Generic classification of time optimal planar stabilizing feedbacks

The aim of this paper is to show how geometric techniques can solve optimization problems for two dimensional single-input control systems. We are first concerned with the problem of stabilization at the origin in minimum time, for a planar control system affine with respect to the control. A topological equivalence relation on the corresponding time-optimal feedback synthesis, was introduced in a previous paper. The set of equivalence classes can be put in a one to one correspondence with a discrete family of graphs. This provides a classification of the global structure of generic time-optimal stabilizing feedbacks in the plane, analogous to the classification of smooth dynamical systems developed by Peixoto. This analysis permit the study of many control systems encountered in the applications.