Models for piezoelectric beams and structures with piezoelectric patches generally ignore magnetic effects. This is because the magnetic energy has a relatively small effect on the overall dynamics. Piezoelectric beam models are known to be exactly observable and can be exponentially stabilized in the energy space by using a mechanical feedback controller. In this paper, a variational approach is used to derive a model for a piezoelectric beam that includes magnetic effects. It is proved that the partial differential equation model is well-posed. Magnetic effects have a strong effect on the stabilizability of the control system. For almost all system parameters the piezoelectric beam can be strongly stabilized, but it is not exponentially stabilizable in the energy space. Strong stabilization is achieved using only electrical feedback. Furthermore, using the same electrical feedback, an exponentially stable closed-loop system can be obtained for a set of system parameters of zero Lebesgue measure. These results are compared to those of a beam without magnetic effects.
Introduction.Piezoelectric actuators have a unique characteristic of converting mechanical energy to electrical and magnetic energy, and vice versa. Therefore, they could be used as actuators or sensors. Piezoelectric actuators are generally scalable, smaller, less expensive, and more efficient than traditional actuators, and hence a competitive choice for many tasks in industry, particularly those involving control of structures. Piezoelectric materials have been employed in civil, industrial, automotive, aeronautic, and space structures.In modeling piezoelectric systems, three major effects and their interrelations need to be considered: mechanical, electrical, and magnetic. Mechanical effects are generally modeled through Kirchhoff, Euler-Bernoulli, or Mindlin-Timoshenko small displacement assumptions; see, for instance, [3], [12], [28], [39]. To include electrical and magnetic effects, there are mainly three approaches: electrostatic, quasi-static, and fully dynamic [29]. Electrostatic and quasi-static approaches are widely usedsee, for instance, [10], [12], [14], [17], [25], [28], [29], [35]. These models completely exclude magnetic effects and their coupling with electrical and mechanical effects. In a electrostatic approach, electrical effects are stationary, even though the mechanical equations are dynamic. In the case of a quasi-static approach, magnetic effects are still ignored but electric charges have time dependence. The electromechanical coupling is not dynamic.A piezoelectric beam is an elastic beam with electrodes at its top and bottom surfaces, insulated at the edges (to prevent fringing effects) and connected to an external electric circuit. (See Figure 1). These are the simplest structures on which to study the interaction between the electrical and mechanical energy in these systems. It