Abnormal extremals for generic planar systems

The aim of the paper is to analyze the generic properties of abnormal extremals. We prove that these are finite concatenations of bang arcs (that correspond to constant ± control). Moreover, the switchings (discontinuity points of the control) happen exactly when the abnormal extremal crosses the set of zeros of the function Δ_A = F∧G. All possible generic singularities of the synthesis on the plane occurring along (projections of) abnormal extremals are classified in an other article by the authors. However, one has to show singularities indeed appear for some generic system, in particular those corresponding to the new projection singularities, namely bifold and ribbon. The set of possible singularities is formed of 28 (equivalence classes of) singular points, but not all sequences of singularities can be realized. We are able to prove that the generic sequences of singularities along abnormal extremals can be classified with a recognizable set of words. As a by product, we obtain the existence of systems presenting singular points corresponding to projection singularities of ribbon type