184 research outputs found
Well-posed infinite horizon variational problems on a compact manifold
We give an effective sufficient condition for a variational problem with
infinite horizon on a compact Riemannian manifold M to admit a smooth optimal
synthesis, i. e. a smooth dynamical system on M whose positive
semi-trajectories are solutions to the problem. To realize the synthesis we
construct a well-projected to M invariant Lagrange submanifold of the
extremals' flow in the cotangent bundle T*M. The construction uses the
curvature of the flow in the cotangent bundle and some ideas of hyperbolic
dynamics
Jacobi fields in optimal control: one-dimensional variations
We study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have much more singularities than the control. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps
Controllability on the group of diffeomorphisms
Given a compact manifold M, we prove that any bracket generating family of vector fields on M, which is invariant under multiplication by smooth functions, generates the connected component of the identity of the group of dieomorphisms
of M
Generalized Ricci Curvature Bounds for Three Dimensional Contact Subriemannian manifolds
Measure contraction property is one of the possible generalizations of Ricci
curvature bound to more general metric measure spaces. In this paper, we
discover sufficient conditions for a three dimensional contact subriemannian
manifold to satisfy this property.Comment: 49 page
Controllability on infinite-dimensional manifolds
Following the unified approach of A. Kriegl and P.W. Michor (1997) for a
treatment of global analysis on a class of locally convex spaces known as
convenient, we give a generalization of Rashevsky-Chow's theorem for control
systems in regular connected manifolds modelled on convenient
(infinite-dimensional) locally convex spaces which are not necessarily
normable.Comment: 19 pages, 1 figur
Geometric Approach to Pontryagin's Maximum Principle
Since the second half of the 20th century, Pontryagin's Maximum Principle has
been widely discussed and used as a method to solve optimal control problems in
medicine, robotics, finance, engineering, astronomy. Here, we focus on the
proof and on the understanding of this Principle, using as much geometric ideas
and geometric tools as possible. This approach provides a better and clearer
understanding of the Principle and, in particular, of the role of the abnormal
extremals. These extremals are interesting because they do not depend on the
cost function, but only on the control system. Moreover, they were discarded as
solutions until the nineties, when examples of strict abnormal optimal curves
were found. In order to give a detailed exposition of the proof, the paper is
mostly self\textendash{}contained, which forces us to consider different areas
in mathematics such as algebra, analysis, geometry.Comment: Final version. Minors changes have been made. 56 page
The rolling problem: overview and challenges
In the present paper we give a historical account -ranging from classical to
modern results- of the problem of rolling two Riemannian manifolds one on the
other, with the restrictions that they cannot instantaneously slip or spin one
with respect to the other. On the way we show how this problem has profited
from the development of intrinsic Riemannian geometry, from geometric control
theory and sub-Riemannian geometry. We also mention how other areas -such as
robotics and interpolation theory- have employed the rolling model.Comment: 20 page
On the Alexandrov Topology of sub-Lorentzian Manifolds
It is commonly known that in Riemannian and sub-Riemannian Geometry, the
metric tensor on a manifold defines a distance function. In Lorentzian
Geometry, instead of a distance function it provides causal relations and the
Lorentzian time-separation function. Both lead to the definition of the
Alexandrov topology, which is linked to the property of strong causality of a
space-time. We studied three possible ways to define the Alexandrov topology on
sub-Lorentzian manifolds, which usually give different topologies, but agree in
the Lorentzian case. We investigated their relationships to each other and the
manifold's original topology and their link to causality.Comment: 20 page
Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing
We study controllability issues for the 2D Euler and Navier-
Stokes (NS) systems under periodic boundary conditions. These systems
describe motion of homogeneous ideal or viscous incompressible fluid on
a two-dimensional torus T^2. We assume the system to be controlled by
a degenerate forcing applied to fixed number of modes.
In our previous work [3, 5, 4] we studied global controllability by
means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing
viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic
control theory have been used for that study. In [3] criteria for
global controllability of nite-dimensional Galerkin approximations of
2D and 3D NS systems have been established. It is almost immediate
to see that these criteria are also valid for the Galerkin approximations
of the Euler systems. In [5, 4] we established a much more intricate suf-
cient criteria for global controllability in finite-dimensional observed
component and for L2-approximate controllability for 2D NS system.
The justication of these criteria was based on a Lyapunov-Schmidt
reduction to a finite-dimensional system. Possibility of such a reduction
rested upon the dissipativity of NS system, and hence the previous
approach can not be adapted for Euler system.
In the present contribution we improve and extend the controllability
results in several aspects: 1) we obtain a stronger sufficient condition for
controllability of 2D NS system in an observed component and for L2-
approximate controllability; 2) we prove that these criteria are valid for
the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability
in projection on any finite-dimensional subspace and establish a
sufficient criterion for such controllability
Energy minimization problem in two-level dissipative quantum control: meridian case
International audienceWe analyze the energy-minimizing problem for a two-level dissipative quantum system described by the Kossakowsky-Lindblad equation. According to the Pontryagin Maximum Principle (PMP), minimizers can be selected among normal and abnormal extremals whose dynamics are classified according to the values of the dissipation parameters. Our aim is to improve our previous analysis concerning 2D solutions in the case where the Hamiltonian dynamics are integrable
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