A. Alla and M. Falcone, An Adaptive POD Approximation Method for the Control of Advection-Diffusion Equations, Internat. Ser. Numer. Math, vol.164, pp.1-17, 2013.
DOI : 10.1007/978-3-0348-0631-2_1
URL : https://hal.archives-ouvertes.fr/hal-00800433

A. Alla, M. Falcone, and D. Kalise, An Efficient Policy Iteration Algorithm for Dynamic Programming Equations, PAMM, vol.13, issue.1, 2013.
DOI : 10.1002/pamm.201310226
URL : https://hal.archives-ouvertes.fr/hal-01068295

N. Altmüller and L. Grüne, Distributed and boundary model predictive control for the heat equation, GAMM-Mitt, pp.131-145, 2012.

M. Bardi and I. Capuzzo-dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, 2008.
DOI : 10.1007/978-0-8176-4755-1

R. E. Bellman, I. Glicksberg, and O. A. Gross, On the ???bang-bang??? control problem, Quarterly of Applied Mathematics, vol.14, issue.1, pp.11-18, 1956.
DOI : 10.1090/qam/78516

P. Benner, S. Görner, and J. Saak, Numerical Solution of Optimal Control Problems for Parabolic Systems, Lecture Notes in Computational Science and Engineering, vol.52, pp.151-169, 2006.
DOI : 10.1007/3-540-33541-2_9

S. Dubljevic, N. H. El-farra, P. Mhaskar, and P. D. Christofides, Predictive control of parabolic PDEs with state and control constraints, International Journal of Robust and Nonlinear Control, vol.50, issue.16, pp.749-772, 2006.
DOI : 10.1002/rnc.1097

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, 1998.

M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi Equations, Applied mathematics, 2014.
DOI : 10.1137/1.9781611973051

M. Falcone, D. Kalise, and A. Kröner, A semi-lagrangian scheme for Lp-penalized minimum time problems, 2014.
URL : https://hal.archives-ouvertes.fr/hal-01089877

H. O. Fattorini, Time and norm optimal controls: A survey of recent results and open problems, Acta Mathematica Scientia, vol.31, issue.6, pp.2203-2218, 2011.
DOI : 10.1016/S0252-9602(11)60394-9

]. R. Ferretti, Internal approximation schemes for optimal control problems in Hilbert spaces, J. Math. Systems Estim. Control, vol.7, issue.1, pp.1-25, 1997.

E. Hernández, D. Kalise, and E. Otárola, Numerical approximation of the LQR problem in??a??strongly damped wave equation, Computational Optimization and Applications, vol.47, issue.2, pp.161-178, 2010.
DOI : 10.1007/s10589-008-9213-6

E. Hernández, D. Kalise, and E. Otárola, A locking-free scheme for the LQR control of a Timoshenko beam, Journal of Computational and Applied Mathematics, vol.235, issue.5, pp.1383-1393, 2011.
DOI : 10.1016/j.cam.2010.08.025

A. Kröner, K. Kunisch, and H. Zidani, Optimal feedback control for undamped wave equations by solving a HJB equation, ESAIM: Control, Optimisation and Calculus of Variations, vol.21, issue.2, 2013.
DOI : 10.1051/cocv/2014033

K. Kunisch, S. Volkwein, and L. Xie, HJB-POD-Based Feedback Design for the Optimal Control of Evolution Problems, SIAM Journal on Applied Dynamical Systems, vol.3, issue.4, pp.701-722, 2004.
DOI : 10.1137/030600485

K. Kunisch and L. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM: Control Optim, Calc. Var, vol.19, issue.2, pp.460-485, 2013.

K. Kunisch and L. Xie, POD-based feedback control of the burgers equation by solving the evolutionary HJB equation, Computers & Mathematics with Applications, vol.49, issue.7-8, pp.1113-1126, 2005.
DOI : 10.1016/j.camwa.2004.07.022

I. Lasiecka, Finite element approximation of time optimal control problems for parabolic equations with Dirichelt boundary conditions, Lecture Notes in Control and Information Sciences, vol.38, pp.354-361, 1982.
DOI : 10.1007/BFb0006155

J. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, 1972.
DOI : 10.1007/978-3-642-65161-8

S. Micu, I. Roventa, and M. Tucsnak, Time optimal boundary controls for the heat equation, Journal of Functional Analysis, vol.263, issue.1, pp.25-49, 2012.
DOI : 10.1016/j.jfa.2012.04.009
URL : https://hal.archives-ouvertes.fr/hal-00639717

J. P. Raymond and H. Zidani, Pontryagin's Principle for Time-Optimal Problems, Journal of Optimization Theory and Applications, vol.196, issue.2, pp.375-402, 1999.
DOI : 10.1023/A:1021793611520

J. C. Reyes and T. Stykel, A balanced truncation-based strategy for optimal control of evolution problems, Math. Contr. Rel. Field, vol.265, issue.4, pp.671-692, 2011.

S. Volkwein, Proper Orthogonal Decomposition: Applications in Optimization and Control, Lecture notes

M. Wheeler, A Priori $L_2 $ Error Estimates for Galerkin Approximations to Parabolic Partial Differential Equations, SIAM Journal on Numerical Analysis, vol.10, issue.4, pp.723-759, 1973.
DOI : 10.1137/0710062

E. Zuazua, Approximate controllability for semilinear heat equations with globally Lipschitz nonlinearities, Control Cybern. Engineering, vol.28, pp.665-683, 1999.