Title: Time minimization problem on the group of motions of a plane with admissible control in a half-disk

Authors: Alexey Mashtakov

Journal title: Sbornik Mathematics

Year: 2022

Abstract:

We study a time minimization problem for a model of a car that can move forward on a plane and turn in place. Trajectories of this system are used in image processing for the detection of salient lines. The problem is a modification of a well-known sub-Riemannian
problem in the roto-translation group, where one of the controls is restricted to be non-negative. The problem is of interest in geometric control theory as a model example, in which the set of admissible controls contains zero on the boundary. We prove controllability and existence of optimal trajectories. Then, we apply a necessary optimality condition --- Pontryagin maximum principle to derive a Hamiltonian system for the extremals. Based on the analysis of the Hamiltonian system, we obtain an explicit expression for the extremal
controls and trajectories. We partially investigate the question of optimality of extremals and describe the structure of optimal synthesis.