1. The curvature of optimal control problems with applications to sub-Riemannian geometry
- Author
-
Rizzi, Luca, Geometric Control Design (GECO), Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS)-Inria Saclay - Ile de France, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Scuola Internazionale di Studi Superiori Avanzati (SISSA, Trieste, Italy), and Andrei Agrachev
- Subjects
geometrie sous-Riemannienne ,géométrie Riemannienne ,courbure ,optimal control ,sub-Riemannian geometry ,contrôle optimal ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,curvature ,théorèmes de comparaison ,comparison theorems ,[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] ,Settore MAT/03 - Geometria ,Riemannian geometry ,[MATH.MATH-MG]Mathematics [math]/Metric Geometry [math.MG] - Abstract
Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.; Optimal control theory is an extension of the calculus of variations, and deals with the optimal behaviour of a system under a very general class of constraints. This field has been pioneered by the group of mathematicians led by Lev Pontryagin in the second half of the 50s and nowadays has countless applications to the real worlds (robotics, trains, aerospace, models for human behaviour, human vision, image reconstruction, quantum control, motion of self-propulsed micro-organism). In this thesis we introduce a novel definition of curvature for an optimal control problem. In particular it works for any sub-Riemannian and sub-Finsler structure. Related problems, such as comparison theorems for sub-Riemannian manifolds, LQ optimal control problem and Popp's volume and are also investigated.
- Published
- 2014