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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 & 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.  doi: 10.1007/s00220-006-0002-8.  Google Scholar

[2]

A. A. Agrachev and A. V.Sarychev, Solid controllability in fluid dynamics,, in, 6 (2008), 1.   Google Scholar

[3]

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

[4]

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

[5]

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

[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.   Google Scholar

[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.  doi: 10.1353/ajm.2004.0016.  Google Scholar

[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, 26 (2009), 329.   Google Scholar

[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.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[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.  doi: 10.1006/jdeq.1994.1097.  Google Scholar

[11]

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

[12]

R. V. Gamkrelidze, "Principles of Optimal Control Theory,", Revised edition, (1978).   Google Scholar

[13]

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

[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.  doi: 10.1007/BF01442394.  Google Scholar

[15]

G. Lebeau, Contrôle de l'equation de Schrödinger,, (French) [Control of the Schrödinger equation], 71 (1992), 267.   Google Scholar

[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.  doi: 10.1137/070709578.  Google Scholar

[17]

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

[18]

H. J. Sussmann and W. Liu, Lie bracket extensions and averaging the single-bracket case,, in, (1993), 109.   Google Scholar

[19]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar

[20]

E. Zuazua, Remarks on the controllability of the Schrödinger equation,, in, 33 (2003), 193.   Google Scholar

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.  doi: 10.1007/s00220-006-0002-8.  Google Scholar

[2]

A. A. Agrachev and A. V.Sarychev, Solid controllability in fluid dynamics,, in, 6 (2008), 1.   Google Scholar

[3]

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

[4]

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

[5]

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

[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.   Google Scholar

[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.  doi: 10.1353/ajm.2004.0016.  Google Scholar

[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, 26 (2009), 329.   Google Scholar

[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.  doi: 10.1007/s00209-006-0005-3.  Google Scholar

[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.  doi: 10.1006/jdeq.1994.1097.  Google Scholar

[11]

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

[12]

R. V. Gamkrelidze, "Principles of Optimal Control Theory,", Revised edition, (1978).   Google Scholar

[13]

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

[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.  doi: 10.1007/BF01442394.  Google Scholar

[15]

G. Lebeau, Contrôle de l'equation de Schrödinger,, (French) [Control of the Schrödinger equation], 71 (1992), 267.   Google Scholar

[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.  doi: 10.1137/070709578.  Google Scholar

[17]

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

[18]

H. J. Sussmann and W. Liu, Lie bracket extensions and averaging the single-bracket case,, in, (1993), 109.   Google Scholar

[19]

T. Tao, "Nonlinear Dispersive Equations. Local and Global Analysis,", CBMS Regional Conference Series in Mathematics, 106 (2006).   Google Scholar

[20]

E. Zuazua, Remarks on the controllability of the Schrödinger equation,, in, 33 (2003), 193.   Google Scholar

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