Regular Synthesis for Time-Optimal Control of Single-Input Real Analytic Systems in the Plane

Abstract

For arbitrary single-input real analytic systems in the plane, in which the control enters linearly, we prove the existence of a regular synthesis for the optimal control problem in which it is desired to minimize the integral of a strictly positive real analytic Lagrangian that does not depend on the control variable. The analysis proceeds by applying our previous results on nondegenerate $\mathcal{C}^\infty $ systems, as well as those on arbitrary real analytic ones, to study the local structure of the time optimal trajectories. The structure of the optimal trajectories for our problem is derived by reparametrization of time. The existence of a synthesis is then proved by using subanalytic set theory.