hager, Multiplier Methods for Nonlinear Optimal Control, Optimal Control of Complex Structures, International Series of Numerical Mathematics, pp.1061-1080, 1990.
DOI : 10.1137/0727063

M. Hinze, A Variational Discretization Concept in Control Constrained Optimization: The Linear-Quadratic Case, Computational Optimization and Applications, vol.30, issue.1, pp.45-61, 2005.
DOI : 10.1007/s10589-005-4559-5

URL : https://hal.archives-ouvertes.fr/hal-01395598

G. Knowles, Finite Element Approximation of Parabolic Time Optimal Control Problems, SIAM Journal on Control and Optimization, vol.20, issue.3, pp.414-427, 1982.
DOI : 10.1137/0320032

I. Lasiecka, Boundary control of parabolic systems: finite-element approximations Ritz-Galerkin approximation of the time optimal boundary control problem for parabolic systems with Dirichlet boundary conditions, Appl. Math. Optim. SIAM J. Control Optim, vol.629, pp.287-333, 1980.

K. Malanowski, C. Büskens, and H. Maurer, Convergence of approximations to nonlinear control problems, Mathematical Programming with Data Perturbation, A. V. Fiacco, pp.253-284, 1997.

R. Mcknight and W. Bosarge, The Ritz???Galerkin Procedure for Parabolic Control Problems, SIAM Journal on Control, vol.11, issue.3, pp.510-524, 1973.
DOI : 10.1137/0311040

C. Meyer and A. Rösch, Superconvergence Properties of Optimal Control Problems, SIAM Journal on Control and Optimization, vol.43, issue.3, pp.970-985, 2004.
DOI : 10.1137/S0363012903431608

G. Stampacchia, Le probl??me de Dirichlet pour les ??quations elliptiques du second ordre ?? coefficients discontinus, Annales de l???institut Fourier, vol.15, issue.1, pp.189-258, 1965.
DOI : 10.5802/aif.204

L. R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of Computation, vol.54, issue.190, pp.483-493, 1990.
DOI : 10.1090/S0025-5718-1990-1011446-7

D. Tiba and F. Tröltzsch, Error estimates for the discretization of state constrained convex control problems, Numerical Functional Analysis and Optimization, vol.45, issue.9-10, pp.1005-1028, 1996.
DOI : 10.1007/BF01189480

F. Tröltzsch, Approximation of nonlinear parabolic boundary problems by the Fourier methodconvergence of optimal controls, Optimization On a convergence of semidiscrete Ritz-Galerkin schemes applied to the boundary control of parabolic equations with non-linear boundary condition Semidiscrete Ritz-Galerkin approximation of nonlinear parabolic boundary control problems-strong convergence of optimal controls, Optimal Control of Partial Differential Equations II International Series of Numerical Mathematics Z. Angew. Math. Mech. Appl. Math. Optim, vol.783940, issue.72, pp.219-232, 1987.