2007/06/10 by Ulysse Serres, Serres, Ulysse · 2 citations
Mathematics · #34K35 #53B40 #93C15 #Differential Geometry (math.DG) #FOS: Mathematics #Optimization and Control (math.OC) #math.DG #math.OC #msc:34K35 #msc:53B40 #msc:93C15
paper · pdf · doi:10.48550/arxiv.0706.1366
27 pages, 1 figure
arxiv created 2007/06/10 · arxiv updated 2009/12/01
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 change 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 for Zermelo's problems the E. Hopf theorem on flatness of Riemannian tori without conjugate points.