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inria-00070625, version 1

## Dubins' problem on surfaces. II. Nonpositive curvature

Mario Sigalotti 1, Yacine Chitour

N° RR-5378 (2004)

Abstract: Let $M$ be a complete, connected, two-dimensional Riemannian manifold with nonpositive Gaussian curvature $K$. We say that $M$ satisfies the 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 Poincaré half-plane does not verify property UCC. In this paper, we provide a complete characterization of the surfaces $M$, 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.

• 1:  APICS (INRIA Sophia Antipolis)
• INRIA
• Domain : Computer Science/Other
• Keywords : CONTROLLABILITY / GEOMETRIC CONTROL / GEODESIC CURVATURE / NONPOSITIVE CURVATURE
• Internal note : RR-5378

• inria-00070625, version 1
• oai:hal.inria.fr:inria-00070625
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• Submitted on: Friday, 19 May 2006 21:02:42
• Updated on: Wednesday, 31 May 2006 14:24:24