Search results for " Control"
showing 10 items of 7691 documents
Two applications of geometric optimal control to the dynamics of spin particle
2013
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.
Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging
2018
In this article, we analyse the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation meth…
A global optimality result with application to orbital transfer
2006
The objective of this note is to present a global optimality result on Riemannian metrics $ds^2=dr^2+(r^2/c^2)(G(\vphi)d\theta^2+d\vphi^2)$. This result can be applied to the averaged energy minimization coplanar orbit transfer problem.
Feedback equivalence and the contrast problem in nuclear magnetic resonance imaging
2013
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.
Description of accessibility sets near an abnormal trajectory and consequences
2002
We describe precisely, under generic conditions, the contact of the accessibility set at time $T$ with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $L^\infty$-sector and the $L^2$-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.
Cotcot: short reference manual
2005
Technical report; This reference introduces the Matlab package COTCOT designed to compute extremals in the case of smooth Hamiltonian systems, and to obtain the associated conjugate points with respect to the index performance of the underlying optimal control problem.
The transcendence required for computing the sphere and wave front in the Martinet sub-Riemannian geometry
1999
Consider a \it{sub-Riemannian geometry} $(U,D,g)$ where $U$ is a neighborhood of $O$ in $\mathbb{R}^3$, $D$ is a \it{Martinet type distribution} identified to $Ker \,\omega$, $\omega =dz-\f{y^2}{2}dx$, $q=(x,y,z)$ and $g$ is a \it{metric on $D$} which can be taken in the normal form : \mbox{$a(q)dx^2+c(q)dy^2$}, \mbox{$a=1+yF(q)$}, \mbox{$c=1+G(q)$}, \mbox{$G_{|x=y=0}=0$}. In a previous article we analyzed the \it{flat case} : \mbox{$a=c=1$} ; we showed that the set of geodesics is integrable using \it{elliptic integrals} of the \it{first and second kind} ; moreover we described the sphere and the wave front near the abnormal direction using the \it{\mbox{exp-log} category}. The objective o…
Optimality of singular trajectories and asymptotics of accessibility sets under generic assumptions
2002
We investigate minimization problems along a singular trajectory of a single-input affine control system with constraint on the control, and then as an application of a sub-Riemannian system of rank 2. Under generic assumptions we get necessary and sufficient conditions for optimality of such a singular trajectory. Moreover we describe precisely the contact of the accessibility sets at time $T$ with the singular direction. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer $\gamma$ into two sectors, bordered by the first Pontryagin's cone along $\gamma$, called the $L^\infty$-sector and the $L^2$-sector.
Geometric and numerical techniques in optimal control of the two and three-body problems
2009
The objective of this article is to present geometric and numerical techniques developed to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used. We concentrate our study on the energy minimization problem. From Pontryagin's maximum principle, the optimal solution can be found solving the shooting equation for smooth Hamiltonian dynamics. A first step in the analysis is to find in the Kepler case an analytical solution for the averaged Hamiltonian, which corresponds to a Riemannian metric. This will allow to compute the solution for the original Kepler problem, using a nume…
Optimal Control Theory and the Swimming Mechanism of the Copepod Zooplankton
2016
In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the motion occurs minimizing the energy dissipated by the fluid drag forces in relation with the concept of efficiency of a stroke. The maximum principle is used to compute periodic controls candidates as minimizing controls and is a decisive tool combined with appropriate numerical simulations using indirect optimal control schemes to determine the most efficient stroke compared with standard computations usi…