Advertisement

Two Applications of Geometric Optimal Control to the Dynamics of Spin Particles

Conference paper
Part of the Mathematics for Industry book series (MFI, volume 1)

Abstract

The purpose of this article is to present the application of methods from geometric optimal control to two problems in the dynamics of spin particles. First, we consider the saturation problem for a single spin system and second, the control of a linear chain of spin particles with Ising couplings. For both problems the minimizers are parameterized using Pontryagin Maximum Principle and the optimal solution is found by a careful analysis of the corresponding equations.

Keywords

Optimal control Bloch equations Saturation problem Dynamics of spins particles 

References

  1. 1.
    Agrachev, A., Sachkov, Y.: Control Theory from the Geometric Viewpoint, vol 87 of Encyclopaedia of Mathematical Sciences. Control Theory and Optimization, II, xiv+412 pp. Springer, Berlin (2004)Google Scholar
  2. 2.
    Agrachev, A., Boscain, U., Sigalotti, M.: A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete Contin. Dyn. Syst. 20(4), 801–822 (2008)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Arnold, V.I.: Mathematical methods of classical mechanics, Translated from the Russian by Vogtmann, K., and Weinstein, A., 2nd edn. Graduate Texts in Mathematics, vol 60, xvi+508 pp. Springer, New York (1989)Google Scholar
  4. 4.
    Birkhoff, G.D.: Dynamical Systems, vol IX. American society colloquium publications (1927)Google Scholar
  5. 5.
    Bolsinov, V., Fomenko, A.T.: Integrable geodesic flows on two-dimensional surfaces. In: Monographs in Contemporary Mathematrics. Kluwer Academic, Dordretcht (2000)Google Scholar
  6. 6.
    Bonnard, B., Chyba, M.: Singular Trajectories and their Role in Control Theory, Vol 40 of Mathématiques & Applications, xvi+357 pp. Springer, Berlin (2003)Google Scholar
  7. 7.
    Bonnard, B., Caillau, J.B., Sinclair, R., Tanaka, M.: Tanaka Conjugate and cut loci of a two-sphere of revolution with application to optimal control. Ann. Inst. H. Poincaré Anal. Non Linéaire 26(4), 1081–1098 (2009)Google Scholar
  8. 8.
    Bonnard, B., Caillau, J.B., Janin, G.: Conjugate-cut loci and injectivity domains on two-spheres of revolution. ESAIM Control Optim. Calc. Var. 19(2), 533–554 (2013)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Bonnard, B., Chyba, M., Marriott, J.: Singular trajectories and the contrast imaging problem in nuclear magnetic resonance. SIAM J. Control Optim. 51(2), 1325–1349 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Bonnard, B., Cots, O., Jassionnesse, L.: Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces. In: Stefani, G., Boscain, U., Gauthier, J.-P., Sarychev, A., Sigalotti, M. (eds.) Geometric Control Theory and sub-Riemannian Geometry. Springer INdAMSeries 5, pp. 53–72. Springer International Publishing, Switzerland (2014). doi: 10.1007/978-3-319-02132-4_4 CrossRefGoogle Scholar
  11. 11.
    Bonnard, B., Cots, O., Pomet, J.-B., Shcherbakova, N.: Riemannian metrics on 2d-manifolds related to the Euler-Poinsot rigid body motion. ESAIM Control Optim. Calc. Var. (to appear 2014)Google Scholar
  12. 12.
    Bonnnard, B., Caillau, J.B.: Metrics with equatorial singularities on the sphere. Annali di Matematica (to appear 2014). DOI 10.1007/s10231-013-0333-y
  13. 13.
    Boscain, U., Charlot, G., Gauthier, J.P., Guérin, S., Jauslin, H.-R.: Optimal control in laser-induced population transfer for two- and three-level quantum systems. J. Math. Phys. 43(5), 2107–2132 (2002)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Boscain, U., Chambrion, T., Charlot, G.: Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. Discrete Contin. Dyn. Syst. Ser. B 5(4), 957–990 (electronic, 2005)Google Scholar
  15. 15.
    Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, vol 80 of Pure and Applied Mathematics, xv+628 pp. Academic Press, Inc., Harcourt Brace Jovanovich Publishers, New York (1978)Google Scholar
  16. 16.
    Itoh, J.-I., Kiyohara, K.: The cut loci and the conjugate loci on ellipsoids. Manuscripta Math. 114(2), 247–264 (2004)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Itoh, J.-I., Jin-ichi, K.: Kiyohara cut loci and conjugate loci on Liouville surfaces. Manuscripta Math. 136(1–2), 115–141 (2011)Google Scholar
  18. 18.
    Jurdjevic, V.: Geometric Control Theory, vol 52 of Cambridge Studies in Advanced Mathematics, xviii+492 pp. Cambridge University Press, Cambridge (1997)Google Scholar
  19. 19.
    Khaneja, N., Glaser, S.J., Steffen, R.: Brockett sub-Riemannian geometry and time optimal control of three spin systems: quantum gates and coherence transfer. Phys. Rev. A 65(3), part A, 032301, 11 pp (2002)Google Scholar
  20. 20.
    Lapert, M., Zhang, Y., Braun, M., Glaser, S.J., Sugny, D.: Singular extremals for the time-optimal control of dissipative spin particles. Phys. Rev. Lett. 104, 083001 (2010)CrossRefGoogle Scholar
  21. 21.
    Lapert, M., Zhang, Y., Janich, M., Glaser, S.J., Sugny, D.: Exploring the physical limits of saturation contrast in magnetic resonance imaging. Nature- Sci. Rep. 2, 589 (2012)Google Scholar
  22. 22.
    Lawden, D.F.: Elliptic Functions and Applications, vol 80 of Applied Mathematical Sciences, xiv+334 pp. Springer, New York (1989)Google Scholar
  23. 23.
    Levitt, M.H.: Spin Dynamics—Basics of Nuclear Magnetic Resonance. Wiley, New York (2001). 686 ppGoogle Scholar
  24. 24.
    Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes, viii+360 pp. Translated from the Russian by Trirogoff, K.N. L. W. Neustadt Interscience Publishers, Wiley, New York (1962)Google Scholar
  25. 25.
    Yuan, H.: Geometry, Optimal Control and Quantum Computing. Harvard Ph.D. thesis (2006)Google Scholar
  26. 26.
    Yuan, H., Zeier, R., Khaneja, N.: Elliptic functions and efficient control of Ising spin chains with unequal couplings. Phys. Rev. A 77, 032340 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.Institut de Mathématiques de Bourgogne, CNRS 5584Université de BourgogneDijonFrance
  2. 2.INRIA Sophia AntipolisSophia Antipolis CedexFrance
  3. 3.University of HawaiiHonoluluUSA

Personalised recommendations