Optimal motions of an elastic rod controlled by a piezoelectric actuator

The longitudinal vibrations of an elastic rod controlled by normal forces in the cross section, which are uniformly distributed along the length over a selected interval, are studied. Such a system can be implemented using an actuator consisting of piezoelectric elements located along the axis of the rod. Criteria for the uncontrollability of individual vibration modes are given. A generalized solution to the initial-boundary value problem is found applying d’Alembert traveling waves, which are determined on the space-time mesh formed by characteristics. Linear combinations of the traveling wave and control functions define the sought displacements and dynamic potential in the energy space. The latter in a certain way relates the momentum density and the force in the cross section. The problem is to transfer the rod to a prescribed state in a fixed time while minimizing the norm of the control force. The optimal motion and the corresponding feedforward control law are found by reducing the original problem to a one-dimensional variational one. The example shows the control of vibrations for certain geometric parameters of the piezoelectric actuator.