Controllability of a quantum particle in a 1D variable domain

Karine Beauchard

ESAIM: Control, Optimisation and Calculus of Variations (2008)

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

Abstract

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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 ] → ℝ + * 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.

How to cite

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Beauchard, Karine. "Controllability of a quantum particle in a 1D variable domain." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2008): 105-147. <http://eudml.org/doc/244959>.

@article{Beauchard2008,
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 $\phi $ 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 $T &gt; 0$, 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},
language = {eng},
number = {1},
pages = {105-147},
publisher = {EDP-Sciences},
title = {Controllability of a quantum particle in a 1D variable domain},
url = {http://eudml.org/doc/244959},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Beauchard, Karine
TI - Controllability of a quantum particle in a 1D variable domain
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2008
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 $\phi $ 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 $T &gt; 0$, 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
UR - http://eudml.org/doc/244959
ER -

References

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  1. [1] F. Albertini and D. D’Alessandro, Notions of controllability for bilinear multilevel quantum systems. IEEE Trans. Automat. Control 48 (2003) 1399–1403. MR2004373
  2. [2] S. Alinhac and P. Gérard, Opérateurs pseudo-différentiels et théorème de Nash-Moser. Intereditions (Paris), collection Savoirs actuels (1991). Zbl0791.47044MR1172111
  3. [3] C. Altafini, Controllability of quantum mechanical systems by root space decomposition of su(n). J. Math. Phys. 43 (2002) 2051–2062. Zbl1059.93016MR1893660
  4. [4] J.M. Ball, J.E. Marsden and M. Slemrod, Controllability for distributed bilinear systems. SIAM J. Control Optim. 20 (1982). Zbl0485.93015MR661034
  5. [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. Zbl1109.49003MR2254931
  6. [6] L. Baudouin and J. Salomon, Constructive solution of a bilinear control problem. C.R. Math. Acad. Sci. Paris 342 (2006) 119–124. Zbl1079.49021MR2193658
  7. [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. Zbl1109.35094MR2158922
  8. [8] K. Beauchard, Local controllability of a 1-D beam equation. SIAM J. Control Optim. (to appear). Zbl1172.35325MR2407015
  9. [9] K. Beauchard, Local Controllability of a 1-D Schrödinger equation. J. Math. Pures Appl. 84 (2005) 851–956. Zbl1124.93009MR2144647
  10. [10] K. Beauchard and J.-M. Coron, Controllability of a quantum particle in a moving potential well. J. Functional Analysis 232 (2006) 328–389. Zbl1188.93017MR2200740
  11. [11] R. Brockett, Lie theory and control systems defined on spheres. SIAM J. Appl. Math. 25 (1973) 213–225. Zbl0272.93003MR327337
  12. [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. Zbl0953.49005MR1760440
  13. [13] J.-M. Coron, Global asymptotic stabilization for controllable systems without drift. Math. Control Signals Systems 5 (1992) 295–312. Zbl0760.93067MR1164379
  14. [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. Zbl0781.76013MR1233425
  15. [15] J.-M. Coron, On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996) 155–188. Zbl0848.76013MR1380673
  16. [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. Zbl1071.76012MR1932962
  17. [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. Zbl1082.93002MR2193655
  18. [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. Zbl1061.93054MR2060480
  19. [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. Zbl0938.93030MR1470445
  20. [20] A.V. Fursikov and O.Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations. Russian Math. Surveys 54 (1999) 565–618. Zbl0970.35116MR1728643
  21. [21] O. Glass, On the controllability of the 1D isentropic Euler equation. J. European Mathematical Society 9 (2007) 427–486. Zbl1139.35014MR2314104
  22. [22] O. Glass, Exact boundary controllability of 3-D Euler equation. ESAIM: COCV 5 (2000) 1–44. Zbl0940.93012MR1745685
  23. [23] O. Glass, On the controllability of the Vlasov-Poisson system. J. Differential Equations 195 (2003) 332–379. Zbl1109.93007MR2016816
  24. [24] G. Gromov, Partial Differential Relations. Springer-Verlag, Berlin-New York-London (1986). Zbl0651.53001MR864505
  25. [25] A. Haraux, Séries lacunaires et contrôle semi-interne des vibrations d’une plaque rectangulaire. J. Math. Pures Appl. 68 (1989) 457–465. Zbl0685.93039MR1046761
  26. [26] L. Hörmander, On the Nash-Moser Implicit Function Theorem. Annales Academiae Scientiarum Fennicae (1985) 255–259. Zbl0591.58003MR802486
  27. [27] T. Horsin, On the controllability of the Burgers equation. ESAIM: COCV 3 (1998) 83–95. Zbl0897.93034MR1612027
  28. [28] R. Ilner, H. Lange and H. Teismann, Limitations on the control of Schrödinger equations. ESAIM: COCV 12 (2006) 615–635. Zbl1162.93316MR2266811
  29. [29] T. Kato, Perturbation Theory for Linear operators. Springer-Verlag, Berlin, New-York (1966). Zbl0148.12601MR203473
  30. [30] W. Krabs, On moment theory and controllability of one-dimensional vibrating systems and heating processes. Springer – Verlag (1992). Zbl0955.93501MR1162111
  31. [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. Zbl0784.93032MR1157485
  32. [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. Zbl1061.35170MR2061430
  33. [33] G. Lebeau, Contrôle de l’équation de Schrödinger. J. Math. Pures Appl. 71 (1992) 267–291. Zbl0838.35013
  34. [34] Machtyngier, Exact controllability for the Schrödinger equation. SIAM J. Contr. Opt. 32 (1994) 24–34. Zbl0795.93018
  35. [35] M. Mirrahimi and P. Rouchon, Controllability of quantum harmonic oscillators. IEEE Trans. Automat. Control 49 (2004) 745–747. MR2057808
  36. [36] E. Sontag, Control of systems without drift via generic loops. IEEE Trans. Automat. Control 40 (1995) 1210–1219. Zbl0837.93019MR1344033
  37. [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). Zbl1007.81019MR1857459
  38. [38] E. Zuazua, Remarks on the controllability of the Schrödinger equation. CRM Proc. Lect. Notes 33 (2003) 193–211. MR2043529

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