Geometric Control Theory and Sub-Riemannian Geometry

Description

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 Front Matter....Pages i-xii Some open problems....Pages 1-13 Geometry of Maslov cycles....Pages 15-35 How to Run a Centipede: a Topological Perspective....Pages 37-51 Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces....Pages 53-72 On the injectivity and nonfocal domains of the ellipsoid of revolution....Pages 73-85 Null controllability in large time for the parabolic Grushin operator with singular potential....Pages 87-102 The rolling problem: overview and challenges....Pages 103-122 Optimal stationary exploitation of size-structured population with intra-specific competition....Pages 123-132 On geometry of affine control systems with one input....Pages 133-152 Remarks on Lipschitz domains in Carnot groups....Pages 153-166 Differential-geometric and invariance properties of the equations of Maximum Principle (MP)....Pages 167-175 Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces....Pages 177-199 Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds....Pages 201-218 The Delauney-Dubins Problem....Pages 219-239 On Local Approximation Theorem on Equiregular Carnot-Carathéodory Spaces....Pages 241-262 On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle....Pages 263-285 On the Alexandrov Topology of sub-Lorentzian Manifolds....Pages 287-311 The regularity problem for sub-Riemannian geodesics....Pages 313-332 A case study in strong optimality and structural stability of bang-singular extremals....Pages 333-350 Approximate controllability of the viscous Burgers equation on the real line....Pages 351-370 Homogeneous affine line fields and affine lines in Lie algebras....Pages 371-384