The Structure of Time-Optimal Trajectories for Single-Input Systems in the Plane: The General Real Analytic Case

For arbitrary single-input real analytic systems in the plane, in which the control enters linearly, we prove that the time-optimal trajectories are finite concatenations of “bang-bang” and singular arcs, with local bounds on the number of switchings. No “nondegeneracy” assumption is made other than real-analyticity. The analysis proceeds by applying our previous results on nondegenerate C^∞ systems to study the behavior of the trajectories on a neighborhood of every point, except for the points in a discrete set of “branch points.” The branch points are then handled by a combination of control-theoretic arguments and the use of subanalytic set theory.