Sub-Finsler structures on the Engel group

Cover Page

Abstract


A one-parameter family of left-invariant sub-Finsler problems on a four-dimensional nilpotent Lie group of depth 3 with two generators is considered. The indicatrix of sub-Finsler structures is a square rotated by an arbitrary angle in the distribution. Methods of optimal control theory are applied. Abnormal and singular normal trajectories are described, and their optimality is proved. Singular trajectories arriving at the boundary of the reachable set in fixed time are characterized. A bang-bang phase flow is constructed, and estimates for the number of switchings on bang-bang trajectories are obtained. The structure of all normal extremals is described. Mixed trajectories are studied.


About the authors

A. A. Ardentov

Ailamazyan Program Systems Institute of the Russian Academy of Sciences

Author for correspondence.
Email: aaa@pereslavl.ru

Russian Federation, 4а, Petra Pervogo street, Veskovo, Pereslavskiy district, Yaroslavskaja region, 152020

Yu. L. Sachkov

Ailamazyan Program Systems Institute of the Russian Academy of Sciences

Email: yusachkov@gmail.com

Russian Federation, 4а, Petra Pervogo street, Veskovo, Pereslavskiy district, Yaroslavskaja region, 152020

References

  1. Pansu P. Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un // Ann. of Math. (2). 1989. V. 129. № 1. P. 1-60.
  2. Берестовский В. Н. Однородные пространства с внутренней метрикой. II // Сиб. мат. журн. 1989. Т. 30. № 2. С. 14-28, 225.
  3. Boscain U., Chambrion T., Charlot G. Nonisotropic 3-Level Quantum Systems: Complete Solutions for Minimum Time and Minimum Energy // Discrete Contin. Dyn. Syst. Ser. B5. 2005. № 4. P. 957-990.
  4. Busemann H. The Isoperimetric Problem in the Minkowski Plane // AJM. 1947. V. 69. P. 863-871.
  5. Barilari D., Boscain U., Le Donne E., Sigalotti M. Sub-Finsler Structures from the Time-Optimal Control Viewpoint for Some Nilpotent Distributions // J. Dyn. Control Syst. 2017. V. 23. P. 547.
  6. Аграчев А. А., Сачков Ю. Л. Геометрическая теория управления. М.: Физматлит, 2005.
  7. Понтрягин Л. С., Болтянский В. Г., Гамкрелидзе Р. В., Мищенко Е. Ф. Математическая теория оптимальных процессов. М.: Наука, 1961.
  8. Agrachev A. A., Gamkrelidze R. V. Symplectic Geometry for Optimal Control. Nonlinear Controllability and Optimal Control // Monogr. Textbooks Pure and Appl. Math. 1990. V. 133. P. 263-277.

Statistics

Views

Abstract - 200

PDF (Russian) - 144

PlumX


Copyright (c) 2019 Russian academy of sciences

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies