The control problem is investigated with reference to a system which consists of a weighty rod leaned by its lower end on a mobile support. System performance from its unstable initial equilibrium to another unstable position with prescribed translation velocity is sought with the assumption of inertia-free support motion with bounded velocity. It is proved that optimal support velocity time dependence belongs to the permis-sible domain wall and realizes bang-bang control with two switches as a maximum. Calculation formulas for support velocity switches points of time in dependence on edge conditions are received. Results of computer simulation of a system performance with taking into account inertia of a support are presented.