Controllability of a quantum particle in a 1D variable domain
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 105-147.

We consider a quantum particle in a 1D infinite square potential well with variable length. It is a nonlinear control system in which the state is the wave function φ of the particle and the control is the length l(t) of the potential well. We prove the following controllability result : given φ 0 close enough to an eigenstate corresponding to the length l=1 and φ f close enough to another eigenstate corresponding to the length l=1, there exists a continuous function l:[0,T] + * with T>0, such that l(0)=1 and l(T)=1, and which moves the wave function from φ 0 to φ f in time T. In particular, we can move the wave function from one eigenstate to another one by acting on the length of the potential well in a suitable way. Our proof relies on local controllability results proved with moment theory, a Nash-Moser implicit function theorem and expansions to the second order.

DOI : https://doi.org/10.1051/cocv:2007047
Classification : 35B37,  35Q55,  93B05,  93C20
Mots clés : controllability, Schrödinger equation, Nash-Moser theorem, moment theory
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     title = {Controllability of a quantum particle in a {1D} variable domain},
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     url = {http://www.numdam.org/articles/10.1051/cocv:2007047/}
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Beauchard, Karine. Controllability of a quantum particle in a 1D variable domain. ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 105-147. doi : 10.1051/cocv:2007047. http://www.numdam.org/articles/10.1051/cocv:2007047/

[1] F. Albertini and D. D'Alessandro, Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48 (2003) 1399-1403. | MR 2004373

[2] S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Intereditions (Paris), collection Savoirs actuels (1991). | MR 1172111 | Zbl 0791.47044

[3] C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys. 43 (2002) 2051-2062. | MR 1893660 | Zbl 1059.93016

[4] J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982). | MR 661034 | Zbl 0485.93015

[5] L. Baudouin, A bilinear optimal control problem applied to a time dependent Hartree-Fock equation coupled with classical nuclear dynamics. Portugaliae Matematica (N.S.) 63 (2006) 293-325. | EuDML 53082 | MR 2254931 | Zbl 1109.49003

[6] L. Baudouin and J. Salomon, Constructive solution of a bilinear control problem. C.R. Math. Acad. Sci. Paris 342 (2006) 119-124. | MR 2193658 | Zbl 1079.49021

[7] L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control. J. Differential Equations 216 (2005) 188-222. | MR 2158922 | Zbl 1109.35094

[8] K. Beauchard, Local controllability of a 1-D beam equation. SIAM J. Control Optim. (to appear). | MR 2407015 | Zbl 1172.35325

[9] K. Beauchard, Local Controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851-956. | MR 2144647 | Zbl 1124.93009

[10] K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well. J. Functional Analysis 232 (2006) 328-389. | MR 2200740

[11] R. Brockett, Lie theory and control systems defined on spheres. SIAM J. Appl. Math. 25 (1973) 213-225. | MR 327337 | Zbl 0272.93003

[12] E. Cancès, C. Le Bris and M. Pilot, Contrôle optimal bilinéaire d'une équation de Schrödinger. C.R. Acad. Sci. Paris, Série I 330 (2000) 567-571. | MR 1760440 | Zbl 0953.49005

[13] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295-312. | MR 1164379 | Zbl 0760.93067

[14] J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris 317 (1993) 271-276. | MR 1233425 | Zbl 0781.76013

[15] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155-188. | MR 1380673 | Zbl 0848.76013

[16] J.-M. Coron, Local Controllability of a 1-D Tank Containing a Fluid Modeled by the shallow water equations. ESAIM: COCV 8 (2002) 513-554. | Numdam | MR 1932962 | Zbl 1071.76012

[17] J.-M. Coron, On the small-time local controllability of a quantum particule in a moving one-dimensional infinite square potential well. C.R. Acad. Sci., Série I 342 (2006) 103-108. | MR 2193655 | Zbl 1082.93002

[18] J.-M. Coron and E. Crépeau, Exact boundary controllability of a nonlinear KdV equation with critical lengths. J. Eur. Math. Soc. 6 (2004) 367-398. | MR 2060480 | Zbl 1061.93054

[19] J.-M. Coron and A. Fursikov, Global exact controllability of the 2D Navier-Stokes equation on a manifold without boundary. Russ. J. Math. Phys. 4 (1996) 429-448. | MR 1470445 | Zbl 0938.93030

[20] A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 565-618. | MR 1728643 | Zbl 0970.35116

[21] O. Glass, On the controllability of the 1D isentropic Euler equation. J. European Mathematical Society 9 (2007) 427-486. | MR 2314104 | Zbl 1139.35014

[22] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1-44. | Numdam | MR 1745685 | Zbl 0940.93012

[23] O. Glass, On the controllability of the Vlasov-Poisson system. J. Differential Equations 195 (2003) 332-379. | MR 2016816 | Zbl 1109.93007

[24] G. Gromov, Partial Differential Relations. Springer-Verlag, Berlin-New York-London (1986). | MR 864505 | Zbl 0651.53001

[25] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl. 68 (1989) 457-465. | MR 1046761 | Zbl 0685.93039

[26] L. Hörmander, On the Nash-Moser Implicit Function Theorem. Annales Academiae Scientiarum Fennicae (1985) 255-259. | MR 802486 | Zbl 0591.58003

[27] T. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83-95. | Numdam | MR 1612027 | Zbl 0897.93034

[28] R. Ilner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations. ESAIM: COCV 12 (2006) 615-635. | Numdam | MR 2266811

[29] T. Kato, Perturbation Theory for Linear operators. Springer-Verlag, Berlin, New-York (1966). | MR 203473 | Zbl 0148.12601

[30] W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes. Springer - Verlag (1992). | MR 1162111 | Zbl 0955.93501

[31] I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls. Differential Integral Equations 5 (1992) 571-535. | MR 1157485 | Zbl 0784.93032

[32] I. Lasiecka, R. Triggiani and X. Zhang, Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carlemann estimates. J. Inverse Ill Posed-Probl. 12 (2004) 183-231. | MR 2061430 | Zbl 1061.35170

[33] G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl. 71 (1992) 267-291. | Zbl 0838.35013

[34] Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Contr. Opt. 32 (1994) 24-34. | Zbl 0795.93018

[35] M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control 49 (2004) 745-747. | MR 2057808

[36] E. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control 40 (1995) 1210-1219. | MR 1344033 | Zbl 0837.93019

[37] G. Turinici, On the controllability of bilinear quantum systems, in Mathematical Models and Methods for Ab Initio Quantum Chemistry, C. Le Bris and M. Defranceschi Eds., Lect. Notes Chemistry 74, Springer (2000). | MR 1857459 | Zbl 1007.81019

[38] E. Zuazua, Remarks on the controllability of the Schrödinger equation. CRM Proc. Lect. Notes 33 (2003) 193-211. | MR 2043529

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