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

Abstract : The objective of this article is to analyze the integrability proper-ties of extremal solutions of Pontryagin Maximum Principle in the time min-imal control of a linear spin system with Ising coupling in relation with con-jugate 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 com-pared with the direct analysis of the Liouville metrics on the sphere where the parametrization of the extremals is obtained by computing a Liouville nor-mal form. Finally, an algebraic approach is presented in the framework of the application of differential Galois theory to integrability.
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Bernard Bonnard, Thierry Combot, Lionel Jassionnesse. Integrability methods in the time minimal coherence transfer for Ising chains of three spins. Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2015, Special issue on optimal control and related fields, 35 (9), pp.4095-4114. ⟨10.3934/dcds.2015.35.4095⟩. ⟨hal-00969285v3⟩

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