The aim of this paper is to describe the Zermelo navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method of evaluating its control curvature. We will show that, up to the change of the Riemannian metric on the manifold, the control curvature of the Zermelo problem has a simple to handle expression which naturally leads to a generalization of the classical Gauss–Bonnet formula in the form of an inequality. This Gauss–Bonnet inequality allows one to generalize the Zermelo problems and obtain a theorem of E. Hopf that establishes the flatness of Riemannian tori without conjugate points.
Key words and phrases
Conjugate points control curvature feedback transformation Gauss–Bonnet formula Riemannian manifold Zermelo navigation problem
2000 Mathematics Subject Classification
34K35 37C10 37E35 53B40 53C22 93C15
This is a preview of subscription content, log in to check access.
A. A. Agrachev, Feedback-invariant optimal control theory and differential geometry, II. Jacobi curves for singular extremals. J. Dynam. Control Systems4 (1998), No. 4, 583–604.zbMATHCrossRefMathSciNetGoogle Scholar
A. A. Agrachev and N. N. Chtcherbakova, Hamiltonian systems of negative curvature are hyperbolic. Russ. Math. Dokl.400 (2005), 295–298.Google Scholar
A. A. Agrachev, N. N. Chtcherbakova, and I. Zelenko, On curvatures and focal points of dynamical lagrangian distributions and their reductions by first integrals. J. Dynam. Control Systems11 (2005), No. 3, 297–327.zbMATHCrossRefMathSciNetGoogle Scholar
A. A. Agrachev and R. V. Gamkrelidze, Feedback-invariant optimal control theory and differential geometry, I. Regular extremals. J. Dynam. Control Systems3 (1997), No. 3, 343–389.zbMATHCrossRefMathSciNetGoogle Scholar
A. A. Agrachev and Yu. L. Sachkov, Control theory from the geometric viewpoint. Encycl. Math. Sci.87, Springer-Verlag (2004).Google Scholar
U. Serres, On the curvature of two-dimensional optimal control systems and Zermelo’s navigation problem. J. Math. Sci.135 (2006), No. 4, 3224–3243.CrossRefMathSciNetGoogle Scholar
S. Sternberg, Minimal coupling and the symplectic mechanics of a classical particle in the presence of a Yang–Mills field. Proc. Natl. Acad. Sci. USA74 (1977), No. 12, 5253–5254.zbMATHCrossRefMathSciNetGoogle Scholar
E. Zermelo, Über das Navigationsproblem bei ruhender oder veränderlicher windverteilung. Z. Angew. Math. Mech.11 (1931), No. 2, 114–124.CrossRefGoogle Scholar