On curvature and feedback classification of two-dimensional optimal control systems

Abstract

The goal of this paper is to extend to two-dimensional optimal control systems with scalar input the classical notion of Gaussian curvature of two-dimensional Riemannian surface using the Cartan's moving frame method. This notion was already introduced by A. A. Agrachev and R. V. Gamkrelidze for more general control systems using a purely variational approach. Then we will see that the ``control'' analogue to Gaussian curvature reflects similar intrinsic properties of the extremal flow. In particular if the curvature is negative, arbitrarily long segment of extremals are locally optimal. Finally, we will define and characterize flat control systems.