LOCAL EXACT CONTROLLABILITY OF SCHR&#214;DINGER EQUATION WITH STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

Jian Zu; Yong Li

doi:10.11948/2016054

2016 Volume 6 Issue 3

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Jian Zu, Yong Li. LOCAL EXACT CONTROLLABILITY OF SCHRÖDINGER EQUATION WITH STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2016, 6(3): 851-864. doi: 10.11948/2016054

Citation:

Jian Zu, Yong Li. LOCAL EXACT CONTROLLABILITY OF SCHRÖDINGER EQUATION WITH STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS[J]. Journal of Applied Analysis & Computation, 2016, 6(3): 851-864. doi: 10.11948/2016054

LOCAL EXACT CONTROLLABILITY OF SCHRÖDINGER EQUATION WITH STURM-LIOUVILLE BOUNDARY VALUE PROBLEMS

Jian Zu^1,2,
Yong Li^1,2

1 School of Mathematics and Statistics, Northeast Normal University, 5268 Renmin Street, 130024 Changchun, P. R. China;
2 Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, 2555 Jingyue Street, 130117 Changchun, P. R. China

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Abstract

Abstract

In this paper, we investigate the controllability of 1D bilinear Schrödinger equation with Sturm-Liouville boundary value condition. The system represents a quantumn particle controlled by an electric field. K. Beauchard and C. Laurent have proved local controllability of 1D bilinear Schrödinger equation with Dirichlet boundary value condition in some suitable Sobolev space based on the classical inverse mapping theorem. Using a similar method, we extend this result to Sturm-Liouville boundary value proplems.
- Controllability /
- Bilinear Schrödinger equation /
- Sturm-Liouville boundary value condition
MSC: 34L15;93B05;35Q55

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References

[1]	F. Albertini and D. D'Alessandro, Notions of controllability for bilinear multilevel quantum systems, IEEE Trans. Automat. Control, 48(2003)(8), 1399-1403. Google Scholar
[2]	C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(N), J. Math. Phys., 43(2002)(5), 2051-2062. Google Scholar
[3]	J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optim., 20(1982)(4), 575-597. Google Scholar
[4]	K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math. Pures Appl., 84(2005)(7), 851-956. Google Scholar
[5]	K. Beauchard, Controllablity of a quantum particle in a 1D variable domain, ESAIM Control Optim. Calc. Var., 14(2008)(1), 105-147. Google Scholar
[6]	K. Beauchard and J. M. Coron, Controllability of a quantum particle in a moving potential well, J. Funct. Anal., 232(2006)(2), 328-389. Google Scholar
[7]	K. Beauchard and C. Laurent, Controllability of 1D linear and nonlinear Schrödinger equations with bilinear control, J. Math. Pures Appl., 94(2010)(9), 520-554. Google Scholar
[8]	U. Boscain, M. Caponigro, T. Chambrion and M. Sigalotti, A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule, Comm. Math. Phys., 311(2012)(2), 423-455. Google Scholar
[9]	B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z., 254(2006)(4), 729-749. Google Scholar
[10]	Y. X. Gao, Y. Li and J. Zhang, Invariant tori of nonlinear Schrödinger equation, J. Differential Equations, 246(2009)(8), 3296-3331. Google Scholar
[11]	A. Haraux, Lacunary series and semi-internal control of the vibrations of a rectangular plate, J. Math. Pures Appl., 68(1989)(4), 457-465. Google Scholar
[12]	R. Illner, H. Lange and H. Teismann, A note on the exact internal control of nonlinear Schrödinger equations, Quantum control:mathematical and numerical challenges[CRM Proc. Lecture Notes, 33], Amer. Math. Soc., Providence, 2003. Google Scholar
[13]	S. G. Ji and Y. Li, Periodic solutions to one-dimensional wave equation with x-dependent coefficients, J. Differential Equations, 229(2006)(2), 466-493. Google Scholar
[14]	M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators, IEEE Trans. Automat. Control, 49(2004)(5), 745-747. Google Scholar
[15]	M. Morancey and V. Nersesyan, Global exact controllability of 1D Schrödinger equations with a polarizability term, C. R. Math. Acad. Sci. Paris, 352(2014)(5), 425-429. Google Scholar
[16]	V. Nersesyan, Growth of Sobolev norms and controllability of the Schrödinger equation, Comm. Math. Phys., 290(2009)(1), 371-387. Google Scholar
[17]	V. Nersesyan, Global approximate controllability for Schrödinger equation in higher Sobolev norms and applications, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27(2010)(3), 901-915. Google Scholar
[18]	V. Ramakrishna and H. Rabitz, Control of molecular dynamics, Systems modelling and optimization[Res. Notes Math. 396], Chapman & Hall/CRC, 1999. Google Scholar
[19]	L. Rosier and B. Y. Zhang, Local exact controllability and stabilizability of the nonlinear Schrödinger equation on a bounded interval, SIAM J. Control Optim., 48(2009)(2), 972-992. Google Scholar
[20]	O. Sáfár, Inverse eigenvalue problems for smooth potential, J. Appl. Anal. Comput., 2(2012)(3), 315-324. Google Scholar
[21]	H. J. Sussmann and V. Jurdjevic, Controllability of nonlinear systems, J. Differential Equations, 12(1972)(1), 95-116. Google Scholar
[22]	G. Turinici, On the controllability of bilinear quantum systems, Mathematical models and methods for ab initio quantum chemistry[Lecture Notes in Chem. 74], Springer, Berlin Heidelberg, 2000. Google Scholar
[23]	G. Turinici and H. Rabitz, Wavefunction controllability for finite-dimensional bilinear quantum systems, J. Phys. A, 36(2003)(10), 2565-2576. Google Scholar
[24]	E. Zuazua, Remarks on the controllability of the Schrödinger equation, Quantum control:mathematical and numerical challenges[CRM Proc. Lecture Notes 33], Amer. Math. Soc., Providence, 2003. Google Scholar