Abstract
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.
Highlights
Over the past decade, the application of geometric optimal control techniques to the dynamics of spin systems with applications to Nuclear Magnetic Resonance (NMR) spectroscopy and quantum information processing [17] has been an intense research direction
Restricting to the case of three spins, the objective of this article is to provide the preliminary work to compute the optimal solutions parametrized by Pontryagin Maximum Principle
Using the seminal work in [13], we define a chart that identifies locally SO(3) to S2 × S1 which enlightens the geometry of the problem and leads to an explicit computation of the extremals using elliptic functions
Summary
The application of geometric optimal control techniques to the dynamics of spin systems with applications to Nuclear Magnetic Resonance (NMR) spectroscopy and quantum information processing [17] has been an intense research direction. Using the seminal work in [13], we define a chart that identifies locally SO(3) to S2 × S1 which enlightens the geometry of the problem and leads to an explicit computation of the extremals using elliptic functions Another approach consists in integrating the system directly on S2. Which consist in steering the third axis of the frame R from e1 to e3, where (ei) is the canonical basis of R3 It can be transformed into a left-invariant control problem to use the geometric framework and the computations in [13] : dR. The optimal solutions to our problem can be parametrized by the Pontryagin Maximum Principle [19], and thanks to the explicit formula given in [13], the solutions can be computed in both the Riemannian and the sub-Riemannian cases using elliptic functions.
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