388 research outputs found
Time Optimal Synthesis for Left--Invariant Control Systems on SO(3)
Consider the control system given by , where ,
and define two perpendicular left-invariant vector
fields normalized so that \|f\|=\cos(\al) and \|g\|=\sin(\al), \al\in
]0,\pi/4[. In this paper, we provide an upper bound and a lower bound for
, the maximum number of switchings for time-optimal trajectories.
More precisely, we show that N_S(\al)\leq N(\al)\leq N_S(\al)+4, where
N_S(\al) is a suitable integer function of \al which for \al\to 0 is of
order The result is obtained by studying the time optimal
synthesis of a projected control problem on , where the projection is
defined by an appropriate Hopf fibration. Finally, we study the projected
control problem on the unit sphere . It exhibits interesting features
which will be partly rigorously derived and partially described by numerical
simulations
Invariant Carnot-Caratheodory metrics on , , and lens spaces
In this paper we study the invariant Carnot-Caratheodory metrics on
, and induced by their Cartan decomposition
and by the Killing form. Beside computing explicitly geodesics and conjugate
loci, we compute the cut loci (globally) and we give the expression of the
Carnot-Caratheodory distance as the inverse of an elementary function. We then
prove that the metric given on projects on the so called lens spaces
. Also for lens spaces, we compute the cut loci (globally).
For the cut locus is a maximal circle without one point. In all other
cases the cut locus is a stratified set. To our knowledge, this is the first
explicit computation of the whole cut locus in sub-Riemannian geometry, except
for the trivial case of the Heisenberg group
Self-adjoint extensions and stochastic completeness of the Laplace-Beltrami operator on conic and anticonic surfaces
We study the evolution of the heat and of a free quantum particle (described
by the Schr\"odinger equation) on two-dimensional manifolds endowed with the
degenerate Riemannian metric , where , and the parameter . For
this metric describes cone-like manifolds (for it is
a flat cone). For it is a cylinder. For it is a
Grushin-like metric. We show that the Laplace-Beltrami operator is
essentially self-adjoint if and only if . In this case the
only self-adjoint extension is the Friedrichs extension , that does
not allow communication through the singular set both for the heat
and for a quantum particle. For we show that for the
Schr\"odinger equation only the average on of the wave function can
cross the singular set, while the solutions of the only Markovian extension of
the heat equation (which indeed is ) cannot. For we
prove that there exists a canonical self-adjoint extension , called
bridging extension, which is Markovian and allows the complete communication
through the singularity (both of the heat and of a quantum particle). Also, we
study the stochastic completeness (i.e., conservation of the norm for the
heat equation) of the Markovian extensions and , proving
that is stochastically complete at the singularity if and only if
, while is always stochastically complete at the
singularity.Comment: 29 pages, 2 figures, accepted versio
Normal forms and invariants for 2-dimensional almost-Riemannian structures
Two-dimensional almost-Riemannian structures are generalized Riemannian
structures on surfaces for which a local orthonormal frame is given by a Lie
bracket generating pair of vector fields that can become collinear.
Generically, there are three types of points: Riemannian points where the two
vector fields are linearly independent, Grushin points where the two vector
fields are collinear but their Lie bracket is not, and tangency points where
the two vector fields and their Lie bracket are collinear and the missing
direction is obtained with one more bracket. In this paper we consider the
problem of finding normal forms and functional invariants at each type of
point. We also require that functional invariants are "complete" in the sense
that they permit to recognize locally isometric structures. The problem happens
to be equivalent to the one of finding a smooth canonical parameterized curve
passing through the point and being transversal to the distribution. For
Riemannian points such that the gradient of the Gaussian curvature is
different from zero, we use the level set of as support of the
parameterized curve. For Riemannian points such that the gradient of the
curvature vanishes (and under additional generic conditions), we use a curve
which is found by looking for crests and valleys of the curvature. For Grushin
points we use the set where the vector fields are parallel. Tangency points are
the most complicated to deal with. The cut locus from the tangency point is not
a good candidate as canonical parameterized curve since it is known to be
non-smooth. Thus, we analyse the cut locus from the singular set and we prove
that it is not smooth either. A good candidate appears to be a curve which is
found by looking for crests and valleys of the Gaussian curvature. We prove
that the support of such a curve is uniquely determined and has a canonical
parametrization
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
Existence of planar curves minimizing length and curvature
In this paper we consider the problem of reconstructing a curve that is
partially hidden or corrupted by minimizing the functional , depending both on length and curvature . We fix
starting and ending points as well as initial and final directions.
For this functional we discuss the problem of existence of minimizers on
various functional spaces. We find non-existence of minimizers in cases in
which initial and final directions are considered with orientation. In this
case, minimizing sequences of trajectories can converge to curves with angles.
We instead prove existence of minimizers for the "time-reparameterized"
functional \int \| \dot\gamma(t) \|\sqrt{1+K_\ga^2} dt for all boundary
conditions if initial and final directions are considered regardless to
orientation. In this case, minimizers can present cusps (at most two) but not
angles
On 2-step, corank 2 nilpotent sub-Riemannian metrics
In this paper we study the nilpotent 2-step, corank 2 sub-Riemannian metrics
that are nilpotent approximations of general sub-Riemannian metrics. We exhibit
optimal syntheses for these problems. It turns out that in general the cut time
is not equal to the first conjugate time but has a simple explicit expression.
As a byproduct of this study we get some smoothness properties of the spherical
Hausdorff measure in the case of a generic 6 dimensional, 2-step corank 2
sub-Riemannian metric
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