Summary: 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 \((p_1,v_1)\) and \((p_2,v_2)\) in TM, there exists, for every \(\epsilon > 0\), a curve \(\gamma\) in M, with geodesic curvature smaller than \(\epsilon\), such that \(\gamma\) connects \(p_1\) to \(p_2\) and, for \(i=1,2\), \(\dot\gamma\) is equal to \(v_i\) at \(p_i\). Property UCC is equivalent to the complete controllability of a family of control systems of Dubins’ type, parameterized by \(\epsilon\). 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 \(\sup_MK<0\) or \(\inf_MK>-\infty\), 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.
For Part I see: J. Geom. Anal. 15, No. 4, 565–587 (2005; Zbl 1090.53042).