Sub-riemannian sphere in Martinet flat case

A. Agrachev; B. Bonnard; M. Chyba; I. Kupka

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Fix two points $x, \bar{x} \in S^{2}$ and two directions (without orientation) $η, \bar{η}$ of the velocities in these points. In this paper we are interested to the problem of minimizing the cost $J [γ] = \int_{0}^{T} (_{γ (t)} (\dot{γ} (t), \dot{γ} (t)) + K_{γ (t)}^{2}_{γ (t)} (\dot{γ} (t), \dot{γ} (t))) d t$ along all smooth curves starting from x with direction η and ending in $\bar{x}$ with direction $\bar{η}$ . Here g is the standard Riemannian metric on S ^2...