A. Alla and M. Falcone, An adaptive POD approximation method for the control of advection-diusion equations, International Series of Numerical Mathematics, vol.164, p.117, 2013.

A. Alla, M. Falcone, and D. Kalise, HJB-POD based feedback design approach for the wave equation, pp.15-24, 2015.

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

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, 29th IEEE Conference on Decision and Control, p.271283, 1991.
DOI : 10.1109/CDC.1990.204046

L. Baudouin and J. Salomon, Constructive solution of a bilinear optimal control problem for a Schrödinger equation, Systems & Control Letters, vol.57, issue.453, p.464, 2008.

K. Beauchard, J. M. Coron, and H. Teismann, Minimal time for the bilinear control of Schrödinger equations, Systems Control Lett, pp.71-87, 2014.

O. Bokanowski, A. Desilles, H. Zidani, and R. , A C++ library for solving HJ equations, 2013.

O. Bokanowski, N. Forcadel, and H. Zidani, Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption, SIAM Journal on Control and Optimization, vol.48, issue.7, p.42924316, 2010.
DOI : 10.1137/090762075
URL : https://hal.archives-ouvertes.fr/hal-00395589

O. Bokanowski, J. Garcke, M. Griebel, and I. Klompmaker, An adaptive sparse grid semi- Lagrangian scheme for rst order Hamilton-Jacobi Bellman equations, Journal of Scientic Computing, vol.55, issue.3, p.575605, 2013.
DOI : 10.1007/s10915-012-9648-x
URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.393.1708

A. Borzi, J. Salomon, and S. Volkwein, Formulation and numerical solution of nite-level quantum optimal control problems, Journal of Computational and Applied Mathematics, vol.216, issue.170, p.197, 2008.

H. Bungartz and M. Griebel, Sparse grids, Acta Numerica, vol.13, p.1123, 2004.
DOI : 10.1017/s0962492904000182

G. Buse, Exploiting many-core architectures for dimensionally adaptive sparse grids, 2015.

E. Carlini, M. Falcone, and R. Ferretti, An ecient algorithm for Hamilton-Jacobi equations in high dimension, Computing and Visualization in Science, vol.7, issue.1, p.1529, 2004.

A. Chkifa, A. Cohen, and C. Schwab, High-Dimensional Adaptive Sparse Polynomial Interpolation and Applications to Parametric PDEs, Foundations of Computational Mathematics, vol.4, issue.1
DOI : 10.1007/s10208-013-9154-z
URL : https://hal.archives-ouvertes.fr/hal-01350602

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

R. Ferretti, Internal approximation schemes for optimal control problems in Hilbert spaces, Journal of Mathematical Systems, Estimation, and Control, vol.7, issue.1, p.125, 1997.

C. Feuersänger, Sparse Grid Methods for Higher Dimensional Approximation, Dissertation, Institut für Numerische Simulation, 2010.

W. H. Fleming and H. M. Soner, Controlled markov processes and viscosity solutions, Stochastic Modelling and Applied Probability, 2006.

G. Friesecke, F. Henneke, and K. Kunisch, Sparse control of quantum systems

J. Garcke, Sparse grids in a nutshell, Sparse grids and applications, Lecture Notes in Computational Science and Engineering, vol.88, pp.2012-5780

J. Garcke and M. Griebel, On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric elds with the sparse grid combination technique, Journal of Computational Physics, vol.165, issue.2, p.694716, 2000.

M. Gerdts, G. Greif, and H. J. Pesch, Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time, Mathematics and Computers in Simulation, vol.79, issue.4, 2008.
DOI : 10.1016/j.matcom.2008.02.014

S. Gombao, Approximation of optimal controls for semilinear parabolic PDE by solving Hamilton-Jacobi-Bellman equations, Proc. of the 15th International Symposium on the Mathematical Theory of Networks and Systems, 2002.

M. Griebel and J. Hamaekers, Sparse grids for the Schrödinger equation, Mathematical Modelling and Numerical Analysis, vol.41, issue.2, p.215247, 2007.

M. Griebel and S. Knapek, Optimized general sparse grid approximation spaces for operator equations, Mathematics of Computation, vol.78, issue.268, p.22232257, 2009.
DOI : 10.1090/S0025-5718-09-02248-0

M. Gugat and G. Leugering, L ? -norm minimal control of the wave equation: on the weakness of the bang-bang principle, ESAIM: Control, Optimisation and Calculus of Variations, p.254283, 2008.

M. Hinze and S. Volkwein, Proper orthogonal decomposition surrogate models for nonlinear dynamical systems: error estimates and suboptimal control., Dimension Reduction of Large- Scale Systems, Lect. Notes Comput. Sci. Eng, vol.45, p.261306, 2005.

C. Hu and C. Shu, A discontinuous Galerkin nite element method for HamiltonJacobi equations, SIAM Journal on Scientic Computing, vol.21, issue.2, p.666690, 1999.

H. Ishii, Uniqueness of unbounded viscosity solution of Hamilton-Jacobi equations, Indiana University Mathematics Journal, vol.33, issue.5, p.721748, 1984.

K. Ito and K. Kunisch, Optimal bilinear control of an abstract Schrödinger equation, SIAM Journal on Control and Optimization, vol.46, issue.1, p.274287, 2007.

D. Kalise and A. Kröner, Optimal feedback control of advection-diusion-reaction system using balanced truncation, Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems (MTNS), p.11961202, 2013.

D. Kalise, A. Kröner, and K. Kunisch, Local Minimization Algorithms for Dynamic Programming Equations, preprint, pp.15-19, 2014.
DOI : 10.1137/15M1010269
URL : https://hal.archives-ouvertes.fr/hal-01120450

A. Kröner and K. Kunisch, A minimum eort optimal control problem for the wave equation, Comput. Optim. Appl, vol.57, issue.1, 2014.

A. Kröner, K. Kunisch, and B. Vexler, Semi-smooth Newton methods for optimal control of the wave equation with control constraints, SIAM Journal on Control and Optimization, vol.49, issue.2, p.830858, 2011.

A. Kröner, K. Kunisch, and H. Zidani, Optimal feedback control of the undamped wave equation by solving a HJB equation, ESAIM: Control, Optimisation and Calculus of Variations, pp.442-464, 2015.

K. Kunisch, P. Trautmann, and B. Vexler, Optimal control of the undamped linear wave equation with measure value controls, preprint, 2014.

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, p.701722, 2004.
DOI : 10.1137/030600485

K. Kunisch and D. Wachsmuth, On time optimal control of the wave equation, its regularization and optimality system, ESAIM: Control, Optimisation and Calculus of Variations, vol.19, issue.2, p.317336, 2013.
DOI : 10.1051/cocv/2012010

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, p.11131126, 2005.
DOI : 10.1016/j.camwa.2004.07.022

J. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, 1972.

M. Mirrahimi, P. Rouchon, and G. Turinici, Lyapunov control of bilinear Schrödinger equations, Automatica, vol.41, issue.11, 2005.

M. Mirrahimi and R. Van-handel, Stabilizing Feedback Controls for Quantum Systems, SIAM Journal on Control and Optimization, vol.46, issue.2, p.445467, 2007.
DOI : 10.1137/050644793

E. Novak and K. Ritter, Global optimization using hyperbolic cross points, State of the Art in Global Optimization, Nonconvex Optim. Appl, vol.7, 1933.

S. Osher and C. W. Shu, High-Order Essentially Nonoscillatory Schemes for Hamilton???Jacobi Equations, SIAM Journal on Numerical Analysis, vol.28, issue.4, p.907922, 1991.
DOI : 10.1137/0728049

D. Püger, Spatially adaptive sparse grids for high-dimensional problems, Dissertation, 2010.

S. A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Doklady Akademii nauk SSSR Russian, Engl. Transl.: Soviet Math. Dokl, vol.148, issue.4, p.240243, 1963.

X. Warin, Adaptive sparse grids for time dependent Hamilton-Jacobi-Bellman equations in stochastic control, 2014.

C. Zenger, Sparse grids, Parallel Algorithms for Partial Dierential Equations, Notes on Numerical Fluid Mechanics, p.241251, 1991.

F. Germany, S. Scai, and . Augustin, Germany E-mail address: garcke@ins.uni-bonn.de INRIA Saclay and CMAP