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Mathematical Control & Related Fields

December 2013 , Volume 3 , Issue 4

Special issue in the honor of Bernard Bonnard. Part II.

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On the application of geometric optimal control theory to Nuclear Magnetic Resonance
Elie Assémat, Marc Lapert, Dominique Sugny and Steffen J. Glaser
2013, 3(4): 375-396 doi: 10.3934/mcrf.2013.3.375 +[Abstract](790) +[PDF](576.7KB)
Abstract:
We present some applications of geometric optimal control theory to control problems in Nuclear Magnetic Resonance (NMR) and Magnetic Resonance Imaging (MRI). Using the Pontryagin Maximum Principle (PMP), the optimal trajectories are found as solutions of a pseudo-Hamiltonian system. This computation can be completed by second-order optimality conditions based on the concept of conjugate points. After a brief physical introduction to NMR, this approach is applied to analyze two relevant optimal control issues in NMR and MRI: the control of a spin 1/2 particle in presence of radiation damping effect and the maximization of the contrast in MRI. The theoretical analysis is completed by numerical computations. This work has been made possible by the central and essential role of B. Bonnard, who has been at the heart of this project since 2009.
Algebraic geometric classification of the singular flow in the contrast imaging problem in nuclear magnetic resonance
Bernard Bonnard, Monique Chyba, Alain Jacquemard and John Marriott
2013, 3(4): 397-432 doi: 10.3934/mcrf.2013.3.397 +[Abstract](616) +[PDF](1410.8KB)
Abstract:
The analysis of the contrast problem in NMR medical imaging is essentially reduced to the analysis of the so-called singular trajectories of the system modeling the problem: a coupling of two spin 1/2 control systems. They are solutions of a constraint Hamiltonian vector field and restricting the dynamics to the zero level set of the Hamiltonian they define a vector field on $B_1 \times B_2$, where $B_1$ and $B_2$ are the Bloch balls of the two spin particles. In this article we classify the behaviors of the solutions in relation with the relaxation parameters using the concept of feedback classification. The optimality status is analyzed using the feedback invariant concept of conjugate points.
Controllability with quadratic drift
Roger Brockett
2013, 3(4): 433-446 doi: 10.3934/mcrf.2013.3.433 +[Abstract](521) +[PDF](346.2KB)
Abstract:
We study the controllability properties of systems of the form $ \dot{x}=Ax+Bu\;;\; \dot{w}=q(x) $ with $q$ being a vector of quadratic functions of $x$. This class of nonlinear systems is interesting because it is both remarkably tractable and because it is the second order approximation to a larger class of nonlinear systems. We not only describe the distribution generated by the vector fields associated with this system but, in important cases, we are able to give a precise description of which points are reachable from a given initial state, distinguishing between those points that are reachable immediately and those that are only reachable after a sufficient length of time.
Sparse stabilization and optimal control of the Cucker-Smale model
Marco Caponigro, Massimo Fornasier, Benedetto Piccoli and Emmanuel Trélat
2013, 3(4): 447-466 doi: 10.3934/mcrf.2013.3.447 +[Abstract](1241) +[PDF](467.3KB)
Abstract:
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
    We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
    We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
Ricci curvatures in Carnot groups
Ludovic Rifford
2013, 3(4): 467-487 doi: 10.3934/mcrf.2013.3.467 +[Abstract](581) +[PDF](275.2KB)
Abstract:
We study metric contraction properties for metric spaces associated with left-invariant sub-Riemannian metrics on Carnot groups. We show that ideal sub-Riemannian structures on Carnot groups satisfy such properties and give a lower bound of possible curvature exponents in terms of the datas.

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