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
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topBonnard, 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/197281>.
@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; optimal control},
language = {eng},
month = {2},
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/197281},
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
DA - 2011/2//
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; optimal control
UR - http://eudml.org/doc/197281
ER -
References
top- A.A. Agrachev and Y.L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences87, Control Theory and Optimization II. Springer-Verlag, Berlin, Germany (2004).
- E.L. Allgower and K.G. Georg, Introduction to numerical continuation methods, SIAM Classics in Applied Maths45. Society for Industrial and Applied Mathematics, Philadelphia, USA (2003).
- D. Bao, C. Robles and Z. Shen, Zermelo navigation on Riemannian manifolds. J. Differential Geom.66 (2004) 377–435.
- A.G. Bliss, Lectures on the Calculus of Variations. University of Chicago Press, Chicago, USA (1946).
- 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.
- 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.
- B. Bonnard and D. Sugny, Geometric optimal control and two-level dissipative quantum systems. Control Cybern. (to appear).
- B. Bonnard, L. Faubourg and E. Trélat, Mécanique céleste et contrôle des véhicules spatiaux. Springer, Berlin, Germany (2005).
- 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.
- B. Bonnard, J.-B. Caillau and E. Trélat, Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV13 (2007) 207–236.
- 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.
- 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.
- B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems. J. Math. Phys. (to appear).
- U. Boscain and P. Mason, Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys.47 (2006) 062101.
- H.-P. Breuer and F. Petruccione, The theory of open quantum systems. Oxford University Press, London, UK (2002).
- D. D'Alessandro, Introduction to quantum control and dynamics, Applied Mathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, USA (2008).
- M.P. do Carmo, Riemannian geometry. Birkhauser, Boston, USA (1992).
- R. Dujol, Contribution du contrôle orbital des transferts mono-entrée en mécanique spatiale. Ph.D. Thesis, ENSEEIHT-INP, France (2006).
- J. Gergaud and T. Haberkorn, Homotopy method for minimum consumption orbit transfer problem. ESAIM: COCV12 (2006) 294–310.
- T. Haberkhorn, Transfert orbital avec minimisation de la consommation : résolution par homotopie différentielle. Ph.D. Thesis, ENSEEIHT-INP, France (2004).
- N. Khaneja, R. Brockett and S.J. Glaser, Time optimal control of spin systems. Phys. Rev. A. 63 (2001) 032308.
- 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.
- D.F. Lawden, Elliptic functions and applications. Springer Verlag, New York, USA (1989).
- 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.
- 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).
- A. Sarychev, The index of the second variation of a control system. Math. Sbornik41 (1982) 383–401.
- 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.
- 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.
- 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.
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