On the Local Structure of Optimal Trajectories in R3

We analyze the structure of a control function u(t) corresponding to an optimal trajectory for the system $\dot q =f(q)+u\, g(q)$ in a three-dimensional manifold, near a point where some nondegeneracy conditions are satisfied. The kind of optimality which is studied includes time-optimality. The control turns out to be the concatenation of some bang and some singular arcs. Studying the index of the second variation of the switching times, the number of such arcs is bounded by four.