On the brachistochronic motion of the Chaplygin sleigh

Abstract

This paper deals with the brachistochronic motion of the Chaplygin sleigh on a horizontal plane surface. Two cases are considered, such as: the case when the magnitude of the horizontal reaction force at the contact point of the knife edge with the surface is unbounded, and the case when the magnitude of this force is bounded. The problem is solved by applying Pontryagin's maximum principle and singular optimal control theory. The angular acceleration of the sleigh is taken for a control variable. The problem considered is reduced to solving the corresponding two-point boundary value problem. In order to solve the obtained boundary value problem, an appropriate numerical procedure based on the shooting method is presented. Also, the paper analyzes the influence of the bounded magnitude of the reaction force on the structure of the controller sequence. It is shown that in the case of unbounded magnitude of the reaction force, the control is singular, and in the case of bounded magnitude, the control is, in a general case, a combination of singular and bang-bang controls.