Shortest Paths For Sub Riemannian Metrics On Rank Two Distributions
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Shortest Paths For Sub Riemannian Metrics On Rank Two Distributions
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Author : Wensheng Liu
language : en
Publisher:
Release Date : 1995
Shortest Paths For Sub Riemannian Metrics On Rank Two Distributions written by Wensheng Liu and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Geodesics categories.
Shortest Paths For Sub Riemannian Metrics On Rank Two Distributions
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Author : Wensheng Liu
language : en
Publisher: American Mathematical Soc.
Release Date : 1995
Shortest Paths For Sub Riemannian Metrics On Rank Two Distributions written by Wensheng Liu and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1995 with Mathematics categories.
A sub-Riemannian manifold ([italic capitals]M, E, G) consists of a finite-dimensional manifold [italic capital]M, a rank-two bracket generating distribution [italic capital]E on [italic capital]M, and a Riemannian metric [italic capital]G on [italic capital]E. All length-minimizing arcs on ([italic capitals]M, E, G) are either normal extremals or abnormal extremals. Normal extremals are locally optimal, i.e., every sufficiently short piece of such an extremal is a minimizer. The question whether every length-minimizer is a normal extremal was recently settled by R. G. Montgomery, who exhibited a counterexample. The present work proves that regular abnormal extremals are locally optimal, and, in the case that [italic capital]E satisfies a mild additional restriction, the abnormal minimizers are ubiquitous rather than exceptional. All the topics of this research report (historical notes, examples, abnormal extremals, Hamiltonians, nonholonomic distributions, sub-Riemannian distance, the relations between minimality and extremality, regular abnormal extremals, local optimality of regular abnormal extremals, etc.) are presented in a very clear and effective way.
Sub Riemannian Geometry
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Author : Andre Bellaiche
language : en
Publisher: Birkhäuser
Release Date : 2012-12-06
Sub Riemannian Geometry written by Andre Bellaiche and has been published by Birkhäuser this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
Sub-Riemannian geometry (also known as Carnot geometry in France, and non-holonomic Riemannian geometry in Russia) has been a full research domain for fifteen years, with motivations and ramifications in several parts of pure and applied mathematics, namely: control theory classical mechanics Riemannian geometry (of which sub-Riemannian geometry constitutes a natural generalization, and where sub-Riemannian metrics may appear as limit cases) diffusion on manifolds analysis of hypoelliptic operators Cauchy-Riemann (or CR) geometry. Although links between these domains had been foreseen by many authors in the past, it is only in recent years that sub- Riemannian geometry has been recognized as a possible common framework for all these topics. This book provides an introduction to sub-Riemannian geometry and presents the state of the art and open problems in the field. It consists of five coherent and original articles by the leading specialists: Andr Bellache: The tangent space in sub-Riemannian geometry Mikhael Gromov: Carnot-Carathodory spaces seen from within Richard Montgomery: Survey of singular geodesics Hctor J. Sussmann: A cornucopia of four-dimensional abnormal sub-Riemannian minimizers Jean-Michel Coron: Stabilization of controllable systems.
Sub Riemannian Geometry
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Author : Ovidiu Calin
language : en
Publisher: Cambridge University Press
Release Date : 2009-04-20
Sub Riemannian Geometry written by Ovidiu Calin and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2009-04-20 with Mathematics categories.
A comprehensive text and reference on sub-Riemannian and Heisenberg manifolds using a novel and robust variational approach.
Geometric Control Theory And Sub Riemannian Geometry
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Author : Gianna Stefani
language : en
Publisher: Springer
Release Date : 2014-06-05
Geometric Control Theory And Sub Riemannian Geometry written by Gianna Stefani and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-05 with Mathematics categories.
Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.
A Tour Of Subriemannian Geometries Their Geodesics And Applications
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Author : Richard Montgomery
language : en
Publisher: American Mathematical Soc.
Release Date : 2002
A Tour Of Subriemannian Geometries Their Geodesics And Applications written by Richard Montgomery and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
Subriemannian geometries can be viewed as limits of Riemannian geometries. They arise naturally in many areas of pure (algebra, geometry, analysis) and applied (mechanics, control theory, mathematical physics) mathematics, as well as in applications (e.g., robotics). This book is devoted to the study of subriemannian geometries, their geodesics, and their applications. It starts with the simplest nontrivial example of a subriemannian geometry: the two-dimensional isoperimetric problem reformulated as a problem of finding subriemannian geodesics. Among topics discussed in other chapters of the first part of the book are an elementary exposition of Gromov's idea to use subriemannian geometry for proving a theorem in discrete group theory and Cartan's method of equivalence applied to the problem of understanding invariants of distributions. The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail Berry's phase in quantum mechanics, the problem of a falling cat righting herself, that of a microorganism swimming, and a phase problem arising in the $N$-body problem. He shows that all these problems can be studied using the same underlying type of subriemannian geometry. The reader is assumed to have an introductory knowledge of differential geometry. This book that also has a chapter devoted to open problems can serve as a good introduction to this new, exciting area of mathematics.
A Comprehensive Introduction To Sub Riemannian Geometry
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Author : Andrei Agrachev
language : en
Publisher: Cambridge University Press
Release Date : 2019-10-31
A Comprehensive Introduction To Sub Riemannian Geometry written by Andrei Agrachev and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-10-31 with Mathematics categories.
Provides a comprehensive and self-contained introduction to sub-Riemannian geometry and its applications. For graduate students and researchers.
Contemporary Trends In Nonlinear Geometric Control Theory And Its Applications
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Author : A. Anzaldo-Meneses
language : en
Publisher: World Scientific
Release Date : 2002
Contemporary Trends In Nonlinear Geometric Control Theory And Its Applications written by A. Anzaldo-Meneses and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with Mathematics categories.
Concerns contemporary trends in nonlinear geometric control theory and its applications.
Dynamics Bifurcations And Control
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Author : Fritz Colonius
language : en
Publisher: Springer
Release Date : 2003-07-01
Dynamics Bifurcations And Control written by Fritz Colonius and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003-07-01 with Technology & Engineering categories.
This volume originates from the Third Nonlinear Control Workshop "- namics, Bifurcations and Control", held in Kloster Irsee, April 1-3 2001. As the preceding workshops held in Paris (2000) and in Ghent (1999), it was organized within the framework of Nonlinear Control Network funded by the European Union (http://www.supelec.fr/lss/NCN). The papers in this volume center around those control problems where phenomena and methods from dynamical systems theory play a dominant role. Despite the large variety of techniques and methods present in the c- tributions, a rough subdivision can be given into three areas: Bifurcation problems, stabilization and robustness, and global dynamics of control s- tems. A large part of the fascination in nonlinear control stems from the fact that is deeply rooted in engineering and mathematics alike. The contributions to this volume reflect this double nature of nonlinear control. We would like to take this opportunity to thank all the contributors and the referees for their careful work. Furthermore, it is our pleasure to thank Franchise Lamnabhi-Lagarrigue, the coordinator of our network, for her s- port in organizing the workshop and the proceedings and for the tremendous efforts she puts into this network bringing the cooperation between the d- ferent groups to a new level. In particular, the exchange and the active p- ticipation of young scientists, also reflected in the Pedagogical Schools within the Network, is an asset for the field of nonlinear control.
Two Classes Of Riemannian Manifolds Whose Geodesic Flows Are Integrable
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Author : Kazuyoshi Kiyohara
language : en
Publisher: American Mathematical Soc.
Release Date : 1997
Two Classes Of Riemannian Manifolds Whose Geodesic Flows Are Integrable written by Kazuyoshi Kiyohara and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 1997 with Mathematics categories.
Two classes of manifolds whose geodesic flows are integrable are defined, and their global structures are investigated. They are called Liouville manifolds and Kahler-Liouville manifolds respectively. In each case, the author finds several invariants with which they are partly classified. The classification indicates, in particular, that these classes contain many new examples of manifolds with integrable geodesic flow.