The smooth continuation method in optimal control with an application to quantum systems

¹ Institut de mathématiques de Bourgogne, UMR CNRS 5584, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France.
² Institut Carnot de Bourgogne, UMR CNRS 5209, 9 avenue Alain Savary, BP 47870, 21078 Dijon Cedex, France. Dominique.Sugny@u-bourgogne.fr

Received: 15 April 2009
Revised: 6 October 2009

Abstract

The motivation of this article is double. First of all we provide a geometrical framework to the application of the smooth continuation method in optimal control, where the concept of conjugate points is related to the convergence of the method. In particular, it can be applied to the analysis of the global optimality properties of the geodesic flows of a family of Riemannian metrics. Secondly, this study is used to complete the analysis of two-level dissipative quantum systems, where the system is depending upon three physical parameters, which can be used as homotopy parameters, and the time-minimizing trajectory for a prescribed couple of extremities can be analyzed by making a deformation of the Grushin metric on a two-sphere of revolution.