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September  2012, 2(3): 247-270. doi: 10.3934/mcrf.2012.2.247

Controllability of the cubic Schroedinger equation via a low-dimensional source term

1. 

DiMaD, Università di Firenze, via delle Pandette 9, Firenze, 50127, Italy

Received  May 2011 Revised  November 2011 Published  August 2012

We study controllability of $d$-dimensional defocusing cubic Schroe-din-ger equation under periodic boundary conditions. The control is applied additively, via a source term, which is a linear combination of few complex exponentials (modes) with time-variant coefficients - controls. We manage to prove that controlling $2^d$ modes one can achieve controllability of the equation in each finite-dimensional projection of the evolution space $H^{s}(\mathbb{T}^d), \ s>d/2$, as well as approximate controllability in $H^{s}(\mathbb{T}^d)$. We also present a negative result regarding exact controllability of cubic Schroedinger equation via a finite-dimensional source term.
Citation: Andrey Sarychev. Controllability of the cubic Schroedinger equation via a low-dimensional source term. Mathematical Control and Related Fields, 2012, 2 (3) : 247-270. doi: 10.3934/mcrf.2012.2.247
References:
[1]

A. A. Agrachev and A. V.Sarychev, Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing, Communications in Mathematical Physics, 265 (2006), 673-697. doi: 10.1007/s00220-006-0002-8.

[2]

A. A. Agrachev and A. V.Sarychev, Solid controllability in fluid dynamics, in "Instability in Models Connected with Fluid Flows. I" (eds. C. Bardos and A. Fursikov), Int. Math. Ser. (N. Y.), 6, Springer, (2008), 1-35.

[3]

J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optimization, 20 (1982), 575-597. doi: 10.1137/0320042.

[4]

K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math Pures Appl. (9), 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005.

[5]

K. Beauchard and J.-M.Coron, Controllability of a quantum particle in a moving potential well, J. Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation, Geometric and Functional Analysis, 3 (1993), 107-156.

[7]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz estimates and the nonlinear Schrödinger equation on compact manifolds, American J. of Mathematics, 126 (2004), 569-605. doi: 10.1353/ajm.2004.0016.

[8]

T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 26 (2009), 329-349.

[9]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Mathematische Zeitschrift, 254 (2006), 729-749. doi: 10.1007/s00209-006-0005-3.

[10]

H. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite-dimensional systems, J. of Differential Equations, 112 (1994), 131-153. doi: 10.1006/jdeq.1994.1097.

[11]

H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128. doi: 10.1016/0022-0396(90)90129-D.

[12]

R. V. Gamkrelidze, "Principles of Optimal Control Theory," Revised edition, Mathematical Concepts and Methods in Science and Engineering, Vol. 7, Plenum Press, New York-London, 1978.

[13]

R. Illner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equation, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.

[14]

I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization, 23 (1991), 109-154. doi: 10.1007/BF01442394.

[15]

G. Lebeau, Contrôle de l'equation de Schrödinger, (French) [Control of the Schrödinger equation], J. Math. Pures Appl. (9), 71 (1992), 267-291.

[16]

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), 972-992. doi: 10.1137/070709578.

[17]

A. Shirikyan, Euler equations are not exactly controllable by a finite-dimensional external force, Physica D, 237 (2008), 1317-1323.

[18]

H. J. Sussmann and W. Liu, Lie bracket extensions and averaging the single-bracket case, in "Nonholonomic Motion Planning" (ed. Z. X. Li and J. F. Canny), Kluwer Academic Publishers, (1993), 109-148.

[19]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006.

[20]

E. Zuazua, Remarks on the controllability of the Schrödinger equation, in "Quantum Control: Mathematical and Numerical Challenges," CRM Proceedings and Lecture Notes, 33, Amer. Math. Soc., Providence, RI, (2003), 193-211.

show all references

References:
[1]

A. A. Agrachev and A. V.Sarychev, Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing, Communications in Mathematical Physics, 265 (2006), 673-697. doi: 10.1007/s00220-006-0002-8.

[2]

A. A. Agrachev and A. V.Sarychev, Solid controllability in fluid dynamics, in "Instability in Models Connected with Fluid Flows. I" (eds. C. Bardos and A. Fursikov), Int. Math. Ser. (N. Y.), 6, Springer, (2008), 1-35.

[3]

J. M. Ball, J. E. Marsden and M. Slemrod, Controllability for distributed bilinear systems, SIAM J. Control Optimization, 20 (1982), 575-597. doi: 10.1137/0320042.

[4]

K. Beauchard, Local controllability of a 1-D Schrödinger equation, J. Math Pures Appl. (9), 84 (2005), 851-956. doi: 10.1016/j.matpur.2005.02.005.

[5]

K. Beauchard and J.-M.Coron, Controllability of a quantum particle in a moving potential well, J. Functional Analysis, 232 (2006), 328-389. doi: 10.1016/j.jfa.2005.03.021.

[6]

J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equation, Geometric and Functional Analysis, 3 (1993), 107-156.

[7]

N. Burq, P. Gérard and N. Tzvetkov, Strichartz estimates and the nonlinear Schrödinger equation on compact manifolds, American J. of Mathematics, 126 (2004), 569-605. doi: 10.1353/ajm.2004.0016.

[8]

T. Chambrion, P. Mason, M. Sigalotti and U. Boscain, Controllability of the discrete-spectrum Schrödinger equation driven by an external field, Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 26 (2009), 329-349.

[9]

B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Mathematische Zeitschrift, 254 (2006), 729-749. doi: 10.1007/s00209-006-0005-3.

[10]

H. Fattorini, Relaxation theorems, differential inclusions, and Filippov's theorem for relaxed controls in semilinear infinite-dimensional systems, J. of Differential Equations, 112 (1994), 131-153. doi: 10.1006/jdeq.1994.1097.

[11]

H. Frankowska, A priori estimates for operational differential inclusions, J. Differential Equations, 84 (1990), 100-128. doi: 10.1016/0022-0396(90)90129-D.

[12]

R. V. Gamkrelidze, "Principles of Optimal Control Theory," Revised edition, Mathematical Concepts and Methods in Science and Engineering, Vol. 7, Plenum Press, New York-London, 1978.

[13]

R. Illner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equation, ESAIM Control Optim. Calc. Var., 12 (2006), 615-635.

[14]

I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems, Applied Mathematics and Optimization, 23 (1991), 109-154. doi: 10.1007/BF01442394.

[15]

G. Lebeau, Contrôle de l'equation de Schrödinger, (French) [Control of the Schrödinger equation], J. Math. Pures Appl. (9), 71 (1992), 267-291.

[16]

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), 972-992. doi: 10.1137/070709578.

[17]

A. Shirikyan, Euler equations are not exactly controllable by a finite-dimensional external force, Physica D, 237 (2008), 1317-1323.

[18]

H. J. Sussmann and W. Liu, Lie bracket extensions and averaging the single-bracket case, in "Nonholonomic Motion Planning" (ed. Z. X. Li and J. F. Canny), Kluwer Academic Publishers, (1993), 109-148.

[19]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis," CBMS Regional Conference Series in Mathematics, 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, by the American Mathematical Society, Providence, RI, 2006.

[20]

E. Zuazua, Remarks on the controllability of the Schrödinger equation, in "Quantum Control: Mathematical and Numerical Challenges," CRM Proceedings and Lecture Notes, 33, Amer. Math. Soc., Providence, RI, (2003), 193-211.

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