1,721,014 research outputs found
Any sub-Riemannian metric has points of smoothness
Full text linkWe prove the result stated in the title; it is equivalent to the existence
of a regular point of the sub-Riemannian exponential mapping.
We also prove that the metric is analytic on an open everywhere dense
subset in the case of a complete real-analytic sub-Riemannian manifold
Spectrum of the Second Variation
Full text linkSecond variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the classical Euler identity n=1(1 - x(2)/(n)(2)) = sin x/x. The general case may serve as a rich source of new nice identities
Control on the manifolds of mappings with a view to the deep learning.
No full textDeep learning of the artificial neural networks (ANN) can be treated as a particular class of interpolation problems. The goal is to find a neural network whose input-output map approximates well the desired map on a finite or an infinite training set. Our idea consists of taking as an approximant the input-output map, which arises from a nonlinear continuous-time control system. In the limit such control system can be seen as a network with a continuum of layers, each one labelled by the time variable. The values of the controls at each instant of time are the parameters of the layer
Introduction to optimal control theory
No full textThese are lecture notes of the introductory course in Optimal Control theory treated from the geometric point of view. Optimal Control Problem is reduced to the study of controls (and corresponding trajectories) leading to the boundary of attainable sets. We discuss Pontryagin Maximum Principle, basic existence results, and apply these tools to concrete simple optimal control problems. Special sections are devoted to the general theory of linear time-optimal problems and linear-quadratic problems
On the Local Structure of Optimal Trajectories in R3
No full textWe analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four
Optimality of broken extremals
No full textIn this paper, we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. We prove the optimality of broken normal extremals in many cases that include (but not are exhausted by) the cases of the involutive driftless part of the system for any n > k and of the contact driftless part for n = 3 and k = 2
Nonholonomic tangent spaces: intrinsic construction and rigid dimensions
No full textA nonholonomic space is a smooth manifold equipped with a bracket generating family of vector fields. Its infinitesimal version is a homogeneous space of a nilpotent Lie group endowed with a dilation which measures the anisotropy of the space. We give an intrinsic construction of these infinitesimal objects and classify all rigid (i.e. not deformable) cases. © 2003 American Mathematical Society
Controllability of Navier-Stokes equations by few low modes forcing
No full textInvestigated are 2D and 3D Navier-Stokes equations with periodic boundary conditions, controlled by the low-frequency in spatial variables external force. Using principles of geometric control theory, global controllability is established for finite-dimensional Galerkin's approximations of Navier-Stokes equations. In the case of two spatial variables also obtained is surjectivity of finite-dimensional projections of sets of attainability for initial Navier-Stokes equation. The latter result uses the continuity property, which has independent significance. Demonstrated is continuous dependence of the 2D Navier-Stokes equation solution on external force for the case, when the force space is characterized with weak relaxation topology
Optimal transportation under nonholonomic constraints
No full textWe study Monge's optimal transportation problem, where the cost is given by optimal control cost. We prove the existence and uniqueness of an optimal map under certain regularity conditions on the Lagrangian, absolute continuity of the measures with respect to Lebesgue, and most importantly the absence of sharp abnormal minimizers. In particular, this result is applicable in the case of subriemannian manifolds with a 2-generating distribution and cost given by d^2, where d is the subriemannian distance. Also, we discuss some properties of the optimal plan when abnormal minimizers are present. Finally, we consider some examples of displacement interpolation in the case of Grushin plane
Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry
Full text linkWe prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry
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