Abstract. The aim of this article is to present algorithms to compute the first conjugate time along a smooth extremal curve, where the trajectory ceases to be optimal. It is based on recent theoretical developments of geometric optimal control, and the article contains a review of second order optimality conditions. The computations are related to a test of positivity of the intrinsic second order derivative or a test of singularity of the extremal flow. We derive an algorithm called COTCOT (Conditions of Order Two and COnjugate Times), available on the web, and apply it to the minimal time problem of orbit transfer, and to the attitude control problem of a rigid spacecraft. This algorithm involves both normal and abnormal cases.Mathematics Subject Classification. 49K15, 49-04, 70Q05.

In this article, we study the time optimality Status of singular trajectories for singleinput affine Systems. The basic tools are the construction of semi-normal forms under the action of the transformation group generated by changes of coordinates and feedbacks and the use of an adaptated representation of the input space, which allow to evaluate the accessibility sets. We get necessary and sufficient optimality conditions under generic or codimension one assumptions.

Abstract. This article deals with the local sub-Riemannian g e ometry on R 3 (D g) w h e r e D is the distribution ker !, ! being the Martinet one-form: dz ; 1 2 y 2 dx and g is a Riemannian metric on D: We p r o ve that we can take g as a sum of squares adx 2 + cdy 2 : Then we analyze the at case where a = c = 1 : We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci, and the sub-Riemannian sphere. A direct consequence of our computations is to show that the sphere is not sub-analytic. Some of these computations are generalized to a one parameter deformation of the at case.

The objective of this article is to present a sharp result to determine when the cut locus for a class of metrics on a two-sphere of revolution is reduced to a single branch. This work is motivated by optimal control problems in space and quantum dynamics and gives global optimal results in orbital transfer and for Lindblad equations in quantum control. RésuméLe but de cet article est de présenter une condition suffisante permettant de garantir que le lieu de coupure d'une classe de métriques sur la 2-sphère de révolution est réduit à une branche simple. Ce travail est motivé par des problèmes de contrôle optimal en mécanique spatiale et mécanique quantique. Des résultats globaux d'optimalité sont obtenus en transfert orbital ainsi que dans le cas des équations de Lindblad en contrôle quantique.

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