Manuscript submitted to Website: www.aimSciences.org DCDS Supplement Volume 2007 pp. set by publisher SECOND ORDER OPTIMALITY CONDITIONS WITH APPLICATIONS PROCEEDINGS OF THE SIXTH AIMS INTERNATIONAL CONFERENCE

Abstract. The aim of this article is to present the algorithm to compute the first conjugate point along a smooth extremal curve. Under generic assump-tions, the trajectory ceases to be optimal at such a point. An implementation of this algorithm, called cotcot, is available online and based on recent devel-opments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits. Introduction. The goal of this article is to summarize the algorithm to compute conjugate points along smooth extremal trajectories. Because of space restriction, we only consider the normal case [4]. The algorithm is implemented in the cotcot code1, and an important application to orbital transfer is given. The text is organized in two sections. In section 1, we recall the geometric foun