#### Title

Modeling and stabilization of current-controlled piezoelectric beams with dynamic electromagnetic field,

#### Abstract

Piezoelectric materials can be controlled with current (or charge) as the electrical input,instead of voltage. The main purpose of this paper is to derive the governing equations for a current-controlled piezo-electric beam and to investigate stabilizability. The magnetic permeability in piezo-electric materials is generally neglected in models. However, it has a significant qualitative effect onproperties of the control system such as stabilizability. Besides the consideration of current control,there are several new aspects to the model. Most importantly, a fully dynamic magnetic model isincluded. Also, electrical potential and magnetic vector potential are chosen to be quadratic-throughthickness to include the induced effects of the electromagnetic field. Hamilton’s principle is usedto derive a boundary value problem that models a single piezo-electric beam actuated by a current(or charge) source at the electrodes. Two sets of decoupled system of partial differential equations areobtained; one for stretching of the beam and another one for bending motion. Since current (or charge)controller only affects the stretching motion, attention is focused on control of the stretching equationsin this paper. It is shown that the Lagrangian of the beam is invariant under certain transformations.A Coulomb type gauge condition is used. This gauge condition decouples the electrical potentialequation from the equations of the magnetic potential. A semigroup approach is used to prove thatthe Cauchy problem is well-posed. Unlike voltage actuation, a bounded control operator in the naturalenergy space is obtained. The paper concludes with analysis of stabilizability and comparison withother actuation approaches and models.

#### Disciplines

Acoustics, Dynamics, and Controls | Applied Mathematics | Applied Mechanics | Control Theory | Dynamics and Dynamical Systems | Engineering Mechanics | Mechanics of Materials | Other Engineering Science and Materials | Other Materials Science and Engineering | Partial Differential Equations

#### Recommended Repository Citation

Ozer, Ahmet Ozkan and Morris, Kirsten. (2019). Modeling and stabilization of current-controlled piezoelectric beams with dynamic electromagnetic field,. *ESAIM: Control, Optimisation and Calculus of Variations*.

**Available at:**
https://digitalcommons.wku.edu/math_fac_pub/96