Weak Maximum Principle and Application to Swimming at Low Reynolds Number

We refer to [9, 42, 46] for more details about the general concepts and notations introduced in this section.

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

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.

Maximum Principle and Application to Nuclear Magnetic Resonance and Magnetic Resonance Imaging

In this section we state the Pontryagin maximum principle and we outline the proof. We adopt the presentation from Lee and Markus [64] where the result is presented into two theorems.

Nuclear magnetic resonance: The contrast imaging problem

Starting as a tool for characterization of organic molecules, the use of NMR has spread to areas as diverse as pharmacology, medical diagnostics (medical resonance imaging) and structural biology. Recent advancements on the study of spin dynamics strongly suggest the efficiency of geometric control theory to analyze the optimal synthesis. This paper focuses on a new approach to the contrast imaging problem using tools from geometric optimal control. It concerns the study of an uncoupled two-spin system and the problem is to bring one spin to the origin of the Bloch ball while maximizing the modulus of the magnetization vector of the second spin. It can be stated as a Mayer-type optimal prob…

Numerical Approach to the Optimal Control and Efficiency of the Copepod Swimmer

International audience

Two applications of geometric optimal control to the dynamics of spin particle

To appear in a volume of "Math and Industry", Springer-Verlag; 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.

Historical Part—Calculus of Variations

The calculus of variations is an old mathematical discipline and historically finds its origins in the introduction of the brachistochrone problem at the end of the 17th century by Johann Bernoulli to challenge his contemporaries to solve it. Here, we briefly introduce the reader to the main results.

Méthodes géométriques et analytiques pour étudier l'application exponentielle, la sphère et le front d'onde en géométrie sous-riemannienne dans le cas Martinet

Consider a sub-riemannian geometry (U,D,g) where U is a neighborhood of 0 in R 3 , D is a Martinet type distribution identified to ker ω , ω being the 1-form: , q=(x,y,z) and g is a metric on D which can be taken in the normal form : , a=1+yF(q) , c=1+G(q) , . In a previous article we analyze the flat case : a=c=1 ; we describe the conjugate and cut loci , the sphere and the wave front . The objectif of this article is to provide a geometric and computational framework to analyze the general case. This frame is obtained by analysing three one parameter deformations of the flat case which clarify the role of the three parameters in the gradated normal form of order 0 where: , . More generall…

Sub-Riemannian geometry: one-parameter deformation of the Martinet flat case

Feedback equivalence and the contrast problem in nuclear magnetic resonance imaging

International audience; The theoretical analysis of the contrast problem in NMR imaging is mainly reduced, thanks to the Maximum Principle, to the analysis of the so-called singular trajectories of the control system modeling the problem: a coupling of two Bloch equations representing the evolution of the magnetization vector of each spin particle. They are solutions of a constrained Hamiltonian equation. In this article we describe feedback invariants related to the singular flow to distinguish the different cases occurring in physical experiments.