A generic classification of time optimal planar stabilizing feedback

The aim of this paper is to show how geometric techniques can solve optimization problems for two dimensional single-input control systems. We deal 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 feedback in the plane, analogous to the classification of smooth dynamical systems developed by Peixoto (1977). This analysis permit the study of many control systems encountered in the applications