M. S. Aronna, J. F. Bonnans, A. V. Dmitruk, and P. A. Lotito, Quadratic order conditions for bang-singular extremals, Numer. Algebra, Control Optim, AIMS Journal, special issue dedicated to Professor Helmut Maurer on the occasion of his 65th birthday, pp.511-546, 2012.

M. S. Aronna, J. F. Bonnans, and B. S. Goh, Second order analysis of control-affine problems with scalar state constraint, Mathematical Programming, vol.16, issue.4, 2016.
DOI : 10.1007/s10107-015-0976-0

M. S. Aronna, J. F. Bonnans, and A. Kröner, Optimal control of infinite dimensional bilinear systems: application to the heat and wave equations, Mathematical Programming, vol.37, issue.2, 2016.
DOI : 10.1007/s10107-016-1093-4
URL : https://hal.archives-ouvertes.fr/hal-01273496

J. Ball, J. Marsden, and M. Slemrod, Controllability for Distributed Bilinear Systems, SIAM Journal on Control and Optimization, vol.20, issue.4, pp.575-597, 1982.
DOI : 10.1137/0320042

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proceedings of the American Mathematical Society, vol.63, issue.2, pp.370-373, 1977.
DOI : 10.1090/S0002-9939-1977-0442748-6

L. Baudouin, O. Kavian, and J. Puel, Regularity for a Schr??dinger equation with singular potentials and application to bilinear optimal control, Journal of Differential Equations, vol.216, issue.1, pp.188-222, 2005.
DOI : 10.1016/j.jde.2005.04.006

L. Baudouin and J. Salomon, Constructive solution of a bilinear optimal control problem for a Schr??dinger equation, Systems & Control Letters, vol.57, issue.6, pp.453-464, 2008.
DOI : 10.1016/j.sysconle.2007.11.002

T. Bayen, J. F. Bonnans, and F. J. Silva, Characterization of local quadratic growth for strong minima in the optimal control of semi-linear elliptic equations, Transactions of the American Mathematical Society, vol.366, issue.4, pp.2063-2087, 2014.
DOI : 10.1090/S0002-9947-2013-05961-2
URL : https://hal.archives-ouvertes.fr/inria-00632308

T. Bayen and F. J. Silva, Second Order Analysis for Strong Solutions in the Optimal Control of Parabolic Equations, SIAM Journal on Control and Optimization, vol.54, issue.2, pp.819-844, 2016.
DOI : 10.1137/141000415
URL : https://hal.archives-ouvertes.fr/hal-01313949

K. Beauchard, ContributionàContribution`Contributionà létude de la contrôlabilité et de la stabilisation de l'´ equation de Schrödinger, 2005.

K. Beauchard, J. M. Coron, and H. Teismann, Minimal time for the bilinear control of Schr??dinger equations, Systems & Control Letters, vol.71, pp.1-6, 2014.
DOI : 10.1016/j.sysconle.2014.06.009

K. Beauchard and M. Morancey, Local controllability of 1D Schr??dinger equations with bilinear control and minimal time, Mathematical Control and Related Fields, vol.4, issue.2, pp.125-160, 2014.
DOI : 10.3934/mcrf.2014.4.125

M. Bergounioux and D. Tiba, General Optimality Conditions for Constrained Convex Control Problems, SIAM Journal on Control and Optimization, vol.34, issue.2, pp.698-711, 1996.
DOI : 10.1137/S0363012994261987
URL : https://hal.archives-ouvertes.fr/hal-00023059

J. F. Bonnans and P. Jaisson, Optimal Control of a Parabolic Equation with Time-Dependent State Constraints, SIAM Journal on Control and Optimization, vol.48, issue.7, pp.4550-4571, 2010.
DOI : 10.1137/080744608
URL : https://hal.archives-ouvertes.fr/inria-00348854

J. F. Bonnans and N. P. , Osmolovski? ?, Second-order analysis of optimal control problems with control and initial-final state constraints, J. Convex Anal, vol.17, pp.3-4, 2010.

J. F. Bonnans, Optimal control of a semilinear parabolic equation with singular arcs, Optimization Methods and Software, vol.37, issue.2, pp.964-978, 2014.
DOI : 10.1007/s10107-004-0559-y
URL : https://hal.archives-ouvertes.fr/hal-00740698

J. F. Bonnans, P. Martinon, and V. Grélard, Bocop v1.0.3: A collection of examples, 2012.

J. F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems, Series in Operations Research, 2000.
DOI : 10.1007/978-1-4612-1394-9

J. F. Bonnans and D. Tiba, Control problems with mixed constraints and application to an optimal investment problem, Math. Rep. (Bucur.), vol.11, issue.61 4, pp.293-306, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00533579

U. Boscain, M. Caponigro, and M. Sigalotti, A Weak Spectral Condition for the Controllability of the Bilinear Schr??dinger Equation with Application to the Control of a Rotating Planar Molecule, Communications in Mathematical Physics, vol.75, issue.4, pp.423-455, 2012.
DOI : 10.1007/s00220-012-1441-z

E. Casas, Second Order Analysis for Bang-Bang Control Problems of PDEs, SIAM Journal on Control and Optimization, vol.50, issue.4, pp.2355-2372, 2012.
DOI : 10.1137/120862892

E. Casas, C. Clason, and K. Kunisch, Parabolic Control Problems in Measure Spaces with Sparse Solutions, SIAM Journal on Control and Optimization, vol.51, issue.1, pp.28-63, 2013.
DOI : 10.1137/120872395

E. Casas and F. Tröltzsch, Second-Order Optimality Conditions for Weak and Strong Local Solutions of Parabolic Optimal Control Problems, Vietnam Journal of Mathematics, vol.17, issue.1, pp.181-202, 2016.
DOI : 10.1007/s10013-015-0175-6

A. V. Dmitruk, Quadratic conditions for a weak minimum for singular regimes in optimal control problems, Soviet Math. Doklady, vol.18, issue.2, pp.418-422, 1977.

A. V. Dmitruk, Quadratic conditions for the Pontryagin minimum in an optimal control problem linear with respect to control. II. Theorems on the relaxing of constraints on the equality, Izv. Akad. Nauk SSSR Ser. Mat, vol.51, issue.4, pp.812-832, 1987.

H. O. Fattorini, Infinite dimensional linear control systems, North-Holland Mathematics Studies, vol.201, 2005.

H. O. Fattorini and H. Frankowska, Necessary conditions for infinite-dimensional control problems, Mathematics of Control, Signals, and Systems, vol.25, issue.1, pp.41-67, 1991.
DOI : 10.1007/BF02551380

H. Frankowska and D. Tonon, The Goh necessary optimality conditions for the Mayer problem with control constraints, 52nd IEEE Conference on Decision and Control, pp.538-543, 2013.
DOI : 10.1109/CDC.2013.6759937
URL : https://hal.archives-ouvertes.fr/hal-00917508

G. Friesecke, F. Henneke, and K. Kunisch, Sparse controls in quantum systems, Published online as arXiv, 2015.

B. S. Goh, The Second Variation for the Singular Bolza Problem, SIAM Journal on Control, vol.4, issue.2, pp.309-325, 1966.
DOI : 10.1137/0304026

H. Goldberg and F. Tröltzsch, Second-Order Sufficient Optimality Conditions for a Class of Nonlinear Parabolic Boundary Control Problems, SIAM Journal on Control and Optimization, vol.31, issue.4, pp.1007-1025, 1993.
DOI : 10.1137/0331045

M. Hintermüller, P. A. Markowich, D. Marahrens, and C. Sparber, Optimal Bilinear Control of Gross--Pitaevskii Equations, SIAM Journal on Control and Optimization, vol.51, issue.3, pp.2509-2543, 2013.
DOI : 10.1137/120866233

K. Ito and K. Kunisch, Optimal Bilinear Control of an Abstract Schr??dinger Equation, SIAM Journal on Control and Optimization, vol.46, issue.1, pp.274-287, 2007.
DOI : 10.1137/05064254X

H. J. Kelley, A second variation test for singular extremals, AIAA Journal, vol.2, issue.8, pp.1380-1382, 1964.
DOI : 10.2514/3.2562

X. Li and Y. Yao, Maximum Principle of Distributed Parameter Systems aith Time Lages, Die Grundlehren der mathematischen Wissenschaften, Band Lecture Notes in Control and Information Science, vol.181, 1985.

X. Li and J. Yong, Necessary Conditions for Optimal Control of Distributed Parameter Systems, SIAM Journal on Control and Optimization, vol.29, issue.4, pp.895-908, 1991.
DOI : 10.1137/0329049

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

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol.44, 1983.
DOI : 10.1007/978-1-4612-5561-1

L. Poggiolini and G. Stefani, Sufficient optimality conditions for a bang-singular extremal in the minimum time problem, Control Cybernet, vol.37, issue.2, pp.469-490, 2008.

F. Tröltzsch, Regular Lagrange Multipliers for Control Problems with Mixed Pointwise Control-State Constraints, SIAM Journal on Optimization, vol.15, issue.2, pp.616-634, 2004.
DOI : 10.1137/S1052623403426519

A. Waechter and L. T. Biegler, On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming, Mathematical Programming, vol.10, issue.1, pp.25-57, 2006.
DOI : 10.1007/s10107-004-0559-y