Geometric Control Theory and Sub-Riemannian Geometry

ISBN-13: 9783319021317 / Angielski / Twarda / 2013 / 384 str.

Gianna Stefani; Ugo Boscain; Mario Sigalotti

Opis

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.

Spis treści

1 A. A. Agrachev - Some open problems.- 2 D. Barilari, A. Lerario - Geometry of Maslov cycles.- 3 Y. Baryshnikov, B. Shapiro - How to Run a Centipede: a Topological Perspective.- 4 B. Bonnard, O. Cots, L. Jassionnesse - Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces.- 5 J-B. Caillau, C. Royer - On the injectivity and nonfocal domains of the ellipsoid of revolution.- 6 P. Cannarsa, R. Guglielmi - Null controllability in large time for the parabolic Grushin operator with singular potential.- 7 Y. Chitour, M. Godoy Molina, P. Kokkonen - The rolling problem: overview and challenges.- 8 A. A. Davydov, A. S. Platov - Optimal stationary exploitation of size-structured population with intra-specific competition.- 9 B. Doubrov, I. Zelenko - On geometry of affine control systems with one input.- 10 B. Franchi, V. Penso, R. Serapioni - Remarks on Lipschitz domains in Carnot groups.- 11 R. V. Gamkrelidze - Differential-geometric and invariance properties of the equations of Maximum Principle (MP).- 12 N. Garofalo - Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces.- 13 R. Ghezzi, F. Jean - Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds.- 14 V. Jurdjevic - The Delauney-Dubins Problem.- 15 M. Karmanova, S. Vodopyanov - On Local Approximation Theorem on Equiregular Carnot–Carathéodory spaces.- 16 C. Li - On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle.- 17 I. Markina, S. Wojtowytsch - On the Alexandrov Topology of sub-Lorentzian Manifolds.- 18 R. Monti - The regularity problem for sub-Riemannian geodesics.- 19 L. Poggiolini, G. Stefani - A case study in strong optimality and structural stability of bang–singular extremals.- 20 A. Shirikyan - Approximate controllability of the viscous Burgers equation on the real line.- 21 M. Zhitomirskii - Homogeneous affine line fields and affine line fields in Lie algebras.

O autorze

Prof. Gianna Stefani: From 1997 is Full Professor at University of Florence, Italy.

Prof. Ugo Boscain: Directeur de recherche CNRS (DR2) at the Center of Applied Mathematics and Probability (CMAP) of Ecole Polytechnique; Professeur charge de course in numerical analysis and optimization at Ecole Polytechnique (department of applied mathematics); Deputy team leader of the equipe-INRIA GECO Inria Saclay.

Prof. Jean-Paul Gauthier: Experience of JP Gauthier In Scientific Research (January 2011), Including; Research Team Management and Industrial Collaborations; JP Gauthier has scientific experience in several areas (pluridisciplinary); Honorary Member of Institut Universitaire de France (Promotion 1992).

Prof. Andrey Sarychev: Full Professor (Professore Ordinario di I Fascia) at the Department of Mathematics and Informatics U.Dini (DiMaI), University of Florence, Italy, since January 2013. Prof. Mario Sigalotti: Chargé de recherche de première classe (CR1) - Établissement : INRIA Saclay – Île-de-France - Équipe-projet : GECO.

Blurb

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.