On Quick Action in the Problem of Controlling the Vertical Position of a Pendulum by the Movement of its Base

摘要

In the kinematic formulation, two problems are considered for the time-optimal control of the position of a single-link pendulum fixed on a movable base that can move along the horizontal axis. In the first problem, the initial position of the pendulum is the lower, stable position of the pendulum; it is required to transfer it to the upper, unstable position resting either in an arbitrary or a given place along the horizontal axis. In the second problem, the initial position is, on the contrary, the upper position. It is required to move the pendulum along the horizontal axis, while maintaining its original vertical position at the end. Both problems are considered based on nonlinear equations. The numerical solution of the problems is preceded by an analysis of the optimality conditions arising from the maximum principle in the case of the kinematic control of these objects.