Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Journal of Dynamical and Control Systems
Published in: Journal of Dynamical and Control Systems 3/2017

04-10-2016

Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions

Authors: Davide Barilari, Ugo Boscain, Enrico Le Donne, Mario Sigalotti

Published in: Journal of Dynamical and Control Systems | Issue 3/2017

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

In this paper, we study the sub-Finsler geometry as a time-optimal control problem. In particular, we consider non-smooth and non-strictly convex sub-Finsler structures associated with the Heisenberg, Grushin, and Martinet distributions. Motivated by problems in geometric group theory, we characterize extremal curves, discuss their optimality, and calculate the metric spheres, proving their Euclidean rectifiability.
previous article Hermes-Sussmann and ad-rank Conditions for Local Controllability
next article Switching in Time-Optimal Problem: the 3D Case with 2D Control

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now
Literature
1.
Agrachev A, Bonnard B, Chyba M, Kupka I. Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim Calc Var 1997;2:377–448. (electronic).MathSciNetCrossRefMATH
2.
Agrachëv AA, Gamkrelidze RV. Symplectic geometry for optimal control. Nonlinear controllability and optimal control. Monogr. Textbooks Pure Appl. Math. 1990; 133:263–277.
3.
Agrachev AA, Sigalotti M. On the local structure of optimal trajectories in R 3. SIAM J Control Optim 2003;42(2):513–531. MR 1982281 (2004f:49042).MathSciNetCrossRefMATH
4.
Agrachev AA, Sachkov YL, Vol. 87. Control theory from the geometric viewpoint. Encyclopaedia of Mathematical Sciences. Berlin: Springer; 2004. Control Theory and Optimization, II.
5.
Boscain U, Chambrion T, Charlot G. Nonisotropic 3-level quantum systems: complete solutions for minimum time and minimum energy. Discrete Contin Dyn Syst Ser B 2005;5(4):957–990. (electronic).MathSciNetCrossRefMATH
6.
Berestovskiı̆ VN. Homogeneous manifolds with an intrinsic metric. II. Sibirsk Mat Zh 1989;30(2):14–28, 225.MathSciNet
7.
Berestovskiı̆ VN. The structure of locally compact homogeneous spaces with an intrinsic metric. Sibirsk Mat Zh 1989;30(1):23–34.MathSciNet
8.
Breuillard E, Le Donne E. On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry. Proc Natl Acad Sci USA 2013;110(48): 19220–19226.MathSciNetCrossRefMATH
9.
10.
11.
Cowling MG, Martini A, Vol. 3. Sub-Finsler geometry and finite propagation speed. Trends in harmonic analysis, Springer INdAM Ser. Milan: Springer; 2013, pp. 147–205. MR 3026352.
12.
13.
14.
Gromov M, Vol. 152. Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc; 1999. Based on the 1981 French original, With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
15.
Le Donne E, Nicolussi Golo S. 2015. Regularity properties of spheres in homogeneous groups. Accepted in Transactions of the AMS.
16.
17.
Montgomery D, Zippin L. Topological transformation groups. Huntington: Robert E. Krieger Publishing Co; 1974. Reprint of the 1955 original.MATH
18.
Pansu P. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann Math (2) 1989;129(1):1–60.MathSciNetCrossRefMATH
19.
Sigalotti M. Local regularity of optimal trajectories for control problems with general boundary conditions. J Dyn Control Syst 2005;11(1):91–123. MR 2122468 (2005i:49002).MathSciNetCrossRefMATH
Metadata
Title
Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions
Authors
Davide Barilari
Ugo Boscain
Enrico Le Donne
Mario Sigalotti
Publication date
04-10-2016
Publisher
Springer US
Published in
Journal of Dynamical and Control Systems / Issue 3/2017
Print ISSN: 1079-2724
Electronic ISSN: 1573-8698
DOI
https://doi.org/10.1007/s10883-016-9341-8

Other articles of this Issue 3/2017

Journal of Dynamical and Control Systems 3/2017 Go to the issue

OriginalPaper

Tangential Center Problem for a Family of Non-generic Hamiltonians

OriginalPaper

Homotopy Path Spaces for Families of Admissible Paths

OriginalPaper

First Variation of the Hausdorff Measure of Non-horizontal Submanifolds in Sub-Riemannian Stratified Lie Groups

OriginalPaper

On High Dimensional Schrödinger Equation with Quasi-Periodic Potentials

OriginalPaper

Twin Vector Fields and Independence of Spectra for Quadratic Vector Fields

OriginalPaper

When a Minimal Map Is Totally Transitive on a G-Space

Premium Partners

    Image Credits