632 research outputs found
Nonisotropic 3-level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy
We apply techniques of subriemannian geometry on Lie groups and of optimal
synthesis on 2-D manifolds to the population transfer problem in a three-level
quantum system driven by two laser pulses, of arbitrary shape and frequency. In
the rotating wave approximation, we consider a nonisotropic model i.e. a model
in which the two coupling constants of the lasers are different. The aim is to
induce transitions from the first to the third level, minimizing 1) the time of
the transition (with bounded laser amplitudes),
2) the energy of lasers (with fixed final time). After reducing the problem
to real variables, for the purpose 1) we develop a theory of time optimal
syntheses for distributional problem on 2-D-manifolds, while for the purpose 2)
we use techniques of subriemannian geometry on 3-D Lie groups. The complete
optimal syntheses are computed.Comment: 29 pages, 6 figure
On Time-optimal Trajectories for a Car-like Robot with One Trailer
In addition to the theoretical value of challenging optimal control problmes,
recent progress in autonomous vehicles mandates further research in optimal
motion planning for wheeled vehicles. Since current numerical optimal control
techniques suffer from either the curse of dimens ionality, e.g. the
Hamilton-Jacobi-Bellman equation, or the curse of complexity, e.g.
pseudospectral optimal control and max-plus methods, analytical
characterization of geodesics for wheeled vehicles becomes important not only
from a theoretical point of view but also from a prac tical one. Such an
analytical characterization provides a fast motion planning algorithm that can
be used in robust feedback loops. In this work, we use the Pontryagin Maximum
Principle to characterize extremal trajectories, i.e. candidate geodesics, for
a car-like robot with one trailer. We use time as the distance function. In
spite of partial progress, this problem has remained open in the past two
decades. Besides straight motion and turn with maximum allowed curvature, we
identify planar elastica as the third piece of motion that occurs along our
extr emals. We give a detailed characterization of such curves, a special case
of which, called \emph{merging curve}, connects maximum curvature turns to
straight line segments. The structure of extremals in our case is revealed
through analytical integration of the system and adjoint equations
Extremal polynomials in stratified groups
We introduce a family of extremal polynomials associated with the prolongation of a stratified nilpotent Lie algebra. These polynomials tre related to a new algebraic characterization of abnormal sub-Riemannian extremals in stratified nilpotent Lie groups. They satisfy a set of remarkable structure relations that are used to integrate the adjoint equations, in both normal and abnormal case
Time-Minimal Control of Dissipative Two-level Quantum Systems: the Generic Case
The objective of this article is to complete preliminary results concerning
the time-minimal control of dissipative two-level quantum systems whose
dynamics is governed by Lindblad equations. The extremal system is described by
a 3D-Hamiltonian depending upon three parameters. We combine geometric
techniques with numerical simulations to deduce the optimal solutions.Comment: 24 pages, 16 figures. submitted to IEEE transactions on automatic
contro
Extremal curves in nilpotent Lie groups
We classify extremal curves in free nilpotent Lie groups. The classification
is obtained via an explicit integration of the adjoint equation in Pontryagin
Maximum Principle. It turns out that abnormal extremals are precisely the
horizontal curves contained in algebraic varieties of a specific type. We also
extend the results to the nonfree case.Comment: 30 pages, final versio
Sard Property for the endpoint map on some Carnot groups
In Carnot-Caratheodory or sub-Riemannian geometry, one of the major open
problems is whether the conclusions of Sard's theorem holds for the endpoint
map, a canonical map from an infinite-dimensional path space to the underlying
finite-dimensional manifold. The set of critical values for the endpoint map is
also known as abnormal set, being the set of endpoints of abnormal extremals
leaving the base point. We prove that a strong version of Sard's property holds
for all step-2 Carnot groups and several other classes of Lie groups endowed
with left-invariant distributions. Namely, we prove that the abnormal set lies
in a proper analytic subvariety. In doing so we examine several
characterizations of the abnormal set in the case of Lie groups.Comment: 39 page
Rolling balls and Octonions
In this semi-expository paper we disclose hidden symmetries of a classical
nonholonomic kinematic model and try to explain geometric meaning of basic
invariants of vector distributions
A note on Carnot geodesics in nilpotent Lie groups
We show that strictly abnormal geodesics arise in graded nilpotent Lie
groups. We construct such a group, for which some Carnot geodesics are strictly
abnormal; in fact, they are not normal in any subgroup. In the step-2 case we
also prove that these geodesics are always smooth. Our main technique is based
on the equations for the normal and abnormal curves, that we derive (for any
Lie group) explicitly in terms of the structure constants
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