Mathematics

Let $M$ be a complete, connected, two-dimensional Riemannian manifold with non positive Gaussian curvature $K$. We say that $M$ satisfies the {\it unrestricted complete controllability} property for the Dubins' problem (UCC for short) if the following holds: Given any $(p_1,v_1)$ and $(p_2,v_2)$ in $TM$, there exists a curve $\gamma$ in $M$, with arbitrary small geodesic curvature, 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 the (arbitrary small) prescribed bound on the geodesic curvature. It is well-known that the Poincar'e half-plane does not verify property (UCC). During the talk, we will show that UCC) holds if $M$ is of the first kind. Moreover, the converse will be proved to be true if $\sup_M K Seminars -