Considered is a pendulum suspended on the wheel rolling along a horizontal axis. The control moment is applied between the pendulum and the wheel. The problem of simultaneous stabilization of the vertical position of the pendulum and the position of the wheel on the horizontal axis is considered. It is shown that if the sum of the angle of the pendulum and the angle of rotation of the wheel is taken as the output of the system, then the zero dynamics of the closed system turns out to be stable, although not asymptotically. A method for asymptotic stabilization of the equilibrium position of a closed system is proposed and an estimate for the attraction domain is constructed. The construction of the estimate is reduced to the problem of the solvability of linear matrix inequalities (LMI)
