A time-optimal control of steering an unstable rod

The problem of controlling a system consisting of a weighty rod leaning on a movable support by its bottom end is investigated. Under the assumption that the support moves without inertia with a bounded velocity, a time-optimal motion from an unstable initial equilibrium of the system to another its unstable equilibrium position is found. It is shown that the law of optimal velocity of the support motion belongs to the boundary of the admissible domain and has no more than two switches from one limit value to another. Calculation formulas for determining the time instants of switches of the support velocity are obtained depending on the boundary conditions. The results of computer simulation of the system with account of inertia of the support motion are presented.