Exact controllability for a Schrödinger equation with dynamic boundary conditions. (English) Zbl 1527.35330
Summary: In this paper, we study the controllability of a Schrödinger equation with mixed boundary conditions on disjoint subsets of the boundary: dynamic boundary condition of Wentzell type and Dirichlet boundary condition. The main result of this article is given by new Carleman estimates for the associated adjoint system, where the weight function is constructed specially adapted to the geometry of the domain. Using these estimates, we prove the exact controllability of the system with a boundary control acting only in the part of the boundary where the Dirichlet condition is imposed. Also, we obtain a distributed exact controllability result for the system.
MSC:
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
| 93B05 | Controllability |
| 93B07 | Observability |
| 93C05 | Linear systems in control theory |
| 93C20 | Control/observation systems governed by partial differential equations |
| 35M13 | Initial-boundary value problems for PDEs of mixed type |
Keywords:
Schrödinger equation; dynamic boundary conditions; exact controllability; Carleman estimatesReferences:
| [1] | Aassila, M., Exact controllability of the Schrödinger equation, Appl. Math. Comput., 144 (2003), pp. 89-106, doi:10.1016/S0096-3003(02)00394-6. · Zbl 1031.93019 |
| [2] | Bandrauk, A. D., Molecules in Laser Fields, CRC Press, Boca Raton, FL, 1993. |
| [3] | Baudouin, L. and Mercado, A., An inverse problem for Schrödinger equations with discontinuous main coefficient, Appl. Anal., 87 (2008), pp. 1145-1165, doi:10.1080/00036810802140673. · Zbl 1160.35548 |
| [4] | Baudouin, L., Mercado, A., and Osses, A., A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem, Inverse Problems, 23 (2007), pp. 257-278, doi:10.1088/0266-5611/23/1/014. · Zbl 1111.35104 |
| [5] | Baudouin, L. and Puel, J.-P., Uniqueness and stability in an inverse problem for the Schrödinger equation, Inverse Problems, 23 (2007), pp. 1327-1328, doi:10.1088/0266-5611/23/3/C01. |
| [6] | Benabbas, I. and Teniou, D. E., Observability of wave equation with Ventcel dynamic condition, Evol. Equ. Control Theory, 7 (2018), pp. 545-570, doi:10.3934/eect.2018026. · Zbl 1405.35103 |
| [7] | Burq, N., Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mém. Soc. Math. France, 55 (1993). · Zbl 0930.93007 |
| [8] | Cavalcanti, M. M., Corrêa, W. J., Lasiecka, I., and Lefler, C., Well-posedness and uniform stability for nonlinear Schrödinger equations with dynamic/Wentzell boundary conditions, Indiana Univ. Math. J., 65 (2016), pp. 1445-1502, doi:10.1512/iumj.2016.65.5873. · Zbl 1373.35285 |
| [9] | Corrêa, W. J. and Özsarı, T., Complex Ginzburg-Landau equations with dynamic boundary conditions, Nonlinear Anal. Real World Appl., 41 (2018), pp. 607-641, doi:10.1016/j.nonrwa.2017.12.001. · Zbl 1387.35113 |
| [10] | Ervedoza, S., Zheng, C., and Zuazua, E., On the observability of time-discrete conservative linear systems, J. Funct. Anal., 254 (2008), pp. 3037-3078, doi:10.1016/j.jfa.2008.03.005. · Zbl 1143.65044 |
| [11] | Gal, C. G. and Tebou, L., Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), pp. 324-364, doi:10.1137/15M1032211. · Zbl 1354.93024 |
| [12] | Giusti-Suzor, A. and Mies, F. H., Vibrational trapping and suppression of dissociation in intense laser fields, Phys. Rev. Lett., 68 (1992), 3869. |
| [13] | Ismailov, M. I., Inverse source problem for heat equation with nonlocal Wentzell boundary condition, Results Math., 73 (2018), 68, doi:10.1007/s00025-018-0829-2. · Zbl 1393.35287 |
| [14] | Ismailov, M. I., Tekin, I., and Erkovan, S., An inverse problem for finding the lowest term of a heat equation with Wentzell-Neumann boundary condition, Inverse Probl. Sci. Eng., 27 (2019), pp. 1608-1634, doi:10.1080/17415977.2018.1553968. · Zbl 1467.35349 |
| [15] | Jaffard, S., Contrôle interne exact des vibrations d’une plaque rectangulaire, Port. Math., 47 (1990), pp. 423-429. · Zbl 0718.49026 |
| [16] | Jost, J., Riemannian Geometry and Geometric Analysis, 7th ed., Universitext, Springer, Cham, 2017, doi:10.1007/978-3-319-61860-9. · Zbl 1380.53001 |
| [17] | Khoutaibi, A. and Maniar, L., Null controllability for a heat equation with dynamic boundary conditions and drift terms, Evol. Equ. Control Theory, 9 (2020), pp. 535-559, doi:10.3934/eect.2020023. · Zbl 1445.35197 |
| [18] | Lasiecka, I., Triggiani, R., and Zhang, X., Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. I. H1(Ω)-estimates, J. Inverse Ill-Posed Probl., 12 (2004), pp. 43-123, doi:10.1515/156939404773972761. · Zbl 1057.35042 |
| [19] | Lasiecka, I., Triggiani, R., and Zhang, X., Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. II. L2(Ω)-estimates, J. Inverse Ill-Posed Probl., 12 (2004), pp. 183-231, doi:10.1515/1569394042530919. · Zbl 1061.35170 |
| [20] | Lebeau, G., Contrôle de l’équation de Schrödinger, J. Math. Pures Appl. (9), 71 (1992), pp. 267-291. · Zbl 0838.35013 |
| [21] | Lions, J.-L., Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, , Masson, Paris, 1988. · Zbl 0653.93002 |
| [22] | Machtyngier, E., Exact controllability for the Schrödinger equation, SIAM J. Control Optim., 32 (1994), pp. 24-34, doi:10.1137/S0363012991223145. · Zbl 0795.93018 |
| [23] | Maniar, L., Meyries, M., and Schnaubelt, R., Null controllability for parabolic equations with dynamic boundary conditions, Evol. Equ. Control Theory, 6 (2017), pp. 381-407, doi:10.3934/eect.2017020. · Zbl 1366.93058 |
| [24] | Mercado, A., Osses, A., and Rosier, L., Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights, Inverse Problems, 24 (2008), 015017, 18, doi:10.1088/0266-5611/24/1/015017. · Zbl 1153.35407 |
| [25] | Miller, L., Controllability cost of conservative systems: Resolvent condition and transmutation, J. Funct. Anal., 218 (2005), pp. 425-444, doi:10.1016/j.jfa.2004.02.001. · Zbl 1122.93011 |
| [26] | Nagasawa, M., Schrödinger Equations and Diffusion Theory, Mod. Birkhäuser Class., Birkhäuser/Springer Basel AG, Basel, 1993. · Zbl 0780.60003 |
| [27] | Phung, K.-D., Observability and control of Schrödinger equations, SIAM J. Control Optim., 40 (2001), pp. 211-230, doi:10.1137/S0363012900368405. · Zbl 0995.93037 |
| [28] | Rosier, L. and Zhang, B.-Y., Exact boundary controllability of the nonlinear Schrödinger equation, J. Differential Equations, 246 (2009), pp. 4129-4153, doi:10.1016/j.jde.2008.11.004. · Zbl 1171.35015 |
| [29] | Rosier, L. and Zhang, B.-Y., Null controllability of the complex Ginzburg-Landau equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2009), pp. 649-673, doi:10.1016/j.anihpc.2008.03.003. · Zbl 1170.35095 |
| [30] | Taylor, M. E., Partial Differential Equations I. Basic Theory, 2nd ed., , Springer, New York, 2011, doi:10.1007/978-1-4419-7055-8. · Zbl 1206.35002 |
| [31] | Tucsnak, M. and Weiss, G., Observation and Control for Operator Semigroups, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser Verlag, Basel, 2009, doi:10.1007/978-3-7643-8994-9. · Zbl 1188.93002 |
| [32] | Zhang, M., Yin, J., and Gao, H., Insensitizing controls for the parabolic equations with dynamic boundary conditions, J. Math. Anal. Appl., 475 (2019), pp. 861-873, doi:10.1016/j.jmaa.2019.02.077. · Zbl 1415.93056 |
| [33] | Zuazua, E., Remarks on the controllability of the Schrödinger equation, 33 (2003), pp. 193-211, doi:10.1090/crmp/033/12. |
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