Time-optimal control problems for the swing and the ski

Abstract

This paper is concerned with a class of time-optimal control problems for the swing and the ski. We first consider the motion of a man standing on a swing. For simplicity, we neglect friction and air resistance and assume that the mass of the swinger is concentrated in his baricentre B. For the ski, we use the model considered by Aldo Bressan (1991 BRESSAN , A. , 1991 , On some control problems concerning the ski or swing . Atti Accad. Naz. Lincei, Memoirs , IX-1 , 149 – 196 . [Google Scholar]), with a special approximation of the skier. Our main concern is the existence and the structure of time-optimal controls. As a first step, we introduce an auxiliary control system with control entering linearly and show the relationships between this and the original system. We then perform a detailed study of the auxiliary system, using geometric techniques. In turn, this provides accurate information on the time-optimal controls for the original system.