The curvature of optimal control problems with applications to sub-Riemannian geometry

Courbure des systèmes de contrôle optimal et applications en géométrie sous-Riemannienne

Luca Rizzi (1, 2)

1 GECO - Geometric Control Design
2 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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.

Mots clés: Riemannian geometry; curvature; optimal control; comparison theorems; sub-Riemannian geometry

Mots clés: géométrie Riemannienne; théorèmes de comparaison; geometrie sous-Riemannienne; contrôle optimal; courbure

Domaines: Géométrie différentielle [math.DG]; Optimisation et contrôle [math.OC]; Géométrie métrique [math.MG]

Luca Rizzi. The curvature of optimal control problems with applications to sub-Riemannian geometry. Differential Geometry [math.DG]. Scuola Internazionale di Studi Superiori Avanzati (SISSA, Trieste, Italy), 2014. English. ⟨NNT : ⟩. ⟨tel-01122732⟩