Projection Singularities of Extremals and Morse Property for Minimum Time

For a generic minimum time problem on the plane, we study the projections of the support of extremals (regarded as a two-dimensional object, after normalization) from $\mathbb R^2\times S^1$ to $\mathbb R^2$. Moreover, we study the topology of the reachable set and we give a positive answer to a question of V. I. Arnold: Is the minimum time function, generically, a Morse function in topological sense?