Generic bifurcations of control-affine systems in the plane and their properties

We define bifurcations of control-affine systems in the plane and classify all generic 1-parameter bifurcations with nonvanishing control field. More precisely, we classify topological bifurcations of invariants of usual feedback equivalence. There are six such bifurcations: two bifurcations of equilibrium sets, two bifurcations of critical sets and two bifurcations of pairs of invariants. We also analyze how time optimality properties and stabilizability of generic planar families change when the parameter varies.