ИСТИНА

We consider a vibrating system consisting of a thin rectilinear elastic rod actuated by external loads applied at the ends as well as a normal force, which is distributed piecewise constantly in space. Such a force may be implemented by piezoelectric actuators. The intervals of constancy of this normal force are equal in length, and the force value on each of these sections is considered as an independent control input. We study the longitudinal motions of the rod and the means of control optimization. Based on the eigenmode decomposition, it is shown in the case of uniform rod that the original continuous system is split into several infinite vibrating subsystems each of which is controlled by a certain linearly independent combination of control inputs. It follows that if any of these combinations are taken equal to zero, then the corresponding subsystem becomes uncontrollable. Next, an optimal control problem on a finite time horizon is considered, where the terminal mechanical energy of the rod and energy losses in the control circuit are minimized with some weighting coefficients. We show that for a fixed number of actuators distributed along the rod, approximation of the problem is reduced to the design of linear-quadratic regulators. An example of a uniform rod is presented where finite expressions for the optimal control functions are obtained. Amplitudes of controlled and affected but not minimized modes are derived for approximated suboptimal control.