# Controllablity of a quantum particle in a 1D variable domain

ESAIM: Control, Optimisation and Calculus of Variations (2010)

- Volume: 14, Issue: 1, page 105-147
- ISSN: 1292-8119

## Access Full Article

top## Abstract

top## How to cite

topBeauchard, Karine. "Controllablity of a quantum particle in a 1D variable domain." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 105-147. <http://eudml.org/doc/90860>.

@article{Beauchard2010,

abstract = {
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 $\phi_\{0\}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_\{f\}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb\{R\}^\{*\}_\{+\}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_\{0\}$ to $\phi_\{f\}$ 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.
},

author = {Beauchard, Karine},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Controllability; Schrödinger equation; Nash-Moser theorem; moment theory; controllability},

language = {eng},

month = {3},

number = {1},

pages = {105-147},

publisher = {EDP Sciences},

title = {Controllablity of a quantum particle in a 1D variable domain},

url = {http://eudml.org/doc/90860},

volume = {14},

year = {2010},

}

TY - JOUR

AU - Beauchard, Karine

TI - Controllablity of a quantum particle in a 1D variable domain

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 14

IS - 1

SP - 105

EP - 147

AB -
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 $\phi_{0}$ close enough to an eigenstate corresponding to the length l = 1 and $\phi_{f}$ close enough to another eigenstate corresponding to the length l=1, there exists a continuous function $l:[0,T] \rightarrow \mathbb{R}^{*}_{+}$ with T > 0, such that l(0) = 1 and l(T) = 1, and which moves the wave function from $\phi_{0}$ to $\phi_{f}$ 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.

LA - eng

KW - Controllability; Schrödinger equation; Nash-Moser theorem; moment theory; controllability

UR - http://eudml.org/doc/90860

ER -

## References

top- F. Albertini and D. D'Alessandro, Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control48 (2003) 1399–1403.
- S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Intereditions (Paris), collection Savoirs actuels (1991).
- C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys.43 (2002) 2051–2062. Zbl1059.93016
- J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim.20 (1982). Zbl0485.93015
- 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. Zbl1109.49003
- L. Baudouin and J. Salomon, Constructive solution of a bilinear control problem. C.R. Math. Acad. Sci. Paris342 (2006) 119–124. Zbl1079.49021
- L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potential and application to bilinear optimal control. J. Differential Equations216 (2005) 188–222. Zbl1109.35094
- K. Beauchard, Local controllability of a 1-D beam equation. SIAM J. Control Optim. (to appear). Zbl1124.93009
- K. Beauchard, Local Controllability of a 1-D Schrödinger equation. J. Math. Pures Appl.84 (2005) 851–956. Zbl1124.93009
- K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well. J. Functional Analysis232 (2006) 328–389. Zbl1188.93017
- R. Brockett, Lie theory and control systems defined on spheres. SIAM J. Appl. Math.25 (1973) 213–225. Zbl0272.93003
- E. Cancès, C. Le Bris and M. Pilot, Contrôle optimal bilinéaire d'une équation de Schrödinger. C.R. Acad. Sci. Paris, Série I330 (2000) 567–571. Zbl0953.49005
- J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems5 (1992) 295–312. Zbl0760.93067
- J.-M. Coron, Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels. C. R. Acad. Sci. Paris317 (1993) 271–276.
- J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl.75 (1996) 155–188. Zbl0848.76013
- J.-M. Coron, Local Controllability of a 1-D Tank Containing a Fluid Modeled by the shallow water equations. ESAIM: COCV8 (2002) 513–554. Zbl1071.76012
- 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 I342 (2006) 103–108.
- 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. Zbl1061.93054
- 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. Zbl0938.93030
- A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys54 (1999) 565–618. Zbl0970.35116
- O. Glass, On the controllability of the 1D isentropic Euler equation. J. European Mathematical Society9 (2007) 427–486. Zbl1139.35014
- O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV5 (2000) 1–44. Zbl0940.93012
- O. Glass, On the controllability of the Vlasov-Poisson system. J. Differential Equations195 (2003) 332–379. Zbl1109.93007
- G. Gromov, Partial Differential Relations. Springer-Verlag, Berlin-New York-London (1986). Zbl0651.53001
- A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl.68 (1989) 457–465. Zbl0685.93039
- L. Hörmander, On the Nash-Moser Implicit Function Theorem. Annales Academiae Scientiarum Fennicae (1985) 255–259. Zbl0591.58003
- T. Horsin, On the controllability of the Burgers equation. ESAIM: COCV3 (1998) 83–95. Zbl0897.93034
- R. Ilner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations. ESAIM: COCV 12 (2006) 615–635. Zbl1162.93316
- T. Kato, Perturbation Theory for Linear operators. Springer-Verlag, Berlin, New-York (1966). Zbl0148.12601
- W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes. Springer – Verlag (1992). Zbl0955.93501
- I. Lasiecka and R. Triggiani, Optimal regularity, exact controllability and uniform stabilization of Schrödinger equations with Dirichlet controls. Differential Integral Equations5 (1992) 571–535. Zbl0784.93032
- 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. Zbl1061.35170
- G. Lebeau, Contrôle de l'équation de Schrödinger. J. Math. Pures Appl.71 (1992) 267–291. Zbl0838.35013
- Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Contr. Opt.32 (1994) 24–34. Zbl0795.93018
- M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control49 (2004) 745–747.
- E. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control40 (1995) 1210–1219. Zbl0837.93019
- 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). Zbl0971.81198
- E. Zuazua, Remarks on the controllability of the Schrödinger equation. CRM Proc. Lect. Notes33 (2003) 193–211.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.