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

Published online by Cambridge University Press:  24 March 2010

Bernard Bonnard
Affiliation:
Institut de mathématiques de Bourgogne, UMR CNRS 5584, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France.
Nataliya Shcherbakova
Affiliation:
Institut de mathématiques de Bourgogne, UMR CNRS 5584, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France.
Dominique Sugny
Affiliation:
Institut Carnot de Bourgogne, UMR CNRS 5209, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France. Dominique.Sugny@u-bourgogne.fr
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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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2010

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References

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).
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).
Bao, D., Robles, C. and Shen, Z., Zermelo navigation on Riemannian manifolds. J. Differential Geom. 66 (2004) 377435. CrossRef
A.G. Bliss, Lectures on the Calculus of Variations. University of Chicago Press, Chicago, USA (1946).
Bonnard, B. and Kupka, I., 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) 111159. CrossRef
Bonnard, B. and Sugny, D., Time-minimal control of dissipative two-level quantum systems: the integrable case. SIAM J. Control Optim. 48 (2009) 12891308. CrossRef
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.
Bonnard, B., Caillau, J.-B. and Trélat, E., Second order optimality conditions in the smooth case and applications in optimal control. ESAIM: COCV 13 (2007) 207236. CrossRef
Bonnard, B., Caillau, J.-B., Sinclair, R. and Tanaka, M., Conjugate and cut loci of a two-sphere of revolution with application to optimal control. Ann. Inst. H. Poincaré Anal. Non Linéaire 26 (2009) 10811098. CrossRef
Bonnard, B., Chyba, M. and Sugny, D., Time-minimal control of dissipative two-level quantum systems: the generic case. IEEE Trans. Automat. Contr. 54 (2009) 25952610. CrossRef
B. Bonnard, O. Cots, N. Shcherbakova and D. Sugny, The energy minimization problem for two-level dissipative quantum systems. J. Math. Phys. (to appear).
Boscain, U. and Mason, P., Time minimal trajectories for a spin 1/2 particle in a magnetic field. J. Math. Phys. 47 (2006) 062101. CrossRef
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).
Gergaud, J. and Haberkorn, T., Homotopy method for minimum consumption orbit transfer problem. ESAIM: COCV 12 (2006) 294310. CrossRef
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.
Khaneja, N., Glaser, S.J. and Brockett, R., Sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A (3) 65 (2002) 032301. CrossRef
D.F. Lawden, Elliptic functions and applications. Springer Verlag, New York, USA (1989).
Maurer, H. and Oberle, H.J., Second order sufficient conditions for optimal control problems with free final time: the Riccati approach. SIAM J. Control Optim. 41 (2002) 380403. CrossRef
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).
Sarychev, A., The index of the second variation of a control system. Math. Sbornik 41 (1982) 383401. CrossRef
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.
Vieillard, T., Chaussard, F., Sugny, D., Lavorel, B. and Faucher, O., Field-free molecular alignment of CO2 mixtures in presence of collisional relaxation. J. Raman Spec. 39 (2008) 694. CrossRef
Wu, R., Pechen, A., Rabitz, H., Hsieh, M. and Tsou, B., Control landscapes for observable preparation with open quantum systems. J. Math. Phys. 49 (2008) 022108. CrossRef

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