ESAIM: Control, Optimisation and Calculus of Variations

Research Article

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

Bonnard, Bernarda1, Caillau, Jean-Baptistea2 and Trélat, Emmanuela3

a1 Univ. Dijon, IMB, Bât. Mirande, 9 avenue Alain Savary, 21078 Dijon Cedex, France;

a2 ENSEEIHT-IRIT, UMR CNRS 5505, 2 rue Camichel, 31071 Toulouse, France;

a3 Univ. Orléans, UFR Sciences Mathématiques, Labo. MAPMO, UMR 6628, Route de Chartres, BP 6759, 45067 Orléans Cedex 2, France;


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.

(Received May 19 2005)

(Online publication May 12 2007)

Key Words:

  • Conjugate point;
  • second-order intrinsic derivative;
  • Lagrangian singularity;
  • Jacobi field;
  • orbit transfer;
  • attitude control

Mathematics Subject Classification:

  • 49K15;
  • 49-04;
  • 70Q05