Generic planar systems: Singularities of the extremal time

This paper deals with minimum time stabilization for a single input system on the plane. The optimal and extremal synthesis have been studied in previous papers. Here we give a complete description of topological stable projection singularities for extremals on the plane and for extremals-time tuples on the graph of the minimum time function. It is known that the minimum time function satisfies the Hamilton-Jacobi-Bellmann equation in viscosity sense. We prove that, under generic assumption, we have only Withney Umbrella singularities of the last projection that correspond to Swallowtail singularities for smooth classification.