Optimal chattering modes in the problem of the control of a Timoshenko beam

The linear problem of the control in a plane of the motion of a Timoshenko beam, one end of which is clamped to a rotating disc is considered. The angular acceleration of the disc serves as the control. It is proved that, in the problem of the quenching of the first mode, the optimal control has an infinite number of switchings in a finite time interval (a chattering control). The construction of a suboptimal control with a finite number of switchings is described.