2014, 4(2): 161-186. doi: 10.3934/mcrf.2014.4.161

Internal control of the Schrödinger equation

1. 

CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  November 2012 Revised  September 2013 Published  February 2014

In this paper, we intend to present some already known results about the internal controllability of the linear and nonlinear Schrödinger equations.
    After presenting the basic properties of the equation, we give a self contained proof of the controllability in dimension $1$ using some propagation results. We then discuss how to obtain some similar results on a compact manifold where the zone of control satisfies the Geometric Control Condition. We also discuss some known results and open questions when this condition is not satisfied. Then, we present the links between the controllability and some resolvent estimates. Finally, we discuss the additional difficulties when we consider the nonlinear Schrödinger equation.
Citation: Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161
References:
[1]

S. Alinhac and P. Gérard, Pseudo-Differential Operators and the Nash-Moser Theorem, volume 82 of Graduate Studies in Mathematics,, American Mathematical Society, (2007).

[2]

N. Anantharaman and F. Macía, Semiclassical measures for the Schrödinger equation on the torus,, To appear in the Journal of the European Mathematical Society.., ().

[3]

N. Anantharaman and F. Macià, The dynamics of the Schrödinger flow from the point of view of semiclassical measures,, In Spectral geometry, (2012), 93. doi: 10.1090/pspum/084/1351.

[4]

N. Anantharaman and G. Rivière, Dispersion and controllability for the Schrödinger equation on negatively curved manifolds,, Anal. PDE, 5 (2012), 313. doi: 10.2140/apde.2012.5.313.

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C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01896020.

[7]

N. Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes,, Mém. Soc. Math. France (N.S.)., ().

[8]

N. Burq and P. Gérard, Condition nécéssaire et suffisante pour la contrôlabilite exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749. doi: 10.1016/S0764-4442(97)80053-5.

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N. Burq and M. Zworski, Control theory and high energy eigenfunctions,, In Journées, (2004).

[10]

N. Burq and M. Zworski, Geometric control in the presence of a black box,, J. of American Math. Soc, 17 (2004), 443. doi: 10.1090/S0894-0347-04-00452-7.

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T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, 2003., ().

[12]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits (and erratum),, J. Funct. Anal., 246 (2007), 145. doi: 10.1016/j.jfa.2006.09.012.

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Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique. I. Concentration microlocale,, Comm. Partial Differential Equations, 19 (1994), 1535. doi: 10.1080/03605309408821063.

[14]

J.-M. Coron, Control and Nonlinearity,, Amer Mathematical Society, (2007).

[15]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface,, Math. Z., 254 (2006), 729. doi: 10.1007/s00209-006-0005-3.

[16]

B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time,, SIAM J. Control Optim., 48 (2009), 521. doi: 10.1137/070712067.

[17]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1.

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T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations,, J. Funct. Anal., 263 (2012), 3641. doi: 10.1016/j.jfa.2012.09.003.

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S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, J. Funct. Anal., 254 (2008), 3037. doi: 10.1016/j.jfa.2008.03.005.

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S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1375. doi: 10.3934/dcdsb.2010.14.1375.

[21]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7.

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S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (2004). doi: 10.1007/978-3-642-18855-8.

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P. Gérard, Microlocal Defect Measures,, Comm. Partial Diff. eq., 16 (1991), 1761. doi: 10.1080/03605309108820822.

[24]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators : Pseudo-differential Operators, volume 3., Springer Verlag, (1985).

[26]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088.

[27]

K. Ito, K. Ramdani and M. Tucsnak, A time reversal based algorithm for solving initial data inverse problems,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 641. doi: 10.3934/dcdss.2011.4.641.

[28]

S. Jaffard, Contrôle interne exacte des vibrations d'une plaque rectangulaire,, Portugal. Math., 47 (1990), 423.

[29]

R. Joly and C. Laurent, Stabilisation for the semilinear wave equation with geometric control condition,, Anal. PDE, 6 (2013), 1089. doi: 10.2140/apde.2013.6.1089.

[30]

V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer, (2005).

[31]

J. Lagnese, Control of wave processes with distributed controls supported on a subregion,, SIAM J. Control Optim., 21 (1983), 68. doi: 10.1137/0321004.

[32]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of schrödinger equations with dirichlet control,, Differential Integral Equations, 5 (1992), 521.

[33]

I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 43. doi: 10.1163/156939404773972761.

[34]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval,, ESAIM Control Optim. Calc. Var., 16 (2010), 356. doi: 10.1051/cocv/2009001.

[35]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3,, SIAM J. Math. Anal., 42 (2010), 785. doi: 10.1137/090749086.

[36]

G. Lebeau, Contrôle de l'équation de Schrödinger,, J. Math. Pures Appl., 71 (1992), 267.

[37]

G. Lebeau, Control for hyperbolic equations,, In Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis, (1992), 160. doi: 10.1007/BFb0115024.

[38]

J.-L. Lions, Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribuées, Tom 2,, Masson, (1988).

[39]

E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24. doi: 10.1137/S0363012991223145.

[40]

F. Maciá, High-frequency propagation for the Schrödinger equation on the torus,, J. Funct. Anal., 258 (2010), 933. doi: 10.1016/j.jfa.2009.09.020.

[41]

A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015017.

[42]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554. doi: 10.1137/S036301290139107X.

[43]

L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429. doi: 10.1007/s00205-004-0312-y.

[44]

L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation,, J. Funct. Anal., 218 (2005), 425. doi: 10.1016/j.jfa.2004.02.001.

[45]

L. Miller, Resolvent conditions for the control of unitary groups and their approximations,, J. Spectr. Theory, 2 (2012), 1. doi: 10.4171/JST/20.

[46]

S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering,, Acta Math., 203 (2009), 149. doi: 10.1007/s11511-009-0041-z.

[47]

K.-D. Phung, Observability and control of Schrödinger equations,, SIAM J. Control Optim., 40 (2001), 211. doi: 10.1137/S0363012900368405.

[48]

J. Ralston, Solutions of the wave equation with localized energy,, Comm. Pure Appl. Math., 22 (1969), 807. doi: 10.1002/cpa.3160220605.

[49]

J. Ralston, Approximate eigenfunctions of the Laplacian,, J. Differential Geometry, 12 (1977), 87.

[50]

K. Ramdani, T. Takahashi, G. Tenenbaum and M. Tucsnak, A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator,, J. Funct. Anal., 226 (2005), 193. doi: 10.1016/j.jfa.2005.02.009.

[51]

J. Rauch and M. Taylor, Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains,, Indiana Univ. Math. J., 24 (1974), 79. doi: 10.1512/iumj.1975.24.24004.

[52]

L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients,, Invent. Math., 131 (1998), 493. doi: 10.1007/s002220050212.

[53]

L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129. doi: 10.1016/j.jde.2008.11.004.

[54]

L. Rosier and B.-Y. Zhang, Exact controllability and stabilizability of the nonlinear schrödinger equation on a bounded interval,, SIAM J. Control Optim., 48 (2009), 972. doi: 10.1137/070709578.

[55]

L. Rosier and B.-Y. Zhang, Control and Stabilization of the Nonlinear Schrödinger Equation on Rectangles,, Math. Models Methods Appl. Sci., 20 (2010), 2293. doi: 10.1142/S0218202510004933.

[56]

T. Tao, Nonlinear Dispersive Equations, Local and global Analysis,, Amer Mathematical Society, (2006).

[57]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 115 (1990), 193. doi: 10.1017/S0308210500020606.

[58]

G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation,, Trans. Amer. Math. Soc., 361 (2009), 951. doi: 10.1090/S0002-9947-08-04584-4.

[59]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, (2009). doi: 10.1007/978-3-7643-8994-9.

[60]

E. Trélat, Y. Privat and E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains,, submitted., ().

[61]

Q. Zhou and M. Yamamoto, Hautus condition on the exact controllability of conservative systems,, Internat. J. Control, 67 (1997), 371.

[62]

E. Zuazua, Contrôlabilité exacte en temps arbitrairement petit de quelques modèles de plaques,, volume Appendix A.1 to [38]., ().

[63]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 33.

[64]

E. Zuazua, Remarks on the controllability of the schrödinger equation,, In Quantum control: Mathematical and numerical challenges, (2003), 193.

[65]

C. Zuily, Uniqueness and Nonuniqueness in the Cauchy Problem, volume 33 of Progress in Mathematics,, Birkhäuser Boston Inc., (1983).

show all references

References:
[1]

S. Alinhac and P. Gérard, Pseudo-Differential Operators and the Nash-Moser Theorem, volume 82 of Graduate Studies in Mathematics,, American Mathematical Society, (2007).

[2]

N. Anantharaman and F. Macía, Semiclassical measures for the Schrödinger equation on the torus,, To appear in the Journal of the European Mathematical Society.., ().

[3]

N. Anantharaman and F. Macià, The dynamics of the Schrödinger flow from the point of view of semiclassical measures,, In Spectral geometry, (2012), 93. doi: 10.1090/pspum/084/1351.

[4]

N. Anantharaman and G. Rivière, Dispersion and controllability for the Schrödinger equation on negatively curved manifolds,, Anal. PDE, 5 (2012), 313. doi: 10.2140/apde.2012.5.313.

[5]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary,, SIAM J. Control Optim., 30 (1992), 1024. doi: 10.1137/0330055.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, Part I,, Geom. Funct. Anal., 3 (1993), 107. doi: 10.1007/BF01896020.

[7]

N. Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes,, Mém. Soc. Math. France (N.S.)., ().

[8]

N. Burq and P. Gérard, Condition nécéssaire et suffisante pour la contrôlabilite exacte des ondes,, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), 749. doi: 10.1016/S0764-4442(97)80053-5.

[9]

N. Burq and M. Zworski, Control theory and high energy eigenfunctions,, In Journées, (2004).

[10]

N. Burq and M. Zworski, Geometric control in the presence of a black box,, J. of American Math. Soc, 17 (2004), 443. doi: 10.1090/S0894-0347-04-00452-7.

[11]

T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, 2003., ().

[12]

H. Christianson, Semiclassical non-concentration near hyperbolic orbits (and erratum),, J. Funct. Anal., 246 (2007), 145. doi: 10.1016/j.jfa.2006.09.012.

[13]

Y. Colin de Verdière and B. Parisse, Équilibre instable en régime semi-classique. I. Concentration microlocale,, Comm. Partial Differential Equations, 19 (1994), 1535. doi: 10.1080/03605309408821063.

[14]

J.-M. Coron, Control and Nonlinearity,, Amer Mathematical Society, (2007).

[15]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface,, Math. Z., 254 (2006), 729. doi: 10.1007/s00209-006-0005-3.

[16]

B. Dehman and G. Lebeau, Analysis of the HUM Control Operator and Exact Controllability for Semilinear Waves in Uniform Time,, SIAM J. Control Optim., 48 (2009), 521. doi: 10.1137/070712067.

[17]

B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation,, Ann. Sci. École Norm. Sup., 36 (2003), 525. doi: 10.1016/S0012-9593(03)00021-1.

[18]

T. Duyckaerts and L. Miller, Resolvent conditions for the control of parabolic equations,, J. Funct. Anal., 263 (2012), 3641. doi: 10.1016/j.jfa.2012.09.003.

[19]

S. Ervedoza, C. Zheng and E. Zuazua, On the observability of time-discrete conservative linear systems,, J. Funct. Anal., 254 (2008), 3037. doi: 10.1016/j.jfa.2008.03.005.

[20]

S. Ervedoza and E. Zuazua, A systematic method for building smooth controls for smooth data,, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1375. doi: 10.3934/dcdsb.2010.14.1375.

[21]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 17 (2000), 583. doi: 10.1016/S0294-1449(00)00117-7.

[22]

S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (2004). doi: 10.1007/978-3-642-18855-8.

[23]

P. Gérard, Microlocal Defect Measures,, Comm. Partial Diff. eq., 16 (1991), 1761. doi: 10.1080/03605309108820822.

[24]

A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457.

[25]

L. Hörmander, The Analysis of Linear Partial Differential Operators : Pseudo-differential Operators, volume 3., Springer Verlag, (1985).

[26]

V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088.

[27]

K. Ito, K. Ramdani and M. Tucsnak, A time reversal based algorithm for solving initial data inverse problems,, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 641. doi: 10.3934/dcdss.2011.4.641.

[28]

S. Jaffard, Contrôle interne exacte des vibrations d'une plaque rectangulaire,, Portugal. Math., 47 (1990), 423.

[29]

R. Joly and C. Laurent, Stabilisation for the semilinear wave equation with geometric control condition,, Anal. PDE, 6 (2013), 1089. doi: 10.2140/apde.2013.6.1089.

[30]

V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer, (2005).

[31]

J. Lagnese, Control of wave processes with distributed controls supported on a subregion,, SIAM J. Control Optim., 21 (1983), 68. doi: 10.1137/0321004.

[32]

I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of schrödinger equations with dirichlet control,, Differential Integral Equations, 5 (1992), 521.

[33]

I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part I: $H^1(\Omega)$-estimates,, J. Inverse Ill-Posed Probl., 12 (2004), 43. doi: 10.1163/156939404773972761.

[34]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on an interval,, ESAIM Control Optim. Calc. Var., 16 (2010), 356. doi: 10.1051/cocv/2009001.

[35]

C. Laurent, Global controllability and stabilization for the nonlinear Schrödinger equation on some compact manifolds of dimension 3,, SIAM J. Math. Anal., 42 (2010), 785. doi: 10.1137/090749086.

[36]

G. Lebeau, Contrôle de l'équation de Schrödinger,, J. Math. Pures Appl., 71 (1992), 267.

[37]

G. Lebeau, Control for hyperbolic equations,, In Analysis and optimization of systems: state and frequency domain approaches for infinite-dimensional systems (Sophia-Antipolis, (1992), 160. doi: 10.1007/BFb0115024.

[38]

J.-L. Lions, Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribuées, Tom 2,, Masson, (1988).

[39]

E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24. doi: 10.1137/S0363012991223145.

[40]

F. Maciá, High-frequency propagation for the Schrödinger equation on the torus,, J. Funct. Anal., 258 (2010), 933. doi: 10.1016/j.jfa.2009.09.020.

[41]

A. Mercado, A. Osses and L. Rosier, Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/1/015017.

[42]

L. Miller, Escape function conditions for the observation, control, and stabilization of the wave equation,, SIAM J. Control Optim., 41 (2002), 1554. doi: 10.1137/S036301290139107X.

[43]

L. Miller, How violent are fast controls for Schrödinger and plate vibrations?, Arch. Ration. Mech. Anal., 172 (2004), 429. doi: 10.1007/s00205-004-0312-y.

[44]

L. Miller, Controllability cost of conservative systems: Resolvent condition and transmutation,, J. Funct. Anal., 218 (2005), 425. doi: 10.1016/j.jfa.2004.02.001.

[45]

L. Miller, Resolvent conditions for the control of unitary groups and their approximations,, J. Spectr. Theory, 2 (2012), 1. doi: 10.4171/JST/20.

[46]

S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering,, Acta Math., 203 (2009), 149. doi: 10.1007/s11511-009-0041-z.

[47]

K.-D. Phung, Observability and control of Schrödinger equations,, SIAM J. Control Optim., 40 (2001), 211. doi: 10.1137/S0363012900368405.

[48]

J. Ralston, Solutions of the wave equation with localized energy,, Comm. Pure Appl. Math., 22 (1969), 807. doi: 10.1002/cpa.3160220605.

[49]

J. Ralston, Approximate eigenfunctions of the Laplacian,, J. Differential Geometry, 12 (1977), 87.

[50]

K. Ramdani, T. Takahashi, G. Tenenbaum and M. Tucsnak, A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator,, J. Funct. Anal., 226 (2005), 193. doi: 10.1016/j.jfa.2005.02.009.

[51]

J. Rauch and M. Taylor, Exponential Decay of Solutions to Hyperbolic Equations in Bounded Domains,, Indiana Univ. Math. J., 24 (1974), 79. doi: 10.1512/iumj.1975.24.24004.

[52]

L. Robbiano and C. Zuily, Uniqueness in the Cauchy problem for operators with partially holomorphic coefficients,, Invent. Math., 131 (1998), 493. doi: 10.1007/s002220050212.

[53]

L. Rosier and B.-Y. Zhang, Exact boundary controllability of the nonlinear Schrödinger equation,, J. Differential Equations, 246 (2009), 4129. doi: 10.1016/j.jde.2008.11.004.

[54]

L. Rosier and B.-Y. Zhang, Exact controllability and stabilizability of the nonlinear schrödinger equation on a bounded interval,, SIAM J. Control Optim., 48 (2009), 972. doi: 10.1137/070709578.

[55]

L. Rosier and B.-Y. Zhang, Control and Stabilization of the Nonlinear Schrödinger Equation on Rectangles,, Math. Models Methods Appl. Sci., 20 (2010), 2293. doi: 10.1142/S0218202510004933.

[56]

T. Tao, Nonlinear Dispersive Equations, Local and global Analysis,, Amer Mathematical Society, (2006).

[57]

L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,, Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, 115 (1990), 193. doi: 10.1017/S0308210500020606.

[58]

G. Tenenbaum and M. Tucsnak, Fast and strongly localized observation for the Schrödinger equation,, Trans. Amer. Math. Soc., 361 (2009), 951. doi: 10.1090/S0002-9947-08-04584-4.

[59]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups,, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, (2009). doi: 10.1007/978-3-7643-8994-9.

[60]

E. Trélat, Y. Privat and E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains,, submitted., ().

[61]

Q. Zhou and M. Yamamoto, Hautus condition on the exact controllability of conservative systems,, Internat. J. Control, 67 (1997), 371.

[62]

E. Zuazua, Contrôlabilité exacte en temps arbitrairement petit de quelques modèles de plaques,, volume Appendix A.1 to [38]., ().

[63]

E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 33.

[64]

E. Zuazua, Remarks on the controllability of the schrödinger equation,, In Quantum control: Mathematical and numerical challenges, (2003), 193.

[65]

C. Zuily, Uniqueness and Nonuniqueness in the Cauchy Problem, volume 33 of Progress in Mathematics,, Birkhäuser Boston Inc., (1983).

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