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September  2015, 35(9): 4095-4114. doi: 10.3934/dcds.2015.35.4095

Integrability methods in the time minimal coherence transfer for Ising chains of three spins

Bernard Bonnard 1, , Thierry Combot 1, and  Lionel Jassionnesse 1,

1. 

Institut de mathématiques de Bourgogne, UMR 5584 CNRS Université de Bourgogne, Faculté des Sciences Mirande 9 avenue Alain Savary, France, France, France

Received  April 2014 Revised  October 2014 Published  April 2015

The objective of this article is to analyze the integrability properties of extremal solutions of Pontryagin Maximum Principle in the time minimal control of a linear spin system with Ising coupling in relation with conjugate and cut loci computations. Restricting to the case of three spins, the problem is equivalent to analyze a family of almost-Riemannian metrics on the sphere $S^{2}$, with Grushin equatorial singularity. The problem can be lifted into a SR-invariant problem on $SO(3)$, this leads to a complete understanding of the geometry of the problem and to an explicit parametrization of the extremals using an appropriate chart as well as elliptic functions. This approach is compared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville normal form. Finally, an algebraic approach is presented in the framework of the application of differential Galois theory to integrability.
Keywords: Ising chains of spins, Liouville metrics, SR-geometry, differential Galois theory and integrability., Geometric optimal control.
Mathematics Subject Classification: Primary: 49K15, 70H06; Secondary: 33E0.
Citation: Bernard Bonnard, Thierry Combot, Lionel Jassionnesse. Integrability methods in the time minimal coherence transfer for Ising chains of three spins. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4095-4114. doi: 10.3934/dcds.2015.35.4095
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show all references

References:
[1]

Discrete Contin. Dyn. Syst., 20 (2008), 801-822. doi: 10.3934/dcds.2008.20.801.  Google Scholar

[2]

Translated from the Russian by K. Vogtmann and A. Weinstein. Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.  Google Scholar

[3]

Cours Spécialisés [Specialized Courses], 8. Société Mathématique de France, Paris; EDP Sciences, Les Ulis, 2001. viii+170 pp.  Google Scholar

[4]

American society colloquium publications, vol. IX, 1927. Google Scholar

[5]

Monographs in contemporary mathematrics, Kluwer Academic, 2000. doi: 10.1007/978-1-4615-4307-7.  Google Scholar

[6]

Ann. Mat. Pura Appl., 193 (2014), 1353-1382. doi: 10.1007/s10231-013-0333-y.  Google Scholar

[7]

Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series, 5 (2014), 53-72. doi: 10.1007/978-3-319-02132-4_4.  Google Scholar

[8]

ESAIM Control Optim. Calc. Var., 20 (2014), 864-893. doi: 10.1051/cocv/2013087.  Google Scholar

[9]

Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 957-990. doi: 10.3934/dcdsb.2005.5.957.  Google Scholar

[10]

Manuscripta Math., 114 (2004), 247-264. doi: 10.1007/s00229-004-0455-z.  Google Scholar

[11]

Manuscripta Math., 136 (2011), 115-141. doi: 10.1007/s00229-011-0433-1.  Google Scholar

[12]

Ph.D thesis, Université de Bourgogne, 2014. Google Scholar

[13]

Studies in Advanced Mathematics, 52. Cambridge University Press, Cambridge, 1997. xviii+492 pp.  Google Scholar

[14]

Phys. Rev. A (3), 65 (2002), part A, 032301, 11 pp. doi: 10.1103/PhysRevA.65.032301.  Google Scholar

[15]

J. Symbolic Comput., 2 (1986), 3-43. doi: 10.1016/S0747-7171(86)80010-4.  Google Scholar

[16]

Applied Mathematical Sciences, 80. Springer-Verlag, New York, 1989. xiv+334 pp. doi: 10.1007/978-1-4757-3980-0.  Google Scholar

[17]

(2001) Wiley (686 pages). Google Scholar

[18]

Differential algebra, complex analysis and orthogonal polynomials, Contemp. Math., Amer. Math. Soc., Providence, RI, 509 (2010), 143-220. doi: 10.1090/conm/509/09980.  Google Scholar

[19]

Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, Inc. New York-London, 1962.  Google Scholar

[20]

J. Symbolic Comput., 16 (1993), 9-36. doi: 10.1006/jsco.1993.1032.  Google Scholar

[21]

Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-642-55750-7.  Google Scholar

[22]

Ph.D thesis, Harvard, 2006.  Google Scholar

[23]

Phys. Rev. A, 77, (2008), 032340. doi: 10.1103/PhysRevA.77.032340.  Google Scholar

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