# American Institute of Mathematical Sciences

• Previous Article
Time-inconsistent optimal control problems and the equilibrium HJB equation
• MCRF Home
• This Issue
• Next Article
Numerical null controllability of semi-linear 1-D heat equations: Fixed point, least squares and Newton methods
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
 [1] Andrey Sarychev. Errata: Controllability of the cubic Schroedinger equation via a low-dimensional source term. Mathematical Control & Related Fields, 2014, 4 (2) : 261-261. doi: 10.3934/mcrf.2014.4.261 [2] Lianwen Wang. Approximate controllability and approximate null controllability of semilinear systems. Communications on Pure & Applied Analysis, 2006, 5 (4) : 953-962. doi: 10.3934/cpaa.2006.5.953 [3] Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control & Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375 [4] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [5] Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 [6] Hans Weinberger. The approximate controllability of a model for mutant selection. Evolution Equations & Control Theory, 2013, 2 (4) : 741-747. doi: 10.3934/eect.2013.2.741 [7] Shi Jin, Dongsheng Yin. Computational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction. Kinetic & Related Models, 2011, 4 (1) : 295-316. doi: 10.3934/krm.2011.4.295 [8] Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431 [9] Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397 [10] Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784 [11] Valentin Keyantuo, Mahamadi Warma. On the interior approximate controllability for fractional wave equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3719-3739. doi: 10.3934/dcds.2016.36.3719 [12] Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations & Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167 [13] Moncef Aouadi, Taoufik Moulahi. Approximate controllability of abstract nonsimple thermoelastic problem. Evolution Equations & Control Theory, 2015, 4 (4) : 373-389. doi: 10.3934/eect.2015.4.373 [14] Hugo Leiva, Nelson Merentes, José L. Sánchez. Approximate controllability of semilinear reaction diffusion equations. Mathematical Control & Related Fields, 2012, 2 (2) : 171-182. doi: 10.3934/mcrf.2012.2.171 [15] Hugo Leiva, Jahnett Uzcategui. Approximate controllability of discrete semilinear systems and applications. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1803-1812. doi: 10.3934/dcdsb.2016023 [16] Eduardo Martínez. Classical field theory on Lie algebroids: Multisymplectic formalism. Journal of Geometric Mechanics, 2018, 10 (1) : 93-138. doi: 10.3934/jgm.2018004 [17] Kirill D. Cherednichenko, Alexander V. Kiselev, Luis O. Silva. Functional model for extensions of symmetric operators and applications to scattering theory. Networks & Heterogeneous Media, 2018, 13 (2) : 191-215. doi: 10.3934/nhm.2018009 [18] Viorel Niţică. Stable transitivity for extensions of hyperbolic systems by semidirect products of compact and nilpotent Lie groups. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 1197-1204. doi: 10.3934/dcds.2011.29.1197 [19] Danijela Damjanović. Central extensions of simple Lie groups and rigidity of some abelian partially hyperbolic algebraic actions. Journal of Modern Dynamics, 2007, 1 (4) : 665-688. doi: 10.3934/jmd.2007.1.665 [20] Heinz Schättler, Urszula Ledzewicz. Perturbation feedback control: A geometric interpretation. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 631-654. doi: 10.3934/naco.2012.2.631

2018 Impact Factor: 1.292