Subanalytic sets and optimal control in the plane

Local bounds on the number of switchings and the existence of a regular synthesis are proved for optimal control problems of the form minimize ??L(x(t))dt subject to x = F(x) + uG(x), x ?? M, |u| ?? 1, where M is an open subset of the plane, F and G are real analytic vector fields on M, and L is a strictly positive real-valued function on M.