# American Institute of Mathematical Sciences

June  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). Google Scholar [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.., (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] N. Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes,, Mém. Soc. Math. France (N.S.)., (). Google Scholar [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. Google Scholar [9] N. Burq and M. Zworski, Control theory and high energy eigenfunctions,, In Journées, (2004). Google Scholar [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. Google Scholar [11] T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, 2003., (). Google Scholar [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. Google Scholar [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. Google Scholar [14] J.-M. Coron, Control and Nonlinearity,, Amer Mathematical Society, (2007). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (2004). doi: 10.1007/978-3-642-18855-8. Google Scholar [23] P. Gérard, Microlocal Defect Measures,, Comm. Partial Diff. eq., 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [24] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457. Google Scholar [25] L. Hörmander, The Analysis of Linear Partial Differential Operators : Pseudo-differential Operators, volume 3., Springer Verlag, (1985). Google Scholar [26] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088. Google Scholar [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. Google Scholar [28] S. Jaffard, Contrôle interne exacte des vibrations d'une plaque rectangulaire,, Portugal. Math., 47 (1990), 423. Google Scholar [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. Google Scholar [30] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer, (2005). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] G. Lebeau, Contrôle de l'équation de Schrödinger,, J. Math. Pures Appl., 71 (1992), 267. Google Scholar [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. Google Scholar [38] J.-L. Lions, Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribuées, Tom 2,, Masson, (1988). Google Scholar [39] E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24. doi: 10.1137/S0363012991223145. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [46] S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering,, Acta Math., 203 (2009), 149. doi: 10.1007/s11511-009-0041-z. Google Scholar [47] K.-D. Phung, Observability and control of Schrödinger equations,, SIAM J. Control Optim., 40 (2001), 211. doi: 10.1137/S0363012900368405. Google Scholar [48] J. Ralston, Solutions of the wave equation with localized energy,, Comm. Pure Appl. Math., 22 (1969), 807. doi: 10.1002/cpa.3160220605. Google Scholar [49] J. Ralston, Approximate eigenfunctions of the Laplacian,, J. Differential Geometry, 12 (1977), 87. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [56] T. Tao, Nonlinear Dispersive Equations, Local and global Analysis,, Amer Mathematical Society, (2006). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [60] E. Trélat, Y. Privat and E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains,, submitted., (). Google Scholar [61] Q. Zhou and M. Yamamoto, Hautus condition on the exact controllability of conservative systems,, Internat. J. Control, 67 (1997), 371. Google Scholar [62] E. Zuazua, Contrôlabilité exacte en temps arbitrairement petit de quelques modèles de plaques,, volume Appendix A.1 to [38]., (). Google Scholar [63] E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 33. Google Scholar [64] E. Zuazua, Remarks on the controllability of the schrödinger equation,, In Quantum control: Mathematical and numerical challenges, (2003), 193. Google Scholar [65] C. Zuily, Uniqueness and Nonuniqueness in the Cauchy Problem, volume 33 of Progress in Mathematics,, Birkhäuser Boston Inc., (1983). Google Scholar

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). Google Scholar [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.., (). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] N. Burq, Contrôle de l'équation des plaques en présence d'obstacles strictement convexes,, Mém. Soc. Math. France (N.S.)., (). Google Scholar [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. Google Scholar [9] N. Burq and M. Zworski, Control theory and high energy eigenfunctions,, In Journées, (2004). Google Scholar [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. Google Scholar [11] T. Cazenave, Semilinear Schrödinger Equations, volume 10 of Courant Lecture Notes in Mathematics,, 2003., (). Google Scholar [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. Google Scholar [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. Google Scholar [14] J.-M. Coron, Control and Nonlinearity,, Amer Mathematical Society, (2007). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [22] S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry,, Springer, (2004). doi: 10.1007/978-3-642-18855-8. Google Scholar [23] P. Gérard, Microlocal Defect Measures,, Comm. Partial Diff. eq., 16 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [24] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire,, J. Math. Pures Appl., 68 (1989), 457. Google Scholar [25] L. Hörmander, The Analysis of Linear Partial Differential Operators : Pseudo-differential Operators, volume 3., Springer Verlag, (1985). Google Scholar [26] V. Isakov, Carleman type estimates in an anisotropic case and applications,, J. Differential Equations, 105 (1993), 217. doi: 10.1006/jdeq.1993.1088. Google Scholar [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. Google Scholar [28] S. Jaffard, Contrôle interne exacte des vibrations d'une plaque rectangulaire,, Portugal. Math., 47 (1990), 423. Google Scholar [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. Google Scholar [30] V. Komornik and P. Loreti, Fourier Series in Control Theory,, Springer, (2005). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] G. Lebeau, Contrôle de l'équation de Schrödinger,, J. Math. Pures Appl., 71 (1992), 267. Google Scholar [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. Google Scholar [38] J.-L. Lions, Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribuées, Tom 2,, Masson, (1988). Google Scholar [39] E. Machtyngier, Exact controllability for the Schrödinger equation,, SIAM J. Control Optim., 32 (1994), 24. doi: 10.1137/S0363012991223145. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [46] S. Nonnenmacher and M. Zworski, Quantum decay rates in chaotic scattering,, Acta Math., 203 (2009), 149. doi: 10.1007/s11511-009-0041-z. Google Scholar [47] K.-D. Phung, Observability and control of Schrödinger equations,, SIAM J. Control Optim., 40 (2001), 211. doi: 10.1137/S0363012900368405. Google Scholar [48] J. Ralston, Solutions of the wave equation with localized energy,, Comm. Pure Appl. Math., 22 (1969), 807. doi: 10.1002/cpa.3160220605. Google Scholar [49] J. Ralston, Approximate eigenfunctions of the Laplacian,, J. Differential Geometry, 12 (1977), 87. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [56] T. Tao, Nonlinear Dispersive Equations, Local and global Analysis,, Amer Mathematical Society, (2006). Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [60] E. Trélat, Y. Privat and E. Zuazua, Optimal observability of wave and Schrödinger equations in ergodic domains,, submitted., (). Google Scholar [61] Q. Zhou and M. Yamamoto, Hautus condition on the exact controllability of conservative systems,, Internat. J. Control, 67 (1997), 371. Google Scholar [62] E. Zuazua, Contrôlabilité exacte en temps arbitrairement petit de quelques modèles de plaques,, volume Appendix A.1 to [38]., (). Google Scholar [63] E. Zuazua, Exact controllability for the semilinear wave equation,, J. Math. Pures Appl., 69 (1990), 33. Google Scholar [64] E. Zuazua, Remarks on the controllability of the schrödinger equation,, In Quantum control: Mathematical and numerical challenges, (2003), 193. Google Scholar [65] C. Zuily, Uniqueness and Nonuniqueness in the Cauchy Problem, volume 33 of Progress in Mathematics,, Birkhäuser Boston Inc., (1983). Google Scholar
 [1] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [2] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [3] Pavel I. Naumkin, Isahi Sánchez-Suárez. On the critical nongauge invariant nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 807-834. doi: 10.3934/dcds.2011.30.807 [4] Younghun Hong. Scattering for a nonlinear Schrödinger equation with a potential. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1571-1601. doi: 10.3934/cpaa.2016003 [5] Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1063-1079. doi: 10.3934/cpaa.2012.11.1063 [6] Dario Bambusi, A. Carati, A. Ponno. The nonlinear Schrödinger equation as a resonant normal form. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 109-128. doi: 10.3934/dcdsb.2002.2.109 [7] Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689 [8] Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791 [9] Mohamad Darwich. On the $L^2$-critical nonlinear Schrödinger Equation with a nonlinear damping. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2377-2394. doi: 10.3934/cpaa.2014.13.2377 [10] Christopher Grumiau, Marco Squassina, Christophe Troestler. On the Mountain-Pass algorithm for the quasi-linear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1345-1360. doi: 10.3934/dcdsb.2013.18.1345 [11] Addolorata Salvatore. Sign--changing solutions for an asymptotically linear Schrödinger equation. Conference Publications, 2009, 2009 (Special) : 669-677. doi: 10.3934/proc.2009.2009.669 [12] Wulong Liu, Guowei Dai. Multiple solutions for a fractional nonlinear Schrödinger equation with local potential. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2105-2123. doi: 10.3934/cpaa.2017104 [13] Xudong Shang, Jihui Zhang. Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2239-2259. doi: 10.3934/cpaa.2018107 [14] Patricio Felmer, César Torres. Radial symmetry of ground states for a regional fractional Nonlinear Schrödinger Equation. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2395-2406. doi: 10.3934/cpaa.2014.13.2395 [15] Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639 [16] Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 [17] Benjamin Dodson. Improved almost Morawetz estimates for the cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 127-140. doi: 10.3934/cpaa.2011.10.127 [18] Kazuhiro Kurata, Tatsuya Watanabe. A remark on asymptotic profiles of radial solutions with a vortex to a nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 597-610. doi: 10.3934/cpaa.2006.5.597 [19] Yue Liu. Existence of unstable standing waves for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2008, 7 (1) : 193-209. doi: 10.3934/cpaa.2008.7.193 [20] Walid K. Abou Salem, Xiao Liu, Catherine Sulem. Numerical simulation of resonant tunneling of fast solitons for the nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1637-1649. doi: 10.3934/dcds.2011.29.1637

2018 Impact Factor: 1.292