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

Bonnard, Bernard; Chyba, Monique

doi:10.1007/978-4-431-54907-9_5

Cited by 8 publications

(26 citation statements)

References 20 publications

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“…We also note, that during preparation of this manuscript article [21] and preprint [22] appeared, where the same sub-Riemannian and almost-Riemannian problems were studied. Thus it is reasonable to indicate explicitly the novelty of some results in this paper.…”

Section: Introductionmentioning

confidence: 96%

“…We integrate the Hamiltonian system for sub-Riemannian geodesics on SO(3) using a well-known approach from mechanics [29,31], and action-angle coordinates in the dual of so(3) induced by a pendulum [24]. In [21] and [22] the authors gave a similar parameterization using the same technique, but omitted details. Here we give a full derivation for parameterization of sub-Riemannian geodesics and use it to obtain a parameterization for almost-Riemannian geodesics.…”

Section: Introductionmentioning

confidence: 99%

“…In [22] necessary and sufficient optimality conditions are formulated for geodesics which start from the singular set. In this paper we show that the sub-Riemannian structure on SO(3) and almost-Riemannian structure on S 2 share a number of discrete symmetries.…”

Section: Introductionmentioning

confidence: 99%

“…This technique allows us only to give necessary conditions, but we state them for some initial points that lie outside the singular set. We use the parameterization of geodesics from this paper and results from [22] to obtain bounds on the cut time for almost-Riemannian geodesics that start from the singular set and for any sub-Riemannian geodesics on SO (3). Thus the second part of this paper may be considered as complementary to [21] and [22], where authors have proved many interesting results.…”

Section: Introductionmentioning

confidence: 99%

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Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Beschastnyi¹,

Sachkov²

2014

Preprint

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In this paper we study geodesics of left-invariant sub-Riemannian metrics on SO(3) and almost-Riemannian metrics on S 2 . These structures are connected with each other, and it is possible to use information about one of them to obtain results about another one. We give an explicit parameterization of sub-Riemannian geodesics on SO(3) and use it to get a parameterization of almost-Riemannian geodesics on S 2 . We use symmetries of the exponential map to obtain some necessary optimality conditions. We present some upper bounds on the cut time in both cases and describe periodic geodesics on SO(3).

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Section: Introductionmentioning

confidence: 96%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Beschastnyi¹,

Sachkov²

2014

Preprint

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“…Optimal control theory was born in its modern version with the Pontryagin Maximum Principle (PMP) in the late 1950's. Its development was originally inspired by problems of space dynamics, but it is now a key tool to study a large spectrum of applications extending from robotics to economics and biology [2,3,4,5,6,7,8,9,10]. Optimal control problems can be solved by two different types of approaches, geometric [2,3,5] and numerical methods [8,11,12] for dynamical systems of low and high dimension, respectively.…”

Section: Introductionmentioning

confidence: 99%

Optimal control theory for applications in Magnetic Resonance Imaging

et al. 2017

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We apply innovative mathematical tools coming from optimal control theory to improve theoretical and experimental techniques in Magnetic Resonance Imaging (MRI). This approach allows us to explore and to experimentally reach the physical limits of the corresponding spin dynamics in the presence of typical experimental imperfections and limitations. We study in this paper two important goals, namely the optimization of image contrast and the maximization of the signal to noise per unit time. We anticipate that the proposed techniques will find practical applications in medical imaging in a near future to help the medical diagnosis.

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Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

et al. 2017

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In order to detect salient lines in spherical images, we consider the problem of minimizing the functional l 0 C(γ (s)) ξ 2 + k 2 g (s) ds for a curve γ on a sphere with fixed boundary points and directions. The total length l is free, s denotes the spherical arclength, and k g denotes the geodesic curvature of γ . Here the smooth external cost C ≥ δ > 0 is obtained from spherical data. We lift this problem to the sub-Riemannian (SR) problem in Lie group SO(3) and show that the spherical projection of certain SR geodesics provides a solution to our curve optimization problem. In fact, this holds only for the geodesics whose spherical projection does not exhibit a cusp. The problem is a spherical extension of a well-known contour perception model, where we extend the model by Boscain and Rossi to the general case ξ > 0, C = 1. For C = 1, we derive SR geodesics and evaluate the first cusp time. We show that these curves have a simpler expression when they are parameterized by spherical arclength rather than by sub-Riemannian arclength. For case C = 1 (data-driven SR geodesics), we solve via a SR Fast Marching method. Finally, we show an experiment of vessel tracking in a spherical image of the retina and study the effect of including the spherical geometry in analysis of vessels curvature.

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

Cited by 8 publications

References 20 publications

Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Sub-Riemannian and almost-Riemannian geodesics on SO(3) and $S^2$

Optimal control theory for applications in Magnetic Resonance Imaging

Tracking of Lines in Spherical Images via Sub-Riemannian Geodesics in $${\text {SO(3)}}$$ SO(3)

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