Optimal conditions with chattering in the inverted two-link pendulum control problem

The plane motion of a two-link inverted mathematical pendulum, attached by a hinge to a moving trolley, is studied. The pendulum is controlled by a bounded force applied to the trolley. The problem of the minimization of the mean square deviation of the pendulum from an unstable equilibrium position is considered. Pontryagin's maximum principle is used. An optimal feedback control, containing special second order trajectories and trajectories with chattering is constructed for a linearized model. It is proved that, before emerging onto a special manifold, the optimal trajectories experience a chattering after a finite period of time and then reach the unstable equilibrium after an infinite time by a specific mode. The global optimality of the solution constructed is proved.