Shooting method in determining global minimum time of brachistochronic motion

Abstract

The paper analyzes the problem of brachistochronic motion for the general case of holonomic scleronomic mechanical system. The problem is formulated as an optimal control task, where the generalized velocities are taken as a control, provided that the value of the total mechanical energy of the system is given. The initial and final values of generalized coordinates are specified. The corresponding two-point boundary value problem of the system of ordinary nonlinear differential equations is obtained, which it is, in a general case, necessary to solve numerically. In solving it by the shooting method it is necessary to evaluate the intervals in which initial values of the conjugate vector are, so as to achieve the specified values of generalized coordinates. This presents a problem, because the conjugate vector coordinates do not represent the physical quantities and their values cannot be easily estimated, and due to the homogeneity of a conjugate system they are directly proportional to the chosen value of the negative coordinates. The work provides the procedures for evaluation of all possible values of the coordinates of the conjugate vector, so outside these intervals there are not certain solutions of the two-point boundary value problem. Also, it is given the procedure for determining the global minimum for the systems with three degrees of freedom, where it is possible to provide three-dimensional graphic representation of the space, where the third dimension is the ultimate moment. In the case of several extremal solutions of the principle of maximum the global optimum is the lowest value of the solutions obtained for the ultimate moment. The procedure is demonstrated on the problem of the brachistochronic spherical rigid body motion, in which the center of mass is not at the fixed point.