Irregularity of Optimal Trajectories in a Control Problem for a Car-like Robot

Abstract

We study the problem to find (a) shortest plane curve(s) joining two given points with given tangent angles and curvatures. The tangent angle and the curvature of the path are continuous and the derivative of the curvature is bounded by $2$. At a regular (i.e. of the class $C^3$) point such a curve must be locally a piece of a clothoid or a line segment (up to isometry a clothoid is given by Fresnel's integrals $x(t)=\int _0^t\cos \tau ^2d\tau $, $y(t)=\int _0^t\sin \tau ^2d\tau $). We prove that if the distance between the initial and final points is greater than $320\sqrt{\pi }$, then a generic shortest curve contains infinitely many switching points.