1,126 research outputs found
Geometric optimal control of the contrast imaging problem in Nuclear Magnetic Resonance
The objective of this article is to introduce the tools to analyze the
contrast imaging problem in Nuclear Magnetic Resonance. Optimal trajectories
can be selected among extremal solutions of the Pontryagin Maximum Principle
applied to this Mayer type optimal problem. Such trajectories are associated to
the question of extremizing the transfer time. Hence the optimal problem is
reduced to the analysis of the Hamiltonian dynamics related to singular
extremals and their optimality status. This is illustrated by using the
examples of cerebrospinal fluid / water and grey / white matter of cerebrum.Comment: 30 pages, 13 figur
-Minimization for Mechanical Systems
Second order systems whose drift is defined by the gradient of a given
potential are considered, and minimization of the -norm of the control is
addressed. An analysis of the extremal flow emphasizes the role of singular
trajectories of order two [25,29]; the case of the two-body potential is
treated in detail. In -minimization, regular extremals are associated with
controls whose norm is bang-bang; in order to assess their optimality
properties, sufficient conditions are given for broken extremals and related to
the no-fold conditions of [20]. An example of numerical verification of these
conditions is proposed on a problem coming from space mechanics
Optimality conditions applied to free-time multi-burn optimal orbital transfers
While the Pontryagin Maximum Principle can be used to calculate candidate
extremals for optimal orbital transfer problems, these candidates cannot be
guaranteed to be at least locally optimal unless sufficient optimality
conditions are satisfied. In this paper, through constructing a parameterized
family of extremals around a reference extremal, some second-order necessary
and sufficient conditions for the strong-local optimality of the free-time
multi-burn fuel-optimal transfer are established under certain regularity
assumptions. Moreover, the numerical procedure for computing these optimality
conditions is presented. Finally, two medium-thrust fuel-optimal trajectories
with different number of burn arcs for a typical orbital transfer problem are
computed and the local optimality of the two computed trajectories are tested
thanks to the second-order optimality conditions established in this paper
Time Minimal Trajectories for a Spin 1/2 Particle in a Magnetic Field
In this paper we consider the minimum time population transfer problem for
the -component of the spin of a (spin 1/2) particle driven by a magnetic
field, controlled along the x axis, with bounded amplitude. On the Bloch sphere
(i.e. after a suitable Hopf projection), this problem can be attacked with
techniques of optimal syntheses on 2-D manifolds. Let be the two
energy levels, and the bound on the field amplitude. For
each couple of values and , we determine the time optimal synthesis
starting from the level and we provide the explicit expression of the time
optimal trajectories steering the state one to the state two, in terms of a
parameter that can be computed solving numerically a suitable equation. For
, every time optimal trajectory is bang-bang and in particular the
corresponding control is periodic with frequency of the order of the resonance
frequency . On the other side, for , the time optimal
trajectory steering the state one to the state two is bang-bang with exactly
one switching. Fixed we also prove that for the time needed to
reach the state two tends to zero. In the case there are time optimal
trajectories containing a singular arc. Finally we compare these results with
some known results of Khaneja, Brockett and Glaser and with those obtained by
controlling the magnetic field both on the and directions (or with one
external field, but in the rotating wave approximation). As byproduct we prove
that the qualitative shape of the time optimal synthesis presents different
patterns, that cyclically alternate as , giving a partial proof of a
conjecture formulated in a previous paper.Comment: 31 pages, 10 figures, typos correcte
String cosmology versus standard and inflationary cosmology
This paper presents a review of the basic, model-independent differences
between the pre-big bang scenario, arising naturally in a string cosmology
context, and the standard inflationary scenario. We use an unconventional
approach in which the introduction of technical details is avoided as much as
possible, trying to focus the reader's attention on the main conceptual aspects
of both scenarios. The aim of the paper is not to conclude in favour either of
one or of the other scenario, but to raise questions that are left to the
reader's meditation. Warnings: the paper does not contain equations, and is not
intended as a complete review of all aspects of string cosmology.Comment: 22 pages, Latex (IOP Style), three figures included using epsfig. To
appear in Class. Quantum Grav. (Topical Review Section). Two misprints
correcte
Cosmological perturbations across a curvature bounce
String-inspired cosmologies, whereby a non-singular curvature bounce is
induced by a general-covariant, -duality-invariant, non-local dilaton
potential, are used to study numerically how inhomogeneities evolve and to
compare the outcome with analytic expressions obtained through different
matching conditions across the bounce. Good agreement is found if continuity
across the bounce is assumed to hold for , the curvature perturbation
on comoving hypersurfaces, rather than for the Bardeen potential.Comment: 36 pages, 5 included figure
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