On Zermelo-like problems: a Gauss-Bonnet inequality and an E. Hopf theorem

Ulysse Serres (1)

1 LAGEP - Laboratoire d'automatique et de génie des procédés

The goal of this paper is to describe Zermelo's navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to changing the Riemannian metric on the manifold the control curvature of Zermelo's problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss-Bonnet formula in an inequality. This Gauss-Bonnet inequality enables to generalize to Zermelo's problems a theorem by E. Hopf establishing the flatness of Riemannian tori without conjugate points.

Mots clés: Conjugate points; control curvature; feedback transformation; Gauss-Bonnet formula; Riemannian manifold; Zermelo's navigation problem

Domaines: Optimisation et contrôle [math.OC]; Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]

Ulysse Serres. On Zermelo-like problems: a Gauss-Bonnet inequality and an E. Hopf theorem. Journal of Dynamical and Control Systems, 2009, 15 (1), http://dx.doi.org/10.1007/s10883-008-9056-6. ⟨10.1007/s10883-008-9056-6⟩. ⟨hal-00705931⟩