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Second order optimality conditions in the smooth case and applications in optimal control

Published online by Cambridge University Press:  12 May 2007

Bernard Bonnard ,
Jean-Baptiste Caillau  and
Emmanuel Trélat
Bernard Bonnard
Affiliation:
Univ. Dijon, IMB, Bât. Mirande, 9 avenue Alain Savary, 21078 Dijon Cedex, France; Bernard.Bonnard@u-bourgogne.fr
Jean-Baptiste Caillau
Affiliation:
ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France; caillau@n7.fr
Emmanuel Trélat
Affiliation:
Univ. Orléans, UFR Sciences Mathématiques, Labo. MAPMO, UMR 6628, Route de Chartres, BP 6759, 45067 Orléans Cedex 2, France; emmanuel.trelat@univ-orleans.fr
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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.

Keywords

Conjugate pointsecond-order intrinsic derivativeLagrangian singularityJacobi fieldorbit transferattitude control
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 2 , April 2007 , pp. 207 - 236
Copyright
© EDP Sciences, SMAI, 2007

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