Dubins' Problem on Surfaces II: Nonpositive Curvature

Let M be a complete, connected, two-dimensional Riemannian manifold with nonpositive Gaussian curvature K. We say that M satisfies the unrestricted complete controllability (UCC) property for the Dubins problem if the following holds: given any (p1, v1) and (p2, v2) in TM , there exists, for every ε > 0, a curve γ in M , with geodesic curvature smaller than ε, such that γ connects p1 to p2 and, for i = 1, 2, γ̇ is equal to vi at pi. Property UCC is equivalent to the complete controllability of a family of control systems of Dubins’ type, parameterized by ε. It is well known that the Poincaré half-plane does not verify property UCC. In this paper, we provide a complete characterization of the two-dimensional nonpositively curved manifolds M , with either uniformly negative or bounded curvature, that satisfy property UCC. More precisely, if supM K < 0 or infM K > −∞, we show that UCC holds if and only if (i) M is of the first kind or (ii) the curvature satisfies a suitable integral decay condition at infinity.