Extremal Synthesis for Generic Planar Systems

For a generic single-input planar control system we analyze the structure of the set of extremals for the time-optimal problem. Generically all extremals are finite concatenations of regular arcs that are bang or correspond to a smooth feedback. Moreover, the support of extremals is a Whitney stratified set. We collect these information in the definition of extremal synthesis. In the cotangent bundle, we give a topological classification of the singularities of the extremal synthesis and study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from ℝ2 × S1 to the plane. With respect to the Whitney classical singularities here we deal with a stratified set with “edges” and “corners,” and along with cusps and folds, we find other stable singularities.