The smooth continuation method in optimal control with an application to quantum systems

Bernard Bonnard; Nataliya Shcherbakova; Dominique Sugny

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 267-292
  • ISSN: 1292-8119

Abstract

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The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.

How to cite

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Bonnard, Bernard, Shcherbakova, Nataliya, and Sugny, Dominique. "The smooth continuation method in optimal control with an application to quantum systems." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 267-292. <http://eudml.org/doc/272928>.

@article{Bonnard2011,
abstract = {The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.},
author = {Bonnard, Bernard, Shcherbakova, Nataliya, Sugny, Dominique},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {optimal control; smooth continuation method; quantum control},
language = {eng},
number = {1},
pages = {267-292},
publisher = {EDP-Sciences},
title = {The smooth continuation method in optimal control with an application to quantum systems},
url = {http://eudml.org/doc/272928},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Bonnard, Bernard
AU - Shcherbakova, Nataliya
AU - Sugny, Dominique
TI - The smooth continuation method in optimal control with an application to quantum systems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 1
SP - 267
EP - 292
AB - The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.
LA - eng
KW - optimal control; smooth continuation method; quantum control
UR - http://eudml.org/doc/272928
ER -

References

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  1. [1] A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences 87, Control Theory and Optimization II. Springer-Verlag, Berlin, Germany (2004). Zbl1062.93001MR2062547
  2. [2] E.L. Allgower and K.G. Georg, Introduction to numerical continuation methods, SIAM Classics in Applied Maths 45. Society for Industrial and Applied Mathematics, Philadelphia, USA (2003). Zbl1036.65047MR2001018
  3. [3] D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds. J. Differential Geom.66 (2004) 377–435. Zbl1078.53073MR2106471
  4. [4] A.G. Bliss, Lectures on the Calculus of Variations. University of Chicago Press, Chicago, USA (1946). Zbl0063.00459MR17881
  5. [5] B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal [Theory of the singularities of the input/output mapping and optimality of singular trajectories in the minimal-time problem]. Forum Math.5 (1993) 111–159. Zbl0779.49025MR1205250
  6. [6] B. Bonnard and D. Sugny, Time-minimal control of dissipative two-level quantum systems: the integrable case. SIAM J. Control Optim.48 (2009) 1289–1308. Zbl1288.81052MR2496977
  7. [7] B. Bonnard and D. Sugny, Geometric optimal control and two-level dissipative quantum systems. Control Cybern. (to appear). Zbl1236.49047MR2779111
  8. [8] B. Bonnard, L. Faubourg and E. Trélat, Mécanique céleste et contrôle des véhicules spatiaux. Springer, Berlin, Germany (2005). Zbl1104.70001
  9. [9] B. Bonnard, R. Dujol and J.-B. Caillau, Smooth approximations of single-input controlled Keplerian trajectories: homotopies and averaging, in Taming heterogeneity and complexity of embedded control, Proceedings of the Joint CTS-HYCON Workshop on Nonlinear and Hybrid Control, Paris, France (2006) 73–95. 
  10. [10] B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV 13 (2007) 207–236. Zbl1123.49014MR2306634
  11. [11] B. Bonnard, J.-B. Caillau, R. Sinclair and M. Tanaka, Conjugate and cut loci of a two-sphere of revolution with application to optimal control. Ann. Inst. H. Poincaré Anal. Non Linéaire26 (2009) 1081–1098. Zbl1184.53036MR2542715
  12. [12] B. Bonnard, M. Chyba and D. Sugny, Time-minimal control of dissipative two-level quantum systems: the generic case. IEEE Trans. Automat. Contr.54 (2009) 2595–2610. Zbl1288.81052MR2571923
  13. [13] B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems. J. Math. Phys. (to appear). Zbl1309.81118MR2742815
  14. [14] U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47 (2006) 062101. Zbl1112.81098MR2239948
  15. [15] H.-P. Breuer and F. Petruccione, The theory of open quantum systems. Oxford University Press, London, UK (2002). Zbl1223.81001MR2012610
  16. [16] D. D'Alessandro, Introduction to quantum control and dynamics, Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, USA (2008). Zbl1139.81001MR2357229
  17. [17] M.P. do Carmo, Riemannian geometry. Birkhauser, Boston, USA (1992). Zbl0752.53001MR1138207
  18. [18] R. Dujol, Contribution du contrôle orbital des transferts mono-entrée en mécanique spatiale. Ph.D. Thesis, ENSEEIHT-INP, France (2006). 
  19. [19] J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem. ESAIM: COCV 12 (2006) 294–310. Zbl1113.49032MR2209355
  20. [20] T. Haberkhorn, Transfert orbital avec minimisation de la consommation : résolution par homotopie différentielle. Ph.D. Thesis, ENSEEIHT-INP, France (2004). 
  21. [21] N. Khaneja, R. Brockett and S.J. Glaser, Time optimal control of spin systems. Phys. Rev. A. 63 (2001) 032308. 
  22. [22] N. Khaneja, S.J. Glaser and R. Brockett, Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A (3) 65 (2002) 032301. MR1891763
  23. [23] D.F. Lawden, Elliptic functions and applications. Springer Verlag, New York, USA (1989). Zbl0689.33001MR1007595
  24. [24] H. Maurer and H.J. Oberle, Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. SIAM J. Control Optim.41 (2002) 380–403. Zbl1012.49018MR1920264
  25. [25] L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, The mathematical theory of optimal processes. L.W. Neustadt Interscience Publishers, John Wiley & Sons, Inc., New York-London (1962). Zbl0117.31702MR166037
  26. [26] A. Sarychev, The index of the second variation of a control system. Math. Sbornik41 (1982) 383–401. Zbl0484.49012
  27. [27] T. Schulte-Herbrüggen, A.K. Spörl, R. Marx, N. Khaneja, J.M. Myers, A.F. Fahmy and S.J. Glaser, Quantum computing implemented via optimal control: Theory and application to spin and pseudo-spin systems, in Lectures on quantum information, D. Bruß and G. Leuchs Eds., Wiley-VCH (2006) 481. Zbl1135.81327
  28. [28] T. Vieillard, F. Chaussard, D. Sugny, B. Lavorel and O. Faucher, Field-free molecular alignment of CO2 mixtures in presence of collisional relaxation. J. Raman Spec. 39 (2008) 694. 
  29. [29] R. Wu, A. Pechen, H. Rabitz, M. Hsieh and B. Tsou, Control landscapes for observable preparation with open quantum systems. J. Math. Phys. 49 (2008) 022108. Zbl1153.81449MR2392845

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