Stabilization of mechanical systems with distributed piezoelectric sensors and actuators

The paper is concerned with the stabilization of an elastic beam subjected to an time-dependent axial forcing. The direct Liapunov method is proposed to establish criteria for the almost sure stochastic stability of the unperturbed (trivial) solution of the structure with closed-loop control. We construct the Liapunov functional as a sum of the modified kinetic energy and the elastic energy of the structure The dissipation of energy described by a viscous model of internal damping in the structure material is not sufficient to extract the energy coming from the parametric excitation. An influence of the damping in the finite bonding layer is described by the effective retardation time of the Voigt-Kelvin model. The distributed control is realized by the piezoelectric sensor and actuator, with the changing widths, glued to the upper and lower beam surface. The paper is devoted to the stability analysis of the closed-loop system described by the stochastic partial differential equation without a finite-dimensional approach. The effective stabilization conditions implying the almost sure stability are the main results. The fluctuating axial force is modelled by the physically realizable ergodic process. The rate velocity feedback is applied to stabilize the panel parametric vibrations. Calculations are performed for the Gaussian process with given mean value and variance as well as for the harmonic process with an amplitude A.