Global minimum time for the brachistohronic motion of a particle in an arbitrary field of potential forces

Abstract

The problem of the brachistochronic motion of a particle in space is considered. Particle М moves in the field of known potential forces. The brachistochrone problem is formulated as an optimal control task, where the particle velocity projections are taken as control variables. The problem considered is reduced to solving the corresponding two-point boundary–value problem (TPBVP).The appropriate numerical procedure to apply in determining the solutions to the TPBVP is based on the shooting method. The paper presents the procedure for estimating the interval of initial values of the conjugate vector coordinates. Based on given estimation, it may be claimed that all solutions to the corresponding TPBVP are certainly located within given intervals, and thereby the global minimum time too for the brachistochronic motion of a particle. In the case of multiple solutions of the principle of maximum, the global minimum is the solution corresponding to the minimum time.