Analytical and numerical construction of the optimal outcome function for a class of time-optimal problems

Abstract

Analytical and numerical algorithms are proposed for constructing the optimal outcome function and its Lebesgue set for the time-optimal control problem with a circular velocity indicatrix. Our approach to the solution of the time-optimal control problem essentially utilizes the specific dynamic properties of the controlled system. The circular vectogram of possible velocities enables us to interpret the cross sections of the controllability set as wavefronts whose source is uniformly distributed on the boundary of the target set. Procedures have been developed for analytical and numerical construction of the evolution of wavefronts based on prior (given the geometry of the target set boundary) identification of their nonsmoothness sets. An essential feature of the construction is the point-to-set distance function. We investigate the differential properties of this function and identify the manifolds on which it loses its smoothness. The proposed wavefront construction algorithms are of independent interest in so far as they enable us to investigate the geometry of the sets and compute their nonconvexity measure. The results are useful not only when studying the evolution of reachability sets of controlled systems, but also for computing the eikonal in geometrical optics and investigating the solutions of the wave equation.