Journal of Mathematical Sciences

, Volume 195, Issue 3, pp 311–335 | Cite as

Energy Minimization Problem in Two-Level Dissipative Quantum Control: Meridian Case

  • B. Bonnard
  • O. Cots
  • N. Shcherbakova


We analyze the energy-minimizing problem for a two-level dissipative quantum system described by the Kossakowsky–Lindblad equation. According to the Pontryagin maximum principle (PMP), minimizers can be selected among normal and abnormal extremals whose dynamics are classified according to the values of the dissipation parameters. Our aim is to improve our previous analysis from [5] concerning 2D solutions in the case where the Hamiltonian dynamics are integrable.


Conjugate Point Pontryagin Maximum Principle Integrable Case Spherical Case Dissipation Parameter 
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  1. 1.
    M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York (1964).zbMATHGoogle Scholar
  2. 2.
    A. Agrachev, U. Boscain, and M. Sigalotti, “A Gauss–Bonnet-like formula on two-dimensional almost Riemannian manifolds,” Discr. Contin. Dynam. Syst. A, 20, 801–822 (2008).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    B. Bonnard and M. Chyba, “Singular trajectories and their role in control theory,” in: Math. Appl., 40, Springer-Verlag, Berlin (2003).Google Scholar
  4. 4.
    B. Bonnard and J.-B. Caillau, Singular Metrics on the Two-Sphere in Space Mechanics, HAL (2008).Google Scholar
  5. 5.
    B. Bonnard, O. Cots, N. Shcherbakova, and D. Sugny, “The energy minimization problem for two-level dissipative quantum systems,” J. Math. Phys., 51 (2010), DOI:  10.1063/1.3479390.MathSciNetGoogle Scholar
  6. 6.
    B. Bonnard, J.-B. Caillau, and O. Cots, “Energy minimization in two-level dissipative quantum control: the integrable case,” in: Proc. 8th AIMS Conf. on Dynamical Systems, Differential Equations and Applications, Dresden (2011), pp. 198–208.Google Scholar
  7. 7.
    U. Boscain and P. Mason, “Time minimal trajectories for a spin 1/2 particle in a magnetic field,” J. Math. Phys., 47, No. 6 (2006).Google Scholar
  8. 8.
    J. B. Caillau, O. Cots, and J. Gergaud, Hampath, Scholar
  9. 9.
    M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston (1992).CrossRefzbMATHGoogle Scholar
  10. 10.
    H. T. Davies, Introduction to Nonlinear Differential and Integral Equations, Dover Publications, New York (1962).Google Scholar
  11. 11.
    D. F. Lawden, Elliptic Functions and Applications, Springer-Verlag, New York (1989).CrossRefzbMATHGoogle Scholar
  12. 12.
    V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton Univ. Press (1960).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne and ENSEEIHT-IRIT INP ToulouseToulouseFrance

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