ИСТИНА

The study is aimed at the optimal control design of longitudinal motions for a rectilinear elastic rod with free ends. Several piezoelectric actuators are attached along its axis. The control is carried out by piezoelectric normal force acting in the cross section. We assume that this force changes piecewise constantly in space, and the intervals of constancy have no gaps and are equal in length. Given an arbitrary initial state and periodic terminal conditions, the optimal control problem is to minimize the mean mechanical energy stored in the rod over a fixed time horizon. For a uniform rod, the minimal admissible horizon is found, and the exact optimal control law is presented.