# On modelling and stabilizability of voltage controlled piezoelectric
material

Matthijs C. de Jong and Jacquelien M. A. Scherpen  M.C. de Jong and J.M.A.
Scherpen are with Jan C. Wilems Center for Systems and Control, ENTEG, Faculty
of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG
Groningen, the Netherlands (email: { matthijs.de.jong,
j.m.a.scherpen}@rug.nl).

###### Abstract

In this paper, we present a new piezoelectric actuator and piezoelectric
composite model and show the well-posedness of these systems. Furthermore, we
show that the piezoelectric composite is stabilizable for certain system
parameters. In this work, we also review several piezoelectric beams,
actuators, and composite models and provide improved definitions of the
different electromagnetic considerations, i.e. fully dynamic electromagnetic
field, quasi-static electric field, and the static electric field assumption.

###### Index Terms:

Modelling, piezoelectric beam, piezoelectric actuator, piezoelectric
composite, Maxwell’s equations, electromagnetic considerations, voltage
control, current control through the boundary

##  I Introduction

1D

    one-dimensional
2D

    two-dimensional
A.S.

    Assymptotically Stabilizable
BIBO

    Bounded Input Bounded Output
CbI

    Control by interconnection
EB

    Euler-Bernoulli
EBBT

    Euler-Bernoulli beam theory
IBP

    integration by parts
IDA

    Interconnection and Damping Assignment
PBC

    Passivity-based control
PD

    Proportional-Derivative
PDE

    Partial Differential Equation
PDEs

    Partial Differential Equations
pH

    port-Hamiltonian
PI

    Proportional-Integral
PID

    Pro-portional-Integral-Derivative
PZT

    lead zirconate titanate
TBT

    Timoshenko beam theory

A piezoelectric actuator is a piece of piezoelectric material sandwiched
between two layers of electrodes. An electric stimulus, such as voltage,
charge, or current, can compress or elongate the actuator in one or more
directions. A specific type of piezoelectric actuator is the piezoelectric
beam, where an electric stimulus acting on the transverse axis incurs
deformation in the longitudinal direction. By bonding a piezoelectric actuator
onto the surface of a mechanical substrate, the deformation of the actuator
incurs shear stress in the substrate, which curves the composition. We refer
to a mechanical substrate with one or more piezoelectric actuators as a
piezoelectric composite, see Fig 1 and is useful in high precision
applications. Piezoelectric beams and composites are considered to be
capacitor dielectrics, see for instance [1].

Figure 1: The piezoelectric actuator consists of a piezoelectric strip,
including the top and bottom electrodes. Bonding the piezoelectric actuator to
a mechanical substrate results in a piezoelectric composite, which can deflect
longitudinally $v$ and transversely $w$ due to the shear stresses between the
two layers.

From a control perspective, roughly two types of applications exist for
piezoelectric actuators and composites, i.e. vibration control and shape
control. The former has applications in acoustic devices as transducer [2] or
suppression of vibrations in mechanical applications [3]. The latter type
envelopes applications, such as flexible wings [4], inflatable space
structures [5], and deformable mirrors [6], where high precision positioning
is of great importance. Often in applications, one side of the composite has a
specific function (e.g. reflecting and focusing electromagnetic waves), such
as inflatable space structures and deformable mirrors.

The dynamics describing the behaviour of a piezoelectric composite originate
from continuum mechanics (mechanical domain) and Maxwell’s equations
(electromagnetic domain) and result in a set of Partial Differential Equation
(PDE). PDE models with different structures and properties are derived by
changing the mechanical or electromagnetic domain assumptions. Several
different types of beam theory can be assumed for the mechanical domain.
Moreover, non-linear phenomena can be incorporated. For instance, it is well-
known that piezoelectric material shows complex non-linear behaviour [7], such
as hysteretic behaviour caused by saturation of the polarization through high
electric field strengths applied to the piezoelectric material. Moreover,
drifting behaviour in strain measurements has been observed, which is known as
creep. In [8], a method which compensates for the ferroelectric hysteretic
behaviour by calculating an inverse operator based on the Prandtl-Ishlinski
theory is proposed. Another approach, based on passivity, can be found in [9,
10, 11] and [12] for Preisach hysteresis models. In [13], the inverse operator
based on the Prandtl-Ishlinski theory is adjusted to compensate for the non-
linear creep phenomena. Using such a systematically calculated inverse
operator mitigates the non-linear behaviour, and the linear behaviour for
piezoelectric beams is recovered, paving the way for stabilizability and other
controller designs.

Different assumptions on the treatment of Maxwell’s equations for the
electromagnetic domain lead to different dynamics and coupling of the
electric, magnetic, and mechanical quantities. These assumptions are often
categorised as a static electric field, a quasi-static electric field, or a
fully dynamic electromagnetic field. Recent efforts show that the treatment of
the electromagnetic domain and choice of input severely alters the
controllability and stabilizability properties of piezoelectric beams and
composites [14, 15, 16]

The traditional choice of actuation is voltage actuation. Voltage-actuated
linear infinite-dimensional piezoelectric beam models are exactly controllable
and exponentially stabilizable in the case of a static or quasi-static
electric field, see [17], [18], and [19]. In [14], the fully dynamic voltage
actuated beam model is shown to be asymptotically stabilizable for almost all
system coefficients and exponentially stabilizable for a small set of system
coefficients. In [20], it has been shown that the fully dynamical beam is
polynomially stabilizable when certain conditions on the physical coefficients
are satisfied.

###  I-A Contributions

In this work, we focus on (linear) voltage-controlled piezoelectric beams,
actuators and composites. We derive the model of a linear, fully dynamic
piezoelectric actuator model, where we employ a similar modelling approach
applied to a piezoelectric beam in [14]. The difference between the
piezoelectric actuator model and the piezoelectric beam model in [14] is the
incorporation of the rotational inertia, resulting in the more general
piezoelectric actuator model in our case. The derived piezoelectric actuator
model is more useful for developing piezoelectric composite models as we show
it can be interconnected with a mechanical substrate in a straightforward
manner. Furthermore, the models presented in this work differ from the non-
linear (current-controlled) piezoelectric actuator and composite from [15],
which are written in the port-Hamiltonian formalism [21], through different
assumptions for the mechanical domain, i.e. in [15] the non-linear Timoshenko
beam theory (TBT) is used. In contrast to our linear voltage-controlled
systems.

The setup of this paper allows us to present a comprehensive modelling
framework, which allows for a comparison of several models of piezoelectric
beams and actuators from the literature in both classical and port-Hamiltonian
(pH) formalism. Therefore, we use a hybrid formalism by writing the underlying
Partial Differential Equations (PDEs) and aligning the variables with the
energy variables of the systems to allow for a comparison of the dynamics and
corresponding energy functionals. Furthermore, we pay special attention to the
treatment of the electromagnetic domain. We present the following key
contributions:

  1. 1.

we derive a more comprehensive, fully dynamic electromagnetic, piezoelectric
actuator model, which takes the rotational inertia into account as opposed to
the piezoelectric beam in [14], and show the system is well-posed.

  2. 2.

we show that the linearized fully dynamic piezoelectric beam in [14] is a
special case of the fully dynamic piezoelectric actuator we derive in this
work through centroidal coordinates. Furthermore, we show that the fully
dynamic nonlinear piezoelectric actuator and composite presented in [15]
exploits the so-called boundary ports of the port-Hamiltonian (pH) framework
to obtain a current input. We show that the model is part of the family of
voltage-controlled piezoelectric actuators, and for the linear case, the model
corresponds to our developed fully dynamic electromagnetic, voltage-controlled
actuator model.

  3. 3.

we derive a well-posed voltage-actuated, fully dynamic electromagnetic,
piezoelectric composite model by interconnecting the piezoelectric actuator
with a purely mechanical substrate.

  4. 4.

we show that the fully dynamic voltage-controlled piezoelectric composite is
useful and asymptotically stabilizable.

  5. 5.

we amplify the different electromagnetic assumptions, i.e. the static electric
field, the quasi-static electric field, and the fully dynamic electromagnetic
field by giving special attention to the effect of these assumptions on the
dynamical equations and corresponding interpretation.

###  I-B Outline

This paper is organised as follows. In Section II, we derive the fully dynamic
piezoelectric actuator model and show that the model is well-posed. In Section
III, we reduce our piezoelectric actuator to a piezoelectric beam and compare
piezoelectric beams presented in the literature. Furthermore, we present an
overview of the different electromagnetic considerations and argue for a
different notion for the quasi-static electric field assumption as presented
in [14]. In Section IV, we make a comparison of the well-posed piezoelectric
actuator derived in Section II to the findings in [15]. Subsequently, in
Section V, we proceed with the proposed piezoelectric actuator model and
derive the mathematical model for a fully dynamic electromagnetic,
piezoelectric composite. In Section VI, we show that the voltage-controlled
piezoelectric composites are asymptotically stabilizable through the feedback
of an electric nature and provide some illustrative simulations to accompany
the stabilizability results in Section VII. Finally, in Section VIII, we give
some concluding remarks and future research directions.

##  II Modelling a voltage actuated piezoelectric actuator with a fully
dynamic electromagnetic field

Let us consider a volume $\Omega$ with length $\ell$, width $2g_{b}$, and
thickness $h=h_{b}-h_{a}$ in the Cartesian coordinate system
$z_{1},z_{2},z_{3}$ with unit vectors $(\bm{z_{1}},\bm{z_{2}},\bm{z_{3}})$, as
depicted in Fig 1. The body of $\Omega$ can be defined as follows

$\displaystyle\Omega:=$ $\displaystyle\\{(z_{1},z_{2},z_{3})\ \mid 0\leq
z_{1}\leq\ell,$ $\displaystyle\ -g_{b}\leq z_{2}\leq g_{b},\ h_{a}\leq
z_{3}\leq h_{b}\\},$

where we assume that the length $\ell$ is significantly larger than the width
and thickness of the volume. We denote the longitudinal deformation along
$z_{1}$ by $v(z_{1},t)$, the transverse deformation along $z_{3}$ by
$w(z_{1},t)$, and the rotation of the beam given by $\phi(z_{1},t)$. From now
on, we omit the dependency of time $t$, when it is clear from the setting. We
consider the one-dimensional (1D) linear Euler-Bernoulli beam theory (EBBT),
see [22] for instance, which leads to the displacement-vector

$\displaystyle\bm{u}=\begin{bmatrix}u_{1}\\\ u_{2}\\\
u_{3}\end{bmatrix}=\begin{bmatrix}v(z_{1})-\bm{z_{3}}\frac{\partial}{\partial
z_{1}}w(z_{1})\\\ 0\\\ w(z_{1})\end{bmatrix},$ (1)

The strain $\bm{\epsilon}$, in the material is a result of (local)
displacements, and is denoted by the vector

$\displaystyle\bm{\epsilon}=\begin{bmatrix}\epsilon_{11}&\epsilon_{22}&\epsilon_{33}&\epsilon_{23}&\epsilon_{31}&\epsilon_{21}\end{bmatrix}^{T}$
(2)

composed of the axial strains $\epsilon_{ii}$ and shear strains
$\epsilon_{ij}$ for $i\neq j$ for $i,j\in 1,\dots,3$. The subscripts $i,j$ are
aligned with the coordinate system $z_{1},z_{2},z_{3}$. Using the linear
strain expression

$\displaystyle\epsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial
z_{j}}+\frac{\partial u_{j}}{\partial z_{i}}\right).$ (3)

on the Euler-Bernoulli (EB) displacement vector (1), we obtain the EB beam
strain

$\displaystyle\epsilon_{11}=\frac{\partial v}{\partial
z_{1}}(z_{1})-\bm{z_{3}}\frac{\partial^{2}w}{\partial z_{1}^{2}}(z_{1}).$ (4)

for the 1D piezoelectric actuator. The stress vector $\bm{\sigma}$ is given by

$\displaystyle\bm{\sigma}=\begin{bmatrix}\sigma_{11}&\sigma_{22}&\sigma_{33}&\sigma_{23}&\sigma_{31}&\sigma_{21}\end{bmatrix}^{T}$
(5)

and is related to (2) by the stiffness matrix $C^{E}$ in Hooke’s law
$\bm{\sigma}=C^{E}\bm{\epsilon}$ [22].

To model the fully dynamic electromagnetic, piezoelectric actuator, we
consider two types of energy; the mechanical energies, composed of the kinetic
co-energy $(T^{\ast})$ and the potential mechanical energy $(V)$, and the
electromagnetic energies, composed of the electric energy $(\mathcal{E})$ and
magnetic energy $(\mathcal{M})$. Let $\rho$ denote the mass density of the
material and let the vectors $\bm{D}$, $\bm{E}$, $\bm{B}$ and $\bm{H}$ denote
the electric displacement, the electric field, the magnetic field, and the
magnetic field intensity, respectively. Then, the aforementioned energies are
given as follows,

$\displaystyle T^{\ast}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\rho(\dot{\bm{u}}\cdot\dot{\bm{u}})~{}d\Omega,$
(6a) $\displaystyle V$
$\displaystyle=\frac{1}{2}\int_{\Omega}\bm{\sigma}\cdot\bm{\epsilon}~{}d\Omega,$
(6b) $\displaystyle\mathcal{M}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\bm{H}\cdot\bm{B}\ d\Omega,$ (6c)
$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{2}\int_{\Omega}\bm{D}\cdot\bm{E}\ d\Omega,$ (6d)

where we omit the spatial dependency from the notation (on $z_{1}$). To
describe the behaviour of the electromagnetic field we require Maxwell’s
equations for dielectrics and the piezoelectric constitutive relations for
permeable material [23, 24]. Let $\mu$ represent magnetic permeability of the
material and denote the volume charge density by $\sigma_{v}$. Then, Maxwell’s
equations [23] are given by the four laws;

$\displaystyle\nabla\times\bm{E}$
$\displaystyle=-\frac{\partial\bm{B}}{\partial t},$ (7a)
$\displaystyle\nabla\cdot\bm{D}$ $\displaystyle=\sigma_{v},$ (7b)
$\displaystyle\nabla\times\bm{H}$
$\displaystyle=\frac{\partial\bm{D}}{\partial t},$ (7c)
$\displaystyle\nabla\cdot\bm{B}$ $\displaystyle=0,$ (7d)

corresponding with Faraday’s law (7a) for time varying magnetic fields,
Gauss’s law (7b) for electric fields, Ampere’s law (7c) that describes the
generation of a magnetic field by current densities, and Gauss’s law (7d) of
magnetism, respectively. Note that the (free) current density $\bm{J}_{f}$ is
neglected in (7c) as is common for dielectrics. However, bound charges are
present as they are attached to the atoms in dielectric material [25, 24, 23].
For the linear isotropic material properties, we consider additionally the two
constitutive relations

$\displaystyle\bm{D}$ $\displaystyle=\varepsilon\bm{E},$ (8a)
$\displaystyle\mu\bm{H}$ $\displaystyle=\bm{B},$ (8b)

In this work we consider piezoelectric beams, actuators and composites that
are actuated by applying an electric potential difference (voltage $V(t)$)
across the $z_{3}$ direction of the piezoelectric layer such that $E_{3}\neq
0$ and $E_{1}=E_{2}=0$. Therefore, (7) and (8) are reduced to scalar equations
with $E_{3},D_{3},H_{2}$ and $B_{2}$ as the remaining nonzero physical
quantities by straightforward calculations. In particular, for Ampere’s law
(7c) and (8b) we obtain the scalar equation

$\displaystyle\mu\dot{D}_{3}$ $\displaystyle=\frac{\partial}{\partial
z_{1}}B_{2}.$ (9)

Let’s define the charge $q(z_{1})$ similarly to [14], by

$\displaystyle q(z_{1})$ $\displaystyle:=\int_{0}^{z_{1}}{D}_{3}(\xi)d\xi,$
(10)

with

$\displaystyle\frac{\partial q}{\partial z_{1}}(z_{1})=D_{3}(z_{1}).$ (11)

Then, integrating (9) from $0$ to $z_{1}$ provides the relation between the
charge $q$ and magnetic field $B_{2}$ as follows,

$\displaystyle\mu\dot{q}(z_{1})=B_{2}(z_{1}).$ (12)

The expressions (11) and (12) are used in writing the energies (6) in scalar
form applicable to the considered voltage actuated piezoelectric actuator. The
explicit forms of the energies (6) of a voltage-actuated piezoelectric
actuator are as follows,

$\displaystyle T^{\ast}$
$\displaystyle=\frac{1}{2}\rho\int_{\Omega}\dot{u}_{1}^{2}+\dot{u}_{3}^{2}~{}d\Omega$
$\displaystyle=\frac{1}{2}\rho\int_{\Omega}\dot{v}^{2}+\bm{z_{3}}^{2}\dot{w}^{2}_{z_{1}}-2\bm{z_{3}}\dot{v}\dot{w}_{z_{1}}+\dot{w}^{2}~{}d\Omega.$
(13a) $\displaystyle V$
$\displaystyle=\frac{1}{2}\int_{\Omega}\sigma_{11}\epsilon_{11}~{}d\Omega$
$\displaystyle=\frac{1}{2}\int_{\Omega}\sigma_{11}\left(v_{z_{1}}-\bm{z_{3}}w_{z_{1}z_{1}}\right)~{}d\Omega,$
(13b) $\displaystyle\mathcal{M}^{\ast}$
$\displaystyle=\frac{1}{2}\int_{\Omega}{\frac{1}{\mu}B_{2}^{2}}d\Omega$
$\displaystyle=\frac{1}{2}\int_{\Omega}{\mu\dot{q}^{2}}d\Omega,$ (13c)
$\displaystyle\mathcal{E}$
$\displaystyle=\frac{1}{2}\int_{\Omega}q_{z_{1}}{E}_{3}\ d\Omega,$ (13d)

where we made use of (8), (11), and (12).

###### Remark 1

In (13), we indicate the (kinetic) co-energies with the use of an asterisk.
The co-energy is the dual of energy and is useful for system elements that
store energy. For linear systems, the energy and co-energy are equal; however,
they are expressed in different variables. For non-linear systems, the co-
energy and energy can differ. Here, we use the notion of co-energy as it is
useful to differentiate between the magnetic and electric energies and their
duals in relation to the Lagrangian equation. Many multi-physical systems have
been derived with the use of a Lagrangian equation as the difference between
the (kinetic) co-energy and potential energy. The (kinetic) co-energies are
associated with a flow (i.e. the rate of change of the generalised
displacements), whereas the potential energy is associated with the
generalised displacement [26, 27].

The coupling between the mechanic and electromagnetic domains is facilitated
by the linear piezoelectric constitutive relations

$\displaystyle\left[\begin{array}[]{c}\bm{\sigma}\\\
\bm{D}\end{array}\right]=$ $\displaystyle\left[\begin{array}[]{cc}C^{E}&-e\\\
e^{T}&\varepsilon\end{array}\right]\left[\begin{array}[]{c}\bm{\epsilon}\\\
\bm{E}\end{array}\right],$ (20)

where $C^{E}$ is the $6\times 6$ stiffness matrix, $e$ denotes the $6\times 3$
piezoelectric constants matrix, and $\varepsilon$ is the $3\times 3$ diagonal
permittivity matrix, see for instance [24, 25, 1]. For the 1D piezoelectric
actuator the piezoelectric constitutive relations (20) are reduced to the
scalar equations

$\displaystyle\begin{bmatrix}\sigma_{11}\\\
D_{3}\end{bmatrix}=\begin{bmatrix}C^{E}_{11}&-\gamma\\\
\gamma&\tfrac{1}{\beta}\end{bmatrix}\begin{bmatrix}\epsilon_{11}\\\
E_{3}\end{bmatrix},$ (21)

where $\gamma:=e_{31}$ and $\beta:=\varepsilon_{33}^{-1}$ denote respectively
the piezoelectric constant and impermittivity.

The mathematical model of the voltage-actuated piezoelectric actuator is
derived by applying Hamilton’s principle to a definite integral that contains
the Lagrangian $\mathcal{L}$ and the impressed forces $W$ that allows the
actuation of the piezoelectric actuator by means of an applied voltage.
Therefore, we define the definite integral

$\displaystyle\mathcal{J}$
$\displaystyle:=\int_{t_{0}}^{t_{1}}\mathcal{L}+W~{}dt.$ (22)

Following Remark 1, and with the generalised displacements coordinate for the
electromagnetic part: the charge (10), we point out that the magnetic energy
(13c) behaves as the (kinetic) co-energy and the electric energy (13d) behaves
as the potential energy for the electromagnetic domain. For the voltage-
controlled piezoelectric actuator, we use the Lagrangian

$\displaystyle\mathcal{L}=(T^{\ast}+\mathcal{M}^{\ast})-(V+\mathcal{E}),$ (23)

to derive the dynamical model. The Lagrangian (23) is similar to the
Lagrangian in [14]. However, the explicit energy expressions differ since we
additionally consider the rotational inertia $I_{0}$, which is typical for the
more general piezoelectric actuator model. Incorporating the rotational
inertia’s $I_{0}$ provides an advantage for deriving the model of a
piezoelectric composite, which we illustrate in Section V.

From (21), we see that the coupling of the mechanical and electromagnetic
domain is done on the one hand through the mechanical stress $\sigma_{11}$
present in (13b) and on the other hand through the electric field $E_{3}$
present in (13d). With the use of the energies (13) and the Lagrangian (23),
we note that the coupling between the mechanical and electromagnetic domain is
present in both the mechanical potential energy $\mathcal{V}$ and the electric
energy $\mathcal{E}$.

To actuate the piezoelectric actuator, we define the external work $W$ next.
Therefore, let $\sigma_{A}$ denote the surface charge and $\tilde{V}$ the
electric scalar potential. Then, the external force [28] is written as
follows,

$\displaystyle W=-\int_{\Omega}\sigma_{A}\frac{\partial\tilde{V}}{\partial
z_{3}}d\Omega,$ (24)

and allow the actuation of the piezoelectric actuator by means of an electric
potential $V$.

To derive the piezoelectric actuator model, we write the definite integral
(22) as follows,

$\displaystyle\begin{split}\mathcal{J}=&\int_{t_{0}}^{t_{1}}\frac{1}{2}\int_{0}^{l}\rho\left(A\dot{v}^{2}+I\dot{w}_{z_{1}}^{2}+2I_{0}\dot{w}_{z_{1}}\dot{v}+A\dot{w}^{2}\right)\\\
&+\mu
A\dot{q}^{2}-{C}\left(Av_{z_{1}}^{2}+Iw_{z_{1}z_{1}}^{2}-2I_{0}v_{z_{1}}w_{z_{1}z_{1}}\right)\\\
&-\gamma\beta\left(Av_{z_{1}}-I_{0}w_{z,z_{1}}\right)q_{z_{1}}\\\
&-A{\beta}q_{z_{1}}^{2}+\gamma\beta\left(Av_{z_{1}}-I_{0}w_{z_{1}z_{1}}\right)q_{z_{1}}\\\
&-Aq_{z_{1}}V(t)~{}dz_{1}~{}dt,\end{split}$ (25)

where we made use of the Lagrangian (23), the energies (13), and work (24).
Furthermore, we made use of the reinforced compliance
$C:=C^{E}_{11}+\gamma^{2}\beta$ and cross-sections and inertias defined as
follows,

$\displaystyle A$
$\displaystyle:=\int_{h_{a}}^{h_{b}}\int_{-g_{b}}^{g_{b}}dz_{2}dz_{3}=2g_{b}(h_{b}-h_{a}),$
$\displaystyle I$
$\displaystyle:=\int_{h_{a}}^{h_{b}}\int_{-g_{b}}^{g_{b}}{z}_{3}^{2}dz_{2}dz_{3}=\frac{2}{3}g_{b}(h_{b}^{3}-h_{a}^{3}),$
(26) $\displaystyle I_{0}$
$\displaystyle:=\int_{h_{a}}^{h_{b}}\int_{-g_{b}}^{g_{b}}\bm{z}_{3}dz_{2}dz_{3}=\left[g_{b}{z}_{3}^{2}\right]_{h_{a}}^{h_{b}}=g_{b}(h_{b}^{2}-h_{a}^{2}).$

Application of Hamilton’s principle [29] to (25) and setting the variation of
admissible displacements $\left\\{v,w_{z_{1}},q\right\\}$ to zero, yields the
dynamical model describing the behaviour of a voltage actuated piezoelectric
actuator with fully dynamic electromagnetic piezoelectric actuator as follows,

$\displaystyle\begin{split}\rho\left(A\ddot{v}-I_{0}\ddot{w}_{z_{1}}\right)&=C\left(A{v}_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right)-{\gamma}\beta
Aq_{z_{1}z_{1}}\\\
\rho\left(I\ddot{w}_{z_{1}}-I_{0}\ddot{v}\right)&=C\left(Iw_{z_{1}z_{1}z_{1}}-I_{0}v_{z_{1}z_{1}}\right)+\gamma\beta
I_{0}q_{z_{1}z_{1}}\\\ \mu A\ddot{q}&=\beta
Aq_{z_{1}z_{1}}-\gamma\beta\left(Av_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right)\\\
\end{split}$ (27a) $\displaystyle\begin{split}v(0)=w(0)=w_{z_{1}}(0)=q(0)=0\\\
{C}Av_{z_{1}}(\ell)-C{I_{0}}w_{{z_{1}}{z_{1}}}(\ell)-\gamma\beta
Aq_{z_{1}}(\ell)=0\\\
CIw_{{z_{1}}{z_{1}}}(\ell)-C{I_{0}}v_{{z_{1}}}(\ell)+\gamma\beta
I_{0}q_{z_{1}}(\ell)=0\\\ A\beta
q_{z_{1}}(\ell)-\gamma\beta(Av_{z_{1}}-I_{0}w_{z_{1}z_{1}})(\ell)=-AV(t),\end{split}$
(27b)

on the spatial domain $z_{1}\in[0,\ell]$ with total energy

$\displaystyle\mathcal{H}_{1}(t)$
$\displaystyle=\frac{1}{2}\int_{0}^{\ell}[\rho\left(A\dot{v}^{2}+I\dot{w}_{z_{1}}^{2}-2I_{0}\dot{v}\dot{w}_{z_{1}}\right)+\mu
A\dot{q}^{2}$
$\displaystyle+C\left(Av_{z_{1}}^{2}+Iw_{z_{1}z_{1}}^{2}-2I_{0}v_{z_{1}}w_{z_{1}z_{1}}\right)$
(28) $\displaystyle\left.+\beta
Aq_{z_{1}}^{2}-2\gamma\beta\left(Av_{z_{1}}-I_{0}w_{z_{1}z_{1}}\right)q_{z_{1}}\right]dz_{1}.$

We refer to the model given by (27) as a piezoelectric actuator, which is
comprised of coupled stretching and bending equations, respectively line (1,3)
and (2).

Before we show how this model relates to the existing literature, we show
first the well-posedness of the derived piezoelectric actuator model (27a)
with boundary conditions (27b). Therefore, we firstly define the operator
associated with the PDE (27) and show that the operator is, in fact, a
generator of a strongly continuous semigroup of contractions, with the use of
the Lumer-Philips Theorem [30].

###### Theorem 1

(Lumer-Phillips theorem [30]) The closed and densely defined operator $A$
generates a strongly continuous semigroup of contractions $T(t)$ on $X$, if
and only if both $A$ and its adjoint $A^{*}$ are dissipative, i.e.

$\displaystyle\begin{matrix}\langle
A\bm{x},\bm{x}\rangle_{X}&\leq&0\quad\text{ for all }\quad
x\in\operatorname{Dom}(A),\\\ \langle
A^{*}{\bm{x}},\bm{x}\rangle_{X}&\leq&0\quad\text{ for all }\quad
x\in\operatorname{Dom}(A^{\ast}).\end{matrix}$ (29)

$\Box$

Next, we establish the well-posedness of the piezoelectric actuator in the
sense of semigroup theory [31].

Without loss of generality, let the length of the beam be $\ell=1$, with the
spatial variable $z\in\left[0,\ell\right]$, and define the space

$H^{1}_{0}(0,1):=\left\\{f\in H^{1}(0,1)\mid f(0)=0\right\\},$

with $H^{1}(0,1)$ denoting the first order Sobolev space, moreover, let
$L^{2}(0,1)$ denote the space of square-integrable functions. Inspired by the
energy functional (II), define the linear space

$\displaystyle\mathcal{X}=\left\\{\bm{x}\in H^{1}_{0}(0,1)\times
H_{0}^{1}(0,1)\times H^{1}_{0}(0,1)\right.$ $\displaystyle\left.\times
L^{2}(0,1)\times L^{2}(0,1)\times L^{2}(0,1)\right\\}$

and inner product

$\displaystyle\begin{split}&\langle x,y\rangle_{\mathcal{X}}:=\\\
&\left\langle\left[\begin{array}[]{l}x_{1}\\\ x_{2}\\\
x_{3}\end{array}\right],\left[\begin{array}[]{ccc}CA&-CI_{0}&-\gamma\beta A\\\
-CI_{0}&CI&\gamma\beta A\\\ -\gamma\beta A&\gamma\beta I_{0}&\beta
A\end{array}\right]\left[\begin{array}[]{l}y_{1}\\\ y_{2}\\\
y_{3}\end{array}\right]\right\rangle_{H_{0}^{1}}\\\
&+\left\langle\left[\begin{array}[]{l}x_{4}\\\ x_{5}\\\
x_{6}\end{array}\right],\left[\begin{array}[]{ccc}\rho A&-\rho I_{0}&\\\ -\rho
I_{0}&\rho I&\\\ &&\mu A\end{array}\right]\left[\begin{array}[]{l}y_{4}\\\
y_{5}\\\ y_{6}\end{array}\right]\right\rangle_{{L}^{2}}\\\
&=\int_{0}^{1}\left[CAx_{1}^{\prime}y_{1}^{\prime}+CIx_{2}^{\prime}y_{2}^{\prime}-CI_{0}\left(x_{1}^{\prime}y_{2}^{\prime}+x_{2}^{\prime}y_{1}^{\prime}\right)\right.\\\
&\quad-\gamma\beta
A\left(x_{1}^{\prime}y_{3}^{\prime}+x_{3}^{\prime}y_{1}^{\prime}\right)+\gamma\beta
I_{0}\left(x_{2}^{\prime}y_{3}^{\prime}+x_{3}^{\prime}y_{2}^{\prime}\right)\\\
&\quad+\beta Ax_{3}^{\prime}y_{3}^{\prime}+\rho Ax_{4}y_{4}+\rho
Ix_{5}y_{5}-pI_{0}\left(x_{4}y_{5}+x_{5}y_{4}\right)\\\ &\quad\left.+\mu
Ax_{6}y_{6}\right]dz_{1},\end{split}$ (30)

where the prime indicates the spatial derivative with respect to $z_{1}$. The
inner product $\langle.\ ,.\rangle_{\mathcal{X}}$ induces the norm
$\|\bm{x}\|^{2}_{\mathcal{X}}=\langle\bm{x},\bm{x}\rangle_{\mathcal{X}}=2\mathcal{H}_{1}(t)$
on $\mathcal{X}$, see (II). For simplicity, denote the spatial variable
$z:=z_{1}$, let $\partial_{z}^{i}:=\frac{\partial^{i}}{\partial z^{i}}$, and
define
$\bm{x}:=\begin{bmatrix}v&w_{z}&q&\dot{v}&\dot{w}_{z}&\dot{q}\end{bmatrix}^{T}$
to be the state. Then, the operator

$\begin{gathered}\mathcal{A}:\operatorname{Dom}(\mathcal{A})\subset\mathcal{X}\rightarrow\mathcal{X},\\\
\mathcal{A}=\left[\begin{array}[]{ccc|ccc}&&&1&&\\\ &&&&1&\\\ &&&&&1\\\
\hline\cr\frac{C}{\rho}\partial_{z}^{2}&&-\frac{\gamma\beta}{\rho}\partial_{z}^{2}&&&\\\
&\frac{C}{\rho}\partial_{z}^{2}&&&&\\\
-\frac{\gamma\beta}{\mu}\partial_{z}^{2}&\frac{\gamma\beta I_{0}}{\mu
A}\partial_{z}^{2}&\frac{\beta}{\mu}\partial_{z}^{2}&&&\end{array}\right]\end{gathered}$
(31)

with

$\displaystyle\operatorname{Dom}(\mathcal{A})=\left\\{\bm{x}\in\mathcal{X}\mid\right.$
$\displaystyle~{}\left.CAx_{1}^{\prime}(1)-C{I_{0}}x_{2}^{\prime}(1)-\gamma
Ax_{3}^{\prime}(1)=0,\right.$
$\displaystyle~{}\left.CIx_{2}^{\prime}(1))-C{I_{0}}x_{1}^{\prime}(1)+\gamma
I_{0}x_{3}^{\prime}(1)=0,\right.$ (32) $\displaystyle~{}\left.\beta
Ax_{3}^{\prime}(1)-\gamma\beta(Ax_{1}^{\prime}(1)-I_{0}x_{2}^{\prime}(1))=-AV(t).\right\\}$

is densely defined in $\mathcal{X}$ and describes the behaviour of a voltage-
actuated piezoelectric actuator. To establish the well-posedness of the
operator (31) in the sense of semigroup theory, we set $V(t)=0$ and make use
of the following Lemma.

###### Lemma 1

The adjoint $\mathcal{A}^{\ast}$ of the operator $\mathcal{A}$, defined in
(31), is skew-adjoint. More specifically,

$\displaystyle\mathcal{A}^{\ast}=-\mathcal{A},$

with $\operatorname{Dom}(\mathcal{A})=\operatorname{Dom}(\mathcal{A}^{\ast})$.

###### Proof:

For any $\bm{x}=\begin{bmatrix}x_{1}&\dots&x_{6}\end{bmatrix}^{T}$ and
$\bm{y}=\begin{bmatrix}y_{1}&\dots&y_{6}\end{bmatrix}^{T}$
$\in\operatorname{Dom}(\mathcal{A})$ we have,

$\displaystyle\begin{split}\langle
A\bm{x},\bm{y}\rangle_{y}&=\int_{0}^{1}\left[CAy_{1}^{\prime}x_{4}^{\prime}-CI_{0}y_{2}^{\prime}x_{4}^{\prime}-\gamma\beta
Ay_{3}^{\prime}x_{4}^{\prime}\right.\\\
&+CIy_{2}^{\prime}x_{5}^{\prime}-CI_{0}y_{1}^{\prime}x_{5}^{\prime}+\gamma\beta
I_{0}y_{3}^{\prime}x_{5}^{\prime}\\\ &+\beta
Ay_{3}^{\prime}x_{6}^{\prime}-\gamma\beta
Ay^{\prime}_{1}x_{6}^{\prime}+\gamma\beta I_{0}y_{2}^{\prime}x_{6}^{\prime}\\\
&+CAy_{4}x_{1}^{\prime\prime}-CI_{0}y_{5}x_{1}^{\prime\prime}-\gamma\beta
Ay_{4}x_{3}^{\prime\prime}\\\ &+\gamma\beta
I_{0}y_{5}x_{3}^{\prime\prime}+CIy_{5}x_{2}^{\prime\prime}-CI_{0}y_{4}x_{2}^{\prime\prime}\\\
&+\left.\beta Ay_{6}x_{3}^{\prime\prime}-\gamma\beta
Ay_{6}x_{1}^{\prime\prime}+\gamma\beta
I_{0}y_{6}x_{2}^{\prime\prime}\right]dz_{1}\\\
=-&\int_{0}^{1}\left[\left(CAy_{4}^{\prime}-CI_{0}y_{5}^{\prime}-\gamma\beta
Ay_{6}^{\prime}\right)x_{1}^{\prime}\right.\\\
&+\left(CIy_{5}^{\prime}-CI_{0}y_{4}^{\prime}+\gamma\beta
I_{0}y_{6}^{\prime}\right)x_{2}^{\prime}\\\ &+\left(\beta
Ay_{6}^{\prime}-\gamma\beta\left(Ay_{4}^{\prime}-I_{0}y_{5}^{\prime}\right)\right)x_{3}^{\prime}\\\
&+\left(CAy_{1}^{\prime\prime}-CI_{0}y_{2}^{\prime\prime}-\gamma\beta
Ay_{3}^{\prime\prime}\right)x_{4}\\\
&+\left(CIy_{2}^{\prime\prime}-CI_{0}y_{1}^{\prime\prime}+\gamma\beta
I_{0}y_{3}^{\prime\prime}\right)x_{5}\\\ &+\left(\beta
Ay_{3}^{\prime\prime}-\gamma\beta\left(Ay_{1}^{\prime\prime}-I_{0}y_{2}^{\prime\prime}\right)x_{6}\right]dz_{1}\\\
&+\left[\left(CAy_{1}^{\prime}-CI_{0}y_{2}^{\prime}-\gamma\beta
Ay_{3}^{\prime}\right)x_{4}\right.\\\
&+\left(CIy_{2}^{\prime}-CI_{0}y_{1}^{\prime}+\gamma\beta
I_{0}y_{3}^{\prime}\right)x_{5}\\\ &+\left(\beta Ay_{3}^{\prime}-\gamma\beta
Ay_{1}^{\prime}+\gamma\beta I_{0}y_{2}^{\prime}\right)x_{6}\\\
&+\left(CAx^{\prime}_{1}+CI_{0}x_{2}^{\prime}-\gamma\beta
Ax_{3}^{\prime}\right)y_{4}\\\
&+\left(CIx_{2}^{\prime}-CI_{0}x_{1}^{\prime}+\gamma\beta
I_{0}x_{3}^{\prime}\right)y_{5}\\\ &\left.+\quad\left(\beta
Ax_{3}^{\prime}-\gamma\beta\left(Ax_{1}^{\prime}-I_{0}x_{2}^{\prime}\right)\right)y_{6}\right]_{0}^{1}\\\
&=\langle\bm{x},-\mathcal{A}\bm{y}\rangle_{\mathcal{X}},\end{split}$ (33)

where we made use of integration by parts (IBP), the domain of $\mathcal{A}$
(II), and imposed the boundary conditions

$\displaystyle y_{4}(0)=y_{5}(0)=y_{6}(0)=0,$ $\displaystyle
CAy_{1}^{\prime}(1)-C{I_{0}}y_{2}^{\prime}(1)-\gamma Ay_{3}^{\prime}(1)=0,$
$\displaystyle CIy_{2}^{\prime}(1))-C{I_{0}}y_{1}^{\prime}(1)+\gamma
I_{0}y_{3}^{\prime}(1)=0,$ $\displaystyle\beta
Ay_{3}^{\prime}(1)-\gamma\beta(Ay_{1}^{\prime}(1)-I_{0}y_{2}^{\prime}(1)=0,$

for $\mathcal{A}^{\ast}$, similar to the boundary conditions of $\mathcal{A}$.
Hence, we conclude the skew-symmetry of $\mathcal{A}^{\ast}$, i.e.
$\mathcal{A}^{\ast}=-\mathcal{A}$ and
$\operatorname{Dom}(\mathcal{A}^{\ast})=\operatorname{Dom}(\mathcal{A})$. ∎

Now we are able to establish the well-posedness of the voltage-actuated
piezoelectric actuator in the absence of control.

###### Theorem 2

The operator $\mathcal{A}$, defined in (31), generates a semigroup of
contractions, satisfying $\|T(t)\|\leq 1$ on $\mathcal{X}$.

###### Proof:

The closed and densely defined operator $\mathcal{A}$ satisfies

$\displaystyle\langle\mathcal{A}\bm{x},\bm{x}\rangle_{\mathcal{X}}=\int_{0}^{1}-\left(\left(Ax_{1}^{\prime\prime}-CI_{0}x_{2}^{\prime\prime}-\gamma\beta
Ax_{3}^{\prime\prime}\right)x_{4}\right.$
$\displaystyle\quad-\left(CIx_{2}^{\prime\prime}-CI_{0}x_{1}^{\prime\prime}+\gamma\beta
I_{0}x_{3}^{\prime\prime}\right)x_{5}$ $\displaystyle\quad-\left(\beta
Ax_{3}^{\prime\prime}-\gamma\beta\left(Ax_{1}^{\prime\prime}-I_{0}x_{2}^{\prime\prime}\right)\right)x_{6}$
$\displaystyle\quad+\left(\left(Ax_{1}^{\prime\prime}-CI_{0}x_{2}^{\prime\prime}-\gamma\beta
Ax_{3}^{\prime\prime}\right)x_{4}\right.$
$\displaystyle\quad+\left(CIx_{2}^{\prime\prime}-CI_{0}x_{1}^{\prime\prime}+\gamma\beta
I_{0}x_{3}^{\prime\prime}\right)x_{5}$ $\displaystyle\quad+\left(\beta
Ax_{3}^{\prime\prime}-\gamma\beta\left(Ax_{1}^{\prime\prime}-I_{0}x_{2}^{\prime\prime}\right)\right)x_{6}$
$\displaystyle\quad+[\left(CAx_{1}^{\prime}-CI_{0}x_{2}^{\prime}-\gamma\beta
Ax_{3}^{\prime}\right)x_{4}$
$\displaystyle\quad+\left(CIx_{2}^{\prime}-CI_{0}x_{1}^{\prime}+\gamma\beta
I_{0}x_{3}^{\prime}\right)x_{5}$ $\displaystyle\quad+\left(\beta
Ax_{3}^{\prime}-\gamma\beta\left(Ax_{1}^{\prime}-I_{0}x_{2}^{\prime}\right)x_{6}\right]_{0}^{1}$
$\displaystyle\quad=0\leq 0$ (34)

where we made use of IBP and the domain of $\mathcal{A}$. Furthermore, from
Lemma 1, we have that

$\displaystyle\langle\mathcal{A}^{\ast}\bm{x},\bm{x}\rangle_{\mathcal{X}}=\langle-\mathcal{A}\bm{x},\bm{x}\rangle_{\mathcal{X}}=-\langle\mathcal{A}\bm{x},\bm{x}\rangle_{\mathcal{X}}=0\leq
0$ (35)

In (II) and (35), we have shown that both $\mathcal{A}$ and
$\mathcal{A}^{\ast}$ are dissipative. Hence, with use of the Lumer-Phillips
Theorem (Theorem 1), we conclude that the operator (31) generates a semigroup
of contractions and is therefore well-posed. ∎

In the sections III, IV, V, we use the piezoelectric actuator and show how
this model relates to the piezoelectric beams, actuators and composites models
in the literature, respectively. Furthermore, we will touch upon the different
electromagnetic considerations for modelling piezoelectric beams, actuators,
and composites. Furthermore, in Sections VI, we show the stabilizability
results of our model and add some simulations for simulation and control
purposes in Section VII.

##  III Comparison of piezoelectric beams

In this section, we relate the derived model by comparing it to existing
piezoelectric beam models. A piezoelectric actuator can either be
interconnected with a mechanical substrate to obtain a piezoelectric
composite, the piezoelectric actuator can be interconnected at the boundary
with its environment for specific applications, or the piezoelectric actuator
can be considered on its own by taking centroidal coordinates (i.e.
$h_{b}=-h_{a}$). The latter ensures that the cross-sectional inertia
$I_{0}\rightarrow 0$, see (II). Taking centroidal coordinates reduces the
piezoelectric actuator (27) to the fully dynamic electromagnetic piezoelectric
beam presented in [14] by substituting $I_{0}=0$ into (27a) and (27b),
respectively. The bending equations become decoupled from the voltage
actuation and the stretching equations (36) by taking centroidal coordinates,
see [32] for more detail. The stretching equations remain influenced by the
control input $V(t)$ and are as follows,

$\displaystyle\begin{split}\rho\ddot{v}&=Cv_{z_{1}z_{1}}-\gamma\beta
q_{z_{1}z_{1}}\\\ \mu\ddot{q}&=\beta q_{z_{1}z_{1}}-\gamma\beta
v_{z_{1}z_{1}}\end{split}$ (36a) $\displaystyle\begin{split}v(0)&=q(0)=0\\\
Cv_{z_{1}}&(1)-\gamma\beta q_{z_{1}}(1)=0\\\ \beta q_{z_{1}}&(1)-\gamma\beta
v_{z_{1}}(1)=-V(t).\end{split}$ (36b)

The decoupling of the bending equations from the stretching equations and
voltage-actuation also holds for other beam theories, such as the Timoshenko
or Rayleigh beam theories [14, 33]. Furthermore, in [33], it is shown that
modelling voltage-controlled piezoelectric beams using either the Euler-
Bernoulli or Timoshenko beam theory results in exactly the same dynamical
equations, due to the decoupling of the stretching and bending equations. The
piezoelectric beam model (36), is similar to the piezoelectric beam model in
[32], which is derived by neglecting the rotational inertia (similar to taking
centroidal coordinates) before applying Hamilton’s principle to (25), see
[14]. The fully dynamic piezoelectric beam (36) is well-posed and
exponentially stabilizable for special cases of the system parameters [14].
For another set of system parameters, the system is strongly stabilizable [14,
34]. In this section, we have shown the relation of the piezoelectric actuator
to the piezoelectric beam. In the next section, we continue with the different
electromagnetic considerations and their connection to the dynamics of
piezoelectric beams and actuators.

###  III-A Electromagnetic considerations

Three considerations exist for the electromagnetic domain: a fully dynamic
electromagnetic field, a quasi-static electric field, and a static electric
field, which we touch upon in this section. In this order, they take less and
less dynamic behaviour of the electromagnetic domain into account.

####  III-A1 The fully dynamic electromagnetic field

For both the voltage actuated piezoelectric actuator (27) and piezoelectric
beam (36) a fully dynamic electromagnetic field is assumed. The
electromagnetic field is produced by accelerating electric charges and by
Maxwell’s correction (i.e. $\tfrac{\partial\bm{D}}{\partial t}$) to Ampere’s
original law, ensuring the continuity equation

$\displaystyle\nabla\cdot\bm{J}=-\frac{\partial\sigma}{\partial t},$ (37)

which amounts to the conversation of charge.

###### Remark 2

The current density is composed of free and bound current densities, i.e.

$\displaystyle\bm{J}=\bm{J}_{f}+\bm{J}_{b}$

For dielectric material, all the charges within the material are bound to the
atoms. Therefore, the current densities present within piezoelectric material
are bound, i.e. $\bm{J}\equiv\bm{J}_{b}$. It is also possible to actuate a
piezoelectric beam or actuator by a current, which allows free charges to flow
through the piezoelectric. In that case $\bm{J}_{f}\neq 0$. However, the scope
of this work encompasses voltage-actuated piezoelectric beams, actuators and
composites.

From (7d), it is known that there exist a magnetic vector potential $\bm{A}$
and electric scalar potential $\varphi$ such that $\bm{B}=\nabla\times\bm{A}$
[23, 24, 14]. Therefore, through Ampere’s law (7c) and the use of vector
identities, the expression for the electric field

$\displaystyle E$ $\displaystyle=-\nabla\varphi-\frac{\partial\bm{A}}{\partial
t}$ (38)

is derived. In (38), the dynamic contribution of the magnetic field to the
electric field for the case of a fully dynamic electromagnetic case is shown.
In piezoelectric beams and actuators, the fully dynamic electromagnetic field
is characterised through Faraday’s law (7a) by

$\displaystyle\frac{\partial B_{2}}{\partial t}$ $\displaystyle=\frac{\partial
E_{3}}{\partial z_{1}}$ (39a) $\displaystyle=\beta\frac{\partial
D_{3}}{\partial z_{1}}-\gamma\beta\frac{\partial\epsilon_{11}}{\partial
z_{1}}$ (39b) $\displaystyle\mu A\frac{\partial^{2}q}{\partial t^{2}}$
$\displaystyle=\beta A\frac{\partial^{2}q}{\partial
z_{1}^{2}}-\gamma\beta\left(A\frac{\partial^{2}v}{\partial
z_{1}^{2}}-I_{0}\frac{\partial^{3}w}{\partial z_{1}^{3}}\right),$ (39c)

where we made use of $E_{1}=E_{2}=0$ in (7a) to obtain (39a). Furthermore, in
(39b), we made use of (21), and in (39c), we used the relations (11) and (10)
on both sides and integrate over the cross-section $A$. The characterising
expression (39c) recurs in the dynamics of the piezoelectric actuator(27) and
beam (36) (with $I_{0}\downarrow 0$).

####  III-A2 The quasi-static electric field

Besides the fully dynamic electromagnetic field assumption, a static electric
field or quasi-static electric field assumption can be used [25, 24, 23]. The
quasi-static electric field is obtained by restricting the magnetic field
intensity to Ampere’s original law for magnetism [23] through,

$\displaystyle\nabla\times\bm{H}\equiv 0,$ (40)

such that the electric displacement becomes static, i.e. $\dot{\bm{D}}\equiv
0$. However, this does not imply that the electric field $\bm{E}$ nor the
magnetic field $\bm{B}$ becomes static. Moreover, (7b) remains valid as
opposed to the definition of the quasi-static electric in [14]. We argue that
there are implicit magnetization and polarization properties present in the
piezoelectric material [24, 25]. Let’s denote the magnetic-field by $\bm{M}$,
the electric susceptibility by $\chi$, and define the polarization
$\bm{P}\equiv\varepsilon_{0}\chi\bm{E}$, such that

$\displaystyle\bm{D}$ $\displaystyle=\varepsilon_{0}\bm{E}+\bm{P},$ (41a)
$\displaystyle\bm{B}$ $\displaystyle=\mu_{0}\left(\bm{H}+\bm{M}\right),$ (41b)

where $\varepsilon_{0}$ and $\mu_{0}$ denote the respective permittivity and
permeability in free space. Furthermore, note that
$\varepsilon=\epsilon_{0}(1+\chi)$. Then, by substituting (41a) and (41b) in
both sides of (7c), we obtain

$\displaystyle\nabla\times\left(\mu_{0}\bm{B}\right)-\underbrace{\nabla\times\bm{M}}_{\bm{J}_{M}}$
$\displaystyle=\left(1+\chi\right)^{-1}\frac{\partial\bm{D}}{\partial
t}+\underbrace{\frac{\partial\bm{P}}{\partial t}}_{\bm{J}_{P}},$ (42)

with $\bm{J}_{M}$ and $\bm{J}_{P}$ denoting the respective magnetic and
polarizing current densities, which are bound electric current densities, i.e.
$\bm{J}_{b}=\bm{J}_{M}+\bm{J}_{P}$. Taking the divergence on both sides of
(42) recovers the continuity equation (37) as follows,

$\displaystyle\begin{split}\nabla\cdot\bm{J}_{P}&=-\left(1+\chi\right)^{-1}\frac{\partial}{\partial
t}\left(\nabla\cdot\bm{D}\right)\\\ &=-\frac{\partial}{\partial
t}\underbrace{\left(1+\chi\right)^{-1}\sigma_{v}}_{\sigma},\end{split}$ (43)

where we made use of (7b). This proves that the continuity equation remains
valid for the quasi-static electric field assumption as opposed to the
definition given in [14], as a consequence of the present bound charges due to
the polarization properties of the material. Furthermore, from (40) we know
there exists a magnetic scalar potential $\Phi_{M}$, such that
$\bm{H}\equiv-\nabla\Phi_{M}$, and subsequently,

$\displaystyle\bm{B}$
$\displaystyle=\mu_{0}\left(\bm{M}-\nabla\Phi_{M}\right),$ (44)

showing the influence of the magnetization on the magnetic field $\bm{B}$. Due
to the coupling of the electromagnetic and mechanical domains by (20), we see
that polarization and magnetization occur implicitly under the quasi-static
electric field assumption. More precisely, in (20), the direct coupling
between the electric field $\bm{E}$ and the strain $\bm{\epsilon}$ for
$\bm{D}\equiv 0$ is the effect of the magnetization and polarization effects.
For the case of piezoelectric beams and actuators, we obtain the constraint
$E_{3}=-\gamma\beta\epsilon_{11}$ from (21) combined with $D_{3}=q_{z}\equiv
0$, which can be substituted in the dynamics (27a) and (36) to obtain the
dynamical model for a piezoelectric actuator and beam model under a quasi-
static electric field assumption, respectively.

####  III-A3 The static electric field

The last simplification of the electromagnetic domain to consider is the
static electric field assumption, which considers the charges to be at rest
and is obtained by restricting the electric field through
$\nabla\times\bm{E}=0$. Therefore, with the use of Poincaré’s theorem [35],
the electric field expression is reduced to the static electric field relation

$\displaystyle E=-\nabla\varphi,$ (45)

see for instance [24, 23, 14]. Then, the magnetic contribution to the electric
field $\bm{E}$ is neglected, i.e. $\dot{\bm{A}}\equiv 0$, for the static
electric field assumption, in contrast to (38) for the fully dynamic case.
Moreover, from Faraday’s law (7a) it follows that the magnetic field $\bm{B}$
becomes stationary ($\dot{\bm{B}}=0$) and without loss of generality
$\bm{B}\equiv 0$. Therefore,

$\displaystyle\frac{\partial B_{2}}{\partial t}=\mu\ddot{q}=0,$

such that the right hand side of (39b) becomes an algabraic constraint. For
piezoelectric beams and actuators, the constraint $\beta\tfrac{\partial
D_{3}}{\partial z_{1}}=\gamma\beta\tfrac{\partial\epsilon_{11}}{\partial
z_{1}}$ can be substituted in both (27) and (36), to obtain the respective
dynamics for a piezoelectric actuator and piezoelectric beam under the static
electric field assumption.

The obtained models for piezoelectric beams for the cases of a static electric
field and quasi-static electric field are quite similar, i.e.

$\displaystyle\rho\ddot{v}$ $\displaystyle=av_{z_{1}z_{1}}$ (46)
$\displaystyle v(0)$ $\displaystyle=0,\quad Cv_{z_{1}}=-\gamma V(t)$ (47)

with $a\equiv C_{11}$ or $a\equiv C$, respectively. Under the quasi-static
electric field assumption, the stiffness is slightly higher (i.e.
$C=C_{11}+\gamma^{2}\beta$) than for the models with a static electric field
assumption. The systems (46) are known to be observable and exponentially
stabilizable, see for instance [19, 18, 17], making these models interesting
for controller design, as is shown in [36, 37]. This concludes the comparison
of piezoelectric beams. In the next section, we continue with the comparison
of piezoelectric actuators.

##  IV Comparison of piezoelectric actuators

In this section, we analyse the nonlinear piezoelectric actuator from [15] and
relate it to the piezoelectric actuator (27a) and to the beam models in [14],
for completeness. The nonlinear piezoelectric actuator derived in [15] is a
current controlled nonlinear fully dynamic piezoelectric actuator model and is
derived by interconnecting the models of a transmission line [27] with the
dynamics of a nonlinear Timoshenko beam within the pH-framework. The resulting
interconnected model describes the dynamics of a nonlinear piezoelectric
actuator. With the use of linearization and reduction of the TBT to the EBBT,
we show that the current-controlled nonlinear piezoelectric actuator derived
in [15] is of the same family as the derived voltage-controlled linear
piezoelectric actuator (27).

To describe the nonlinear piezoelectric actuator model from [15], which is
based on the TBT [22], we define $\phi$ as the transverse rotation of the beam
and define $G$ as the shear-strain coefficient. Now we can describe the PDE of
the nonlinear current controlled nonlinear piezoelectric actuator and its
boundary conditions present underneath the pH nonlinear model presented in
[15] as follows,

$\displaystyle\begin{split}\rho(A\ddot{v}-I_{0}\ddot{\phi})=\frac{\partial}{\partial
z}\left(C\left(Av_{z}-I_{0}\phi_{z}+\tfrac{1}{2}Aw_{z}^{2}\right)-\gamma\beta
Aq_{z}\right)&\\\ \rho A\ddot{w}=\frac{\partial}{\partial
z}\left(\left(C\left(Av_{z}-I_{0}\phi_{z}+\tfrac{1}{2}Aw_{z}^{2}\right)-\gamma\beta
Aq_{z}\right)w_{z}\right.\\\
\left.+\tfrac{1}{2}AG\left(w_{z}-\phi\right)\right)&\\\
\rho(A\ddot{\phi}-I_{0}\ddot{v})=\frac{\partial}{\partial
z}\left(C\left(I\phi_{z}-I_{0}\left(v_{z}-\tfrac{1}{2}w_{z}^{2}\right)\right)+\gamma\beta
I_{0}q_{z}\right.\\\ \left.-\tfrac{1}{2}AG\left(\phi-w_{z}\right)\right)&\\\
\mu A\ddot{q}=\frac{\partial}{\partial z}\left(\beta
Aq_{z}-\gamma\beta\left(Av_{z}-I_{0}\phi_{z}+\tfrac{1}{2}Aw_{z}^{2}\right)\right)&\end{split}$
(48a) $\displaystyle\begin{split}v(0)=w(0)=w_{z}(0)=\phi(0)=0&;\\\ \beta
Aq_{z}(0)-\gamma\beta\left(Av_{z}(0)-I_{0}\phi_{z}(0)\right)=0&\\\
C\left(Av_{z}(1)-I_{0}\phi_{z}(1)+\tfrac{1}{2}Aw_{z}^{2}(1)\right)-\gamma\beta
Aq_{z}(1)=0&\\\
C\left(I\phi_{z}(1)-I_{0}\left(v_{z}(1)-\tfrac{1}{2}w_{z}(1)^{2}\right)\right)+\gamma\beta
I_{0}q_{z}(1)=0&\\\
C\left(Aw_{z}^{3}(1)-I_{0}w_{z}(1)\phi_{z}(1)+Av_{z}(1)w_{z}(1)\right)\quad\quad\quad\quad&\\\
+\tfrac{1}{2}AG\left(w_{z}(1)-\phi(1)\right)-\gamma\beta
Aq_{z}(1){w_{z}(1)}=0&\\\ \dot{q}(1)=I(t)&,\end{split}$ (48b)

with cross-section and inertia as in (II). In the last line of (48b), the
current input is present as boundary input through exploiting the so-called
boundary ports of the pH-formalism; see Remark 3 for more details.

The total energy corresponding to the system described by (48) is

$\displaystyle\mathcal{H}_{2}(t)$
$\displaystyle=\frac{1}{2}\int_{0}^{1}\rho\left(A\dot{v}^{2}+I\dot{\phi}^{2}-2I_{0}\dot{v}\dot{\phi}+A\dot{w}\right)+\mu
A\dot{q}^{2}$
$\displaystyle+C\left(Av_{z}^{2}+I\phi_{z}^{2}+\tfrac{1}{4}Aw_{z}^{4}-2I_{0}\left(v_{z}-\tfrac{1}{2}w_{z}^{2}\right)\phi_{z}\right.$
$\displaystyle\left.+Av_{z}w_{z}^{2}\right)+\tfrac{1}{2}AG\left(w_{z}^{2}+\phi^{2}-2w_{z}\phi\right)+\beta
Aq_{z}^{2}$
$\displaystyle-2\gamma\beta\left(Av_{z}-I_{0}\phi_{z}+\tfrac{1}{2}Aw_{z}^{2}\right)q_{z}~{}dz.$
(49)

###### Remark 3

Within the class of distributed parameter systems of the pH-formalism, the
inputs and outputs of a system are represented by the boundary ports, which
are comprised of the flows (the rate of change of the generalized coordinates)
and efforts (forces) on the boundary and are well established [21]. For the
models (48b) and (27), the voltage input is a consequence of a force balance
and is related to the effort boundary port (within the pH-framework). The
current control input of the piezoelectric model in [15] is associated with
the flow boundary port as the rate of change of the charge $q$. In the
classical formulation, the flow boundary port is closely related to a
Dirichlet boundary condition. In a simple case, we have, for instance,
$q(a)=c$, where c is a constant. Therefore, $\dot{q}(a)=0$ with $a$ on the
boundary. For a control input $u$ we get a Dirichlet boundary condition
$q(1)=u$ with $u\equiv y$ being the output of the single integrator system
with state $\nu$ and

$\displaystyle\begin{split}\dot{\nu}&=I(t)\\\ y&=\nu.\end{split}$ (50)

In summary, the current input of the pH system in [15] is represented by a
single integrator system on the boundary within the classical formalism.
Therefore, we refer to this type of current input as a current-through-the-
boundary input. Piezoelectric beams and composites with this type of current
activation have been investigated in [38, 39].

To make the comparison between the linear well-posed piezoelectric actuator
model (27a) and the nonlinear piezoelectric model (48) we set both inputs to
zero, i.e. $V(t)=0$ and $I(t)=0$, respectively. Furthermore, we linearize the
nonlinear stress-strain relation and reduce the TBT to the EBBT. This can be
done by identifying the difference between the a prioir assumptions the Euler-
Bernoulli beam theory and the Timoshenko beam theory are based upon. These
assumptions are for the Euler-Bernoulli beam theory as follows,

###### Assumption 1

A priori Euler-Bernoulli beam assumptions [22]

  1. 1.

The cross-section is rigid on its plane.

  2. 2.

The cross-section rotates around a neutral surface remaining plane.

  3. 3.

The cross-section remains perpendicular to the neutral surface during
deformation.

Assumption 1.3 restricts the rotation $\phi(z)$ of an Euler-Bernoulli beam,
which is only affected by pure bending, i.e. $\phi(z)\equiv w_{z}(z)$. For the
Timoshenko beam theory, Assumption 1.3 is relaxed, and Assumptions 1.1 and 1.2
remain valid. Therefore, we have that the shear stresses are taken into
account for TBT, and allows for a different rotation of the beam. To reduce
the TBT to the EBBT, we impose Assumption 1.3, through $\phi\equiv w_{z}$ on
the linearized version of (48). Hence, we recover the dynamics that satisfy
the three a priori Euler-Bernoulli beam assumptions and obtain the linear and
reduced piezoelectric actuator model as follows,

$\displaystyle\begin{split}\rho\left(A\ddot{v}-I_{0}\ddot{w}_{z_{1}}\right)&=C\left(A{v}_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right)-\gamma\beta
Aq_{z_{1}z_{1}}\\\
\rho\left(I\ddot{w}_{z_{1}}-I_{0}\ddot{v}\right)&=C\left(Iw_{z_{1}z_{1}z_{1}}-I_{0}v_{z_{1}z_{1}}\right)+\gamma\beta
I_{0}q_{z_{1}z_{1}}\\\ \mu A\ddot{q}&=\beta
Aq_{z_{1}z_{1}}-\gamma\beta\left(Av_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right)\\\
\end{split}$ (51a) $\displaystyle\begin{split}v(0)=w(0)=w_{z_{1}}(0)=0\\\
A\beta q_{z_{1}}(0)-\gamma\beta(Av_{z_{1}}(0)-I_{0}w_{z_{1}z_{1}})(0)=0\\\
{C}Av_{z_{1}}(1)-C{I_{0}}w_{{z_{1}}{z_{1}}}(1)-\gamma\beta Aq_{z_{1}}(1)=0\\\
CIw_{{z_{1}}{z_{1}}}(1)-C{I_{0}}v_{{z_{1}}}(1)+\gamma\beta
I_{0}q_{z_{1}}(1)=0\\\ \dot{q}(1)=0,\end{split}$ (51b)

and total energy function similar to (II). Furthermore, we have that the
internal dynamics and energy function coincide with the well-posed
piezoelectric actuator model (27) derived in this work. Therefore, we conclude
that the models investigated here, i.e. (27a), the piezoelectric beam in [14]
and the nonlinear piezoelectric actuator with current-through-the-boundary
actuation (48) from [15] are from the same ’family’ of voltage-actuated
piezoelectric systems. However, the boundary conditions and the rate of change
of the energy differ, allowing for different behaviour and stabilizability
results. From [15], we know that the spatially discretized non-linear
piezoelectric actuator with Timoshenko beam theory satisfies Brocket’s
necessary conditions for stabilizability. However, for the infinite-
dimensional model (51a), this is an open question.

Interestingly, the derivation of the voltage-controlled models in [14] and
[15] have been derived using different modelling approaches. In [14],
Hamilton’s principle has been applied to a Lagrangian. Whereas in [15], the
model of a transmission line and a nonlinear Timoshenko beam model have been
interconnected in the port-Hamiltonian framework in a power-conserving way.
Moreover, the current actuation of (48) originating from [15] is made possible
by exploiting the boundary ports in the port-Hamiltonian framework and
circumvents the use of a single integrator system (50) on the boundary
required in the classical formalism, see Remark 3.

##  V Derivation and comparison of piezoelectric composites

Figure 2:  (a) Examples of piezoelectric composite structures with the
mechanical substrates and piezoelectric layers, respectively, in white and
grey. The neutral axis is indicated by the dotted line. (1) Single-side
piezoelectric layer allowing bending, (2) Sandwiched mechanical substrate for
increased shear forces, depending on actuation, allows for bending/elongation.
(3) Distributed piezoelectric layers are also possible for single-side
actuation. (b) Bending of a beam around point $P$ with different angles
$d\theta,d\theta^{\prime}$.

A piezoelectric composite is physically obtained by bonding at least one
piezoelectric actuator onto a (purely) mechanical substrate and can be done in
various ways; see Fig 2 for some of the flavours. Therefore, we can say that
the piezoelectric actuator is a subsystem of the piezoelectric composite in a
mathematical sense. In this section, we derive the piezoelectric composite
model using linear Euler-Bernoulli beam theory with a piezoelectric layer on
one side, see Fig 2(1).
Within the classical formalism, it is common to redo the application of
Hamilton’s principle to the Lagrangian associated with the piezoelectric
composite, see for instance [20, 39], whereas, in the pH approach, it is
possible to interconnect different subsystems in a power-conserving way and
obtain the mathematical model of the overall model directly. Then, it must
also be possible to obtain an interconnected system model by interconnecting
subsystems within this hybrid framework through careful considerations. Here,
we show this for interconnecting the piezoelectric actuator with a mechanical
substrate to obtain the piezoelectric composite model. The piezoelectric
actuator model (27) takes into account the rotational inertia’s $I_{0}$, such
that the deformation depending on the geometry of the interconnected
piezoelectric actuator and mechanical substrate are treated properly. In
contrast to the piezoelectric beam (36), where $I_{0}\downarrow 0$ and
requires redoing the derivation of the composite equations through Hamilton’s
principle. For the interconnection, a power-conserving interconnection is
required. The power-conserving interconnection translates to equal speeds at
the interconnection surface ensuring the continuity of velocity and balanced
exchange of force (equal yet opposed direction) [21]. For piezoelectric
composites, the continuity of strain at the interconnection ensures the
continuity of velocity. Moreover, the strain across the normal of the neutral
axis remains the same under bending, which can be seen from the computed
strains

$\displaystyle\varepsilon_{E}$
$\displaystyle=\frac{\bar{EF^{\prime}}-\bar{EF}}{\bar{EF}}=\frac{r_{2}d\theta^{\prime}-r_{2}d\theta}{r_{2}d\theta}=\frac{d\theta^{\prime}}{d\theta}-1,$
$\displaystyle\varepsilon_{A}$
$\displaystyle=\frac{\bar{AB^{\prime}}-\bar{AB}}{\bar{AB}}=\frac{r_{1}d\theta^{\prime}-r_{1}d\theta}{r_{1}d\theta}=\frac{d\theta^{\prime}}{d\theta}-1,$

of the bending composite in Fig 2, even though the length of the curves differ
at different heights, e.g. $\bar{EF}>\bar{AB}$.
The mechanical substrate is a mechanical beam model, which can be derived
using beam theory separately via Hamilton’s principle. However, it can also be
recovered from the piezoelectric actuator model by neglecting the
piezoelectric effects [14]. More precisely, in (20), we see that the coupling
between the mechanical domain $\left(\sigma,\epsilon\right)$ and the
electromagnetic domain $\left(D,E\right)$ is evident through the piezoelectric
constants matrix. By neglecting the effects of the coupling, Hooke’s Law and
the constitutive relations for a dielectric medium are recovered. For the
piezoelectric actuator, the mechanic-electromagnetic coupling is a result of
the piezoelectric constant $\gamma$, see (21). Therefore, by letting
$\gamma\rightarrow 0$, the mechanical and electromagnetic domain become
decoupled, resulting in a purely mechanical substrate model

$\displaystyle\begin{split}\rho\left(A\ddot{v}-I_{0}\ddot{w}_{z_{1}}\right)&=C\left(A{v}_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right)\\\
\rho\left(I\ddot{w}_{z_{1}}-I_{0}\ddot{v}\right)&=C\left(Iw_{z_{1}z_{1}z_{1}}-I_{0}v_{z_{1}z_{1}}\right)\\\
\end{split}$ (52a) $\displaystyle\begin{split}v(0)=w_{z_{1}}(0)=0\\\
{C}Av_{z_{1}}(1)-C{I_{0}}w_{{z_{1}}{z_{1}}}(1)\\\
CIw_{{z_{1}}{z_{1}}}(1)-C{I_{0}}v_{{z_{1}}}(1)=0.\\\ \end{split}$ (52b)

The obtained mechanical substrate model (52) can be interconnected with the
piezoelectric actuator (27) by assuming perfect bonding of the two layers.
Then, we obtain the model for the piezoelectric composite on the spatial
domain $z_{1}\in[0,1]$ as follows,

$\displaystyle\begin{split}\rho_{A}\ddot{v}-\rho_{I_{0}}\ddot{w}_{z_{1}}&=C_{A}v_{z_{1}z_{1}}-C_{I_{0}}w_{z_{1}z_{1}z_{1}}-\gamma\beta
A_{p}q_{z_{1}z_{1}}\\\
\rho_{I}\ddot{w}_{z_{1}}-\rho_{I_{0}}\ddot{v}&=C_{I}w_{z_{1}z_{1}z_{1}}-C_{I_{0}}v_{z_{1}z_{1}}+\gamma\beta
I_{0}q_{z_{1}z_{1}}\\\ \mu A_{p}\ddot{q}&=\beta
A_{p}q_{z_{1}z_{1}}-\gamma\beta\left(Av_{z_{1}z_{1}}-I_{0}w_{z_{1}z_{1}z_{1}}\right),\\\
\end{split}$ (53a) $\displaystyle\begin{split}v(0)=w(0)=w_{z}(0)=q(0)=0;\\\
C_{A}v_{z}(\ell)-C_{I_{0}}w_{zz}(\ell)-\gamma\beta Aq_{z_{1}}(\ell)=0,\\\
C_{I}w_{zz}(\ell)-C_{I_{0}}v_{z}(\ell)+\gamma\beta I_{0}q_{z_{1}}(\ell)=0,\\\
A\beta
q_{z_{1}}(\ell)-\gamma\beta(Av_{z_{1}}(\ell)-I_{0}w_{z_{1}z_{1}}(\ell))=-AV(t),\end{split}$
(53b)

with the concatenated system parameters

$\displaystyle\rho_{A}:=\rho_{p}A_{p}+\rho_{s}A_{s},\qquad$ $\displaystyle
C_{A}:=C_{p}A_{p}+C_{s}A_{s},$ (54)
$\displaystyle\rho_{I}:=\rho_{p}I_{p}+\rho_{s}I_{s},\qquad$ $\displaystyle
C_{I}:=C_{p}I_{p}+C_{s}I_{s},$
$\displaystyle\rho_{I_{0}}:=\rho_{p}I_{0},\qquad$ $\displaystyle
C_{I_{0}}:=C_{11,p}I_{0}.$

where the subscripts $p,s$ indicate the system parameters associated with the
piezoelectric actuator (27) and mechanical substrate (52), respectively.
The total energy corresponding to the composite model (53) is as follows

$\displaystyle\begin{split}\mathcal{H}(t)&=\frac{1}{2}\int_{0}^{\ell}\left[\rho_{A}\dot{v}^{2}+\rho_{I}\dot{w}^{2}_{z_{1}}-2\rho_{I_{0}}\dot{v}\dot{w}_{z_{1}}+\mu
A_{p}\dot{q}^{2}\right.\\\
&\quad\left.+C_{A}v_{z_{1}}^{2}+C_{I}w_{z_{1}z_{1}}^{2}-2C_{I_{0}}v_{z_{1}}w_{z_{1}z_{1}}+\beta
A_{p}q_{z_{1}}^{2}\right.\\\
&\quad\left.-2\gamma\beta\left(Av_{z_{1}}-I_{0}w_{z_{1}z_{1}}\right)q_{z_{1}}\right]dz_{1}.\end{split}$
(55)

This concludes the model derivation of the voltage-actuated piezoelectric
composite. Interestingly, the structure and boundary conditions of the
piezoelectric composite (53) and piezoelectric actuator (27) are similar,
although have different system parameters. Therefore, by following the same
procedure as for the piezoelectric actuator (27), it follows that the
piezoelectric composite (53) is well-posed.

The piezoelectric composites with a static electric field and quasi-static
electric field can be obtained from (53) by the procedures discussed in
Section III-A and result in almost identical models with a difference in the
stiffness of the attached piezoelectric composite, similarly to the
piezoelectric beams (46). More precisely,

$\displaystyle\begin{split}\rho_{A}\ddot{v}-\rho_{I_{0}}\ddot{w}_{z}&=\bar{C}_{A}v_{zz}-\bar{C}_{I_{0}}w_{zzz}\\\
\rho_{I}\ddot{w}_{z}-\rho_{I_{0}}\ddot{v}&=\bar{C}_{I}w_{zzz}-\bar{C}_{I_{0}}v_{zz}\end{split}$
(56a) $\displaystyle\begin{split}v(0)=&w_{z}(0)=0\\\
C_{I}w_{zz}(1)-&C_{I_{0}}v_{z}(1)=0\\\
\bar{C}_{A}v_{z}(1)-&\bar{C}_{I_{0}}I_{0}^{p})w_{zz}(1)=-\gamma
AV(t)\end{split}$ (56b)

with the coefficients $\bar{C}_{A}$, $\bar{C}_{I}$, and $\bar{C}_{I_{0}}$
given in Table I, and energy functional

$\displaystyle\mathcal{H}_{(Q)S}(t)$
$\displaystyle=\frac{1}{2}\int_{0}^{\ell}\left[\rho_{A}\dot{v}^{2}+\rho_{I}\dot{w}^{2}_{z_{1}}-2\rho_{I_{0}}\dot{v}\dot{w}_{z_{1}}\right.$
(57)
$\displaystyle\left.+\bar{C}_{A}v_{z_{1}}^{2}+\bar{C}_{I}w_{z_{1}z_{1}}^{2}-2\bar{C}_{I_{0}}v_{z_{1}}w_{z_{1}z_{1}}\right]~{}dz_{1}.$

The piezoelectric composite models under the static or quasi-static electric
field assumption have direct strain injection through the applied voltage by
the absence of dynamics for the electromagnetic domain.

| Static | Quasi-static
---|---|---
$\bar{C}_{A}$ | $C_{A}-\gamma^{2}\beta A^{p}$ | ${C}_{A}$
$\bar{C}_{I}$ | $C_{I}-\gamma^{2}\beta\frac{(I_{0}^{p})^{2}}{A^{p}}$ | ${C}_{I}$
$\bar{C}_{I_{0}}$ | $C_{I_{0}}-\gamma^{2}\beta I_{0}^{p}$ | ${C}_{I_{0}}$
TABLE I: Coefficients for Static and Quasi-static piezoelectric composite,
corresponding to (56)

In [15], a nonlinear piezoelectric composite is derived with the use of the
Timoshenko beam theory. They show that the spatially discretized system
satisfies Brocket’s necessary conditions [40] for stabilizability.
Furthermore, the nonlinear fully dynamic piezoelectric composite model in [15]
can be linearized, and the Timoshenko beam can be reduced to the Euler-
Bernoulli beam theory, such that an equivalent model to (53) is obtained. In
[15], a quasi-static piezoelectric actuator/composite model is proposed with
current actuation. This model does not rely on the current-through-the-
boundary method we discussed in Remark 3 and is, in fact, a purely current
controlled piezoelectric model with quasi-static electric field assumption.
Thus differing significantly from the piezoelectric composite models proposed
in [39]. In [33], two linear piezoelectric composite models with Euler-
Bernoulli and Timoshenko beam theory are derived. The composites in [33]
assume spatially distributed piezoelectric patches across the length of the
mechanical substrate. The voltage-controlled piezoelectric composite model
derived here (53) assumes a piezoelectric layer across the whole length of the
composite, which is useful for applications where one side of the composite
has a specific function such as reflecting and focusing electromagnetic waves,
e.g. in inflatable space structures and deformable mirrors. In the next
section, we show that the fully dynamic voltage-controlled piezoelectric
composite (53) is useful for applications by investigating its stabilizability
properties.

##  VI Feedback stabilization of piezoelectric composites

To stabilize the fully dynamic voltage-controlled piezoelectric composite
model (53), we investigate a Lyapunov-based control strategy and make use of
the following theorem for infinite dimensional systems.

######  Theorem 3 (LaSalle’s Invariance Principle [41])

Let $\mathcal{V}$ be a continuous Lyapunov function for the strongly
continuous semigroup $T(t)$ on $X$ and let the largest invariant subset be
denoted as

$\mathcal{W}:=\\{\bm{x}\mid\tfrac{d}{dt}\mathcal{V}(\bm{x})=0\\}.$

If $x\in X$ and the orbit

$\tilde{\lambda}(\bm{x})=\bigcup_{t\geq 0}T(t)\bm{x}$

is precompact, then for the distance $d(.,.)$, we have that

$\lim_{t\rightarrow\infty}d(T(t)\bm{x},\mathcal{W})=0.$

Here, by invariance of $\mathcal{W}$ under $T(t)$, we mean
$T(t)\mathcal{W}=\mathcal{W}$ for all $t\geq 0$. $\Box$

Recall the corresponding energy functional $\mathcal{H}(t)$ in (55), with the
change of energy along the trajectories of (53) as follows,

$\displaystyle\frac{d}{dt}\mathcal{H}(t)=-V(t)\dot{q}(1)=-\kappa(\dot{q}(1))^{2}\leq
0$ (58)

for control choice $V(t)=\tfrac{\kappa}{A^{p}}\dot{q}(1)$ with $\kappa>0$.
Then, $\mathcal{H}(t)$ is considered a Lyapunov candidate function to prove
the asymptotic stability of the closed system. Let
$\bm{x}=\operatorname{col}(v_{z},w_{zz},q_{z},\rho_{A}\dot{v}-\rho_{I_{0}}\dot{w}_{z},\rho_{I}\dot{w}_{z}-\rho_{I_{0}}\dot{v},{\mu
A^{p}}\dot{q})$ and denote the closed-loop system as follows,

$\displaystyle\begin{split}\dot{\bm{x}}&=\mathcal{A}\bm{x},t>0\\\
\bm{x}(0)&=\bm{x}_{0}\in\operatorname{Dom}(\mathcal{A})\end{split}$ (59)

with the closed and densely defined operator

$\displaystyle\mathcal{A}\bm{x}$
$\displaystyle=\begin{pmatrix}\tfrac{\rho_{I}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{4}^{\prime}+\tfrac{\rho_{I_{0}}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{5}^{\prime}\\\
\tfrac{\rho_{A}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{5}^{\prime}+\tfrac{\rho_{I_{0}}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{4}^{\prime}\\\
\tfrac{1}{\mu A^{p}}x_{6}^{\prime}\\\
C_{A}x_{1}^{\prime}-C_{I_{0}}x_{2}^{\prime}-\gamma\beta A_{p}x_{3}^{\prime}\\\
C_{I}x_{2}^{\prime}-C_{I_{0}}x_{1}^{\prime}+\gamma\beta I_{0}x_{3}^{\prime}\\\
\beta
A_{p}x_{3}^{\prime}-\gamma\beta\left(Ax_{1}^{\prime}-I_{0}x_{2}^{\prime}\right)\end{pmatrix}$
(60)

on the domain

$\displaystyle\operatorname{Dom}(\mathcal{A})=\\{\bm{x}\in\mathcal{X}_{\mathcal{H}}|$
$\displaystyle\quad C_{A}x_{1}(1)-C_{I_{0}}x_{2}(1)-\gamma\beta Ax_{3}(1)=0,$
(61) $\displaystyle\quad C_{I}x_{2}(1)-C_{I_{0}}x_{1}(1)+\gamma\beta
I_{0}x_{3}(1)=0,$ $\displaystyle\quad A\beta
x_{3}(1)-\gamma\beta(Ax_{1}(1)-I_{0}x_{2}(1))+\kappa{x_{6}}(1)=0\\}.$

It is straightforward to show that the operator (60) generates a strongly
continuous semigroup of contractions $T(t)$, which is well-posed in a similar
fashion as is shown for (31). Furthermore, we are able to prove asymptotic
stability of the closed-loop system. Therefore, define the inner-product
$\langle\bm{x},\bm{x}\rangle_{\mathcal{H}}=2\mathcal{H}(t)$ and assume the
following inequalities on the system parameters.

###### Assumption 2

System parameter inequalities for (59).

$\displaystyle\begin{array}[]{rlrl}\rho_{A}\rho_{I}&\neq\rho_{I_{0}}^{2},&C_{I}&\neq\gamma^{2}\beta\frac{(I_{0}^{p})^{2}}{A^{p}},\\\
C_{A}C_{I}&\neq C_{I_{0}}^{2},&C_{I_{0}}&\neq\gamma^{2}\beta A^{p},\\\
C_{A}&\neq\gamma^{2}\beta A^{p},&C_{I_{0}}&\neq n\gamma^{2}\beta
I_{0}^{p},~{},\end{array}$

with $n\in\left\\{1,2\right\\}$

###### Theorem 4

Let the inequalities in Assumption 2 hold and consider the closed-loop system
(60). Furthermore, consider the Lyapunov candidate functions
$\mathcal{V}=\mathcal{H}(t)$. Then, the closed-loop system is well-posed and
asymptotically stable.

###### Proof:

The closed and densely defined operator $\mathcal{A}$ and its adjoint
$\mathcal{A}^{\ast}$ are dissipative, i.e.

$\displaystyle\langle\mathcal{A}\bm{x},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(x_{6}(1))^{2}\leq 0,$
$\displaystyle\langle\mathcal{A}^{*}{\bm{x}},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(x_{6}(1))^{2}\leq 0,$

where we computed $\mathcal{A}^{\ast}$ in a similar fashion as in Lemma 1.
Hence, by use of the Lummer-Phillips Theorem 1, the operator $\mathcal{A}$
generates a strongly continuous semigroup of contractions and is well-posed.
From the Sobolev embedding Theorem [42] we have that
$\operatorname{Dom}(\mathcal{A})$ is compact in $\mathcal{X}_{\mathcal{H}}$
and thus $\mathcal{A}$ is closed. Therefore, the resolvent of $\mathcal{A}$ is
compact for all $\lambda$ in the resolvent set [43]. Using [41](Theorem 3.65),
we see that the orbit $\tilde{\gamma}(\bm{x})$ is precompact and the limit set
is non-empty. It remains to show that the largest invariant subset

$\mathcal{W}=\\{x\mid\dot{\mathcal{V}}(\bm{x})=0\\},$

with $x_{6}(1)\equiv 0$ contains only the zero vector $\bm{0}$. Therefore, let
Assumption 2 hold and recall the boundary conditions
$v(0)=w(0)=w_{z}(0)=q(0)=0$ and compute that the solution to the ode

$\displaystyle\mathcal{A}\bm{\varphi}(z)=\bm{0},$ (62)

is $\bm{\varphi}(z)\equiv\bm{0}$ for
$\bm{\varphi}\in\operatorname{Dom}(\mathcal{A})$. Hence,
$\bm{0}\in\mathcal{W}$ is the only solution contained in $\mathcal{W}$. Then,
by use of LaSalle’s Invariance Principle Theorem 3, we have that

$\displaystyle\lim_{t\rightarrow\infty}d(T(t)\bm{x},\mathcal{W})=d(T(t)\bm{x},\bm{0})=0,$
(63)

for all $\bm{x}\in\operatorname{Dom}(\mathcal{X})$ and conclude that the
closed-loop system of the voltage-controlled fully dynamic electromagnetic
piezoelectric composite is asymptotically stable. ∎

Furthermore, we show the asymptotic stability of the current-through-the-
boundary-controlled fully dynamic electromagnetic piezoelectric composite
(51a), which is of the same family as the asymptotically stabilizable voltage-
controlled piezoelectric composite (53). In [15], it has been shown that the
spatially discretized current-through-the-boundary-controlled piezoelectric
composite satisfies Brocket’s necessary condition for stabilizability. Here,
we continue to prove the infinite-dimensional system is stabilizable.

###### Theorem 5

Let the inequality $C\neq\gamma^{2}\beta$ hold additionally to Assumption 2
and consider the closed-loop operator (60) with domain

$\displaystyle\begin{split}&\operatorname{Dom}(\mathcal{A}_{\dot{q}})=\\{\bm{x}\in\mathcal{X}_{\mathcal{H}}|\\\
&\quad C_{A}x_{1}(1)-C_{I_{0}}x_{2}(1)-\gamma\beta Ax_{3}(1)=0,\\\ &\quad
C_{I}x_{2}(1)-C_{I_{0}}x_{1}(1)+\gamma\beta I_{0}x_{3}(1)=0,\\\ &\tfrac{1}{\mu
A^{p}}x_{6}+\kappa{x_{3}}(1)=0\\},\end{split}$ (64)

obtained by closing the loop of (51a) though the control
$\dot{q}(1)=\mathcal{I}(t)=-(A^{p}\beta(1-\tfrac{\gamma^{2}\beta}{C_{A}}))^{-1}q_{z}(1)$.
Then, the closed-loop system is well-posed and asymptotically stable.

###### Proof:

The closed and densely defined operator $\mathcal{A}_{\dot{q}}$ and its
adjoint $\mathcal{A}_{\dot{q}}^{\ast}$ are dissipative, i.e.

$\displaystyle\langle\mathcal{A}_{\dot{q}}\bm{x},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(x_{3}(1))^{2}\leq 0,$
$\displaystyle\langle\mathcal{A}_{\dot{q}}^{*}{\bm{x}},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(x_{3}(1))^{2}\leq 0,$

where we made use of the boundary condition from the second row of (64), and
computed the adjoint $\mathcal{A}_{\dot{q}}^{\ast}$ in a similar fashion as in
Lemma 1. Hence, by use of the Lummer-Phillips Theorem 1, the operator
$\mathcal{A}_{\dot{q}}$ on the domain generates a strongly continuous
semigroup of contractions and is well-posed.
Furthermore, from the Sobolev embedding Theorem [42] and using [41] (Theorem
3.65), we see that the orbit $\tilde{\gamma}(\bm{x})$ is precompact. It
remains to show that the largest invariant subset

$\mathcal{W}=\\{x\mid\dot{\mathcal{V}}(\bm{x})=0\\},$

with ${q}_{z}(1)\equiv 0$ contains only the zero vector $\bm{0}$. Therefore,
let Assumption 2 and the inequality $C\neq\gamma^{2}\beta$ hold and recall the
boundary conditions $v(0)=w(0)=w_{z}(0)=q(0)=0$ to compute the solution to the
ode

$\displaystyle\mathcal{A}\bm{\varphi}(z)=\bm{0},$ (65)

is $\bm{\varphi}(z)\equiv\bm{0}$ for
$\bm{\varphi}\in\operatorname{Dom}(\mathcal{A}_{\dot{q}})$. Hence,
$\bm{0}\in\mathcal{W}$ is the only solution contained in $\mathcal{W}$. Then,
by use of LaSalle’s Invariance Principle Theorem 3, we have that

$\displaystyle\lim_{t\rightarrow\infty}d(T(t)\bm{x},\mathcal{W})=d(T(t)\bm{x},\bm{0})=0,$
(66)

for all $\bm{x}\in\operatorname{Dom}(\mathcal{X}_{\dot{q}})$ and conclude that
the closed-loop system of the current-through-the-boundary fully dynamic
electromagnetic composite is asymptotically stable. ∎

Furthermore, for completion, we show the asymptotic stability of the current
controlled piezoelectric composite under static and quasi-static electric
field assumption (56). Recall the corresponding energy functional
$\mathcal{H}_{(Q)S}(t)$ (57), with the change of energy along the trajectories
of (56) as follows,

$\displaystyle\frac{d}{dt}\mathcal{H}_{(Q)S}(t)=-\gamma
A^{p}V(t)\dot{v}(t)=-\kappa(\dot{v}(1))^{2}\leq 0,$ (67)

for control choice $V(t)=\tfrac{\kappa}{\gamma A^{p}}\dot{v}(1)$. Then,
$\mathcal{H}_{(Q)S}(t)$ is considered a Lyapunov candidate function. Let
$\bm{x}=\operatorname{col}(v_{z},w_{zz},\rho_{A}\dot{v}-\rho_{I_{0}}\dot{w}_{z},\rho_{I}\dot{w}_{z}-\rho_{I_{0}}\dot{v})$
and denote the closed-loop systems as follows,

$\displaystyle\begin{split}\dot{\bm{x}}\mathcal{A}_{(Q)S}\bm{x},\quad t>0,\\\
\bm{x}(0)=\bm{x}_{0}\in\operatorname{Dom}(\mathcal{A}_{(Q)S}),\end{split}$
(68)

with the closed and densely defined operator

$\displaystyle\mathcal{A}_{(Q)S}\bm{x}=\begin{pmatrix}\tfrac{\rho_{I}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{3}^{\prime}+\tfrac{\rho_{I_{0}}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{4}^{\prime}\\\
\tfrac{\rho_{A}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{4}^{\prime}+\tfrac{\rho_{I_{0}}}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}x_{3}^{\prime}\\\
\bar{C}_{A}x_{1}^{\prime}-\bar{C}_{I_{0}}x_{2}^{\prime}\\\
\bar{C}_{I}x_{2}^{\prime}-\bar{C}_{I_{0}}x_{1}^{\prime}\end{pmatrix},$ (69)

on the domain

$\displaystyle\begin{split}\operatorname{Dom}(\mathcal{A}_{(Q)S})=\\{\bm{x}\in\mathcal{X}_{(Q)S}&|\\\
\quad\bar{C}_{A}x_{1}(1)-\bar{C}_{I_{0}}x_{2}(1)&=0,\\\
\quad\bar{C}_{I}x_{2}(1)-\bar{C}_{I_{0}}x_{1}(1)+\kappa\left((\right)1)&=0,\\\
\quad A\beta x_{3}(1)-\gamma\beta(Ax_{1}(1)-I_{0}x_{2}(1))&\\\
\quad+\kappa\tfrac{\rho_{I}x_{3}(1)+\rho_{A}x_{4}(1)}{\rho_{A}\rho_{I}-\rho_{I_{0}}^{2}}&=0\\}.\end{split}$
(70)

Define the innerproduct
$\langle\bm{x},\bm{x}\rangle_{(Q)S}=2\mathcal{H}_{(Q)S}(t)$ and assume the
following inequalities on the system parameters.

###### Assumption 3

System parameter inequalities for (69)

$\displaystyle\begin{array}[]{rlrl}\rho_{A}\rho_{I}&\neq\rho_{I_{0}}^{2},&\bar{C}_{A}\bar{C}_{I}&\neq\bar{C}_{I_{0}}^{2}.\\\
\end{array}$

Now we show the asymptotically stabilization of for both the static and quasi-
static piezoelectric composite with the following Theorem.

###### Theorem 6

Let the inequalities in Assumption 3 hold and consider the closed-loop system
69. Furthermore, consider the Lyapunov candidate function
$\mathcal{V}=\mathcal{H}_{(Q)S}$. Then, the closed-loop (quasi-)static
voltage-controlled piezoelectric composite is well-posed and asymptotically
stable.

###### Proof:

The closed and densely defined operator $\mathcal{A}_{(Q)S}$ and its adjoint
$\mathcal{A}_{(Q)S}^{\ast}$ are dissipative, i.e.

$\displaystyle\langle\mathcal{A}_{(Q)S}\bm{x},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(\dot{v}(1))^{2}\leq 0,$
$\displaystyle\langle\mathcal{A}_{(Q)S}^{*}{\bm{x}},\bm{x}\rangle_{X}$
$\displaystyle=-\kappa(\dot{v}(1))^{2}\leq 0,$

where we computed $\mathcal{A}_{(Q)S}^{\ast}$ in a similar fashion as in Lemma
1. Hence, by use of the Lummer-Phillips Theorem 1, the operator
$\mathcal{A}_{(Q)S}$ generates a strongly continuous semigroup of contractions
and is well-posed.
Furthermore, from the Sobolev embedding Theorem [42] and using [41](Theorem
3.65 ), we see that the orbit $\tilde{\gamma}(\bm{x})$ is precompact. It
remains to show that the largest invariant subset

$\mathcal{W}=\\{x\mid\dot{\mathcal{V}}(\bm{x})=0\\},$

with $\dot{v}(1)\equiv 0$ contains only the zero vector $\bm{0}$. Therefore,
let Assumption 3 hold and recall the boundary conditions
$v(0)=w(0)=w_{z}(0)=0$ and compute that the solution to the ode

$\displaystyle\mathcal{A}_{(Q)S}\bm{\varphi}(z)=\bm{0},$ (71)

is $\bm{\varphi}(z)\equiv\bm{0}$ for
$\bm{\varphi}\in\operatorname{Dom}(\mathcal{A}_{(Q)S})$. Hence,
$\bm{0}\in\mathcal{W}$ is the only solution contained in $\mathcal{W}$. Then,
by use of LaSalle’s Invariance Principle Theorem 3, we have that

$\displaystyle\lim_{t\rightarrow\infty}d(T(t)\bm{x},\mathcal{W})=d(T(t)\bm{x},\bm{0})=0,$
(72)

for all $\bm{x}\in\operatorname{Dom}(\mathcal{X}_{(Q)S})$ and conclude that
the closed-loop system is asymptotically stable. ∎

In this section, we have shown that the voltage-actuated piezoelectric
composite is asymptotically stabilizable under any of the three
electromagnetic assumptions. For simulation and control purposes, we include
simulations of the voltage-controlled fully dynamic electromagnetic
piezoelectric composite in the following section.

##  VII Simulation results of the asymptotically stabilizing piezoelectric
composite

For simulation purposes, we consider the fully dynamic electromagnetic
piezoelectric composite with the top layer being a lead zirconate titanate
(PZT)-5***https://support.piezo.com/article/62-material-properties
piezoelectric layer and the mechanical substrate (with centroidal coordinates)
is a steel-304†††https://support.piezo.com/article/62-material-properties#pack
mechanical layer of the same dimensions. An overview of the system parameters
is given in Table II. The open-loop and closed-loop simulations, by closing
the loop in a standard passive manner [44], are obtained using the structure-
preserving spatial discretization method [45] with N=20 segments. The time-
discretization is complimented with the variable-step ode23s-solver (build-in
Matlab® solver). Furthermore, to overcome difficulties with the spatial
discretization of (piezoelectric) models using a mixed Finite-Element method
[45], mentioned in [5], we use a trapezoidal spatial integration and time
integration to compute the longitudinal $(v(z,t))$ and transversal deflection
$w(z,t)$, respectively.
In Fig 3, the open-loop dynamics for the transverse tip deflection is
presented after the actuation with 500V for two seconds. The asymptotic
behaviour of the transverse tip deflection of the closed-loop system
$\eqref{ch2:eq:closedloop_sys}$ with $\kappa=10$ is shown in Fig 59, where it
can be seen that the transversal tip deflection is mitigated. Furthermore, in
Fig 5, the state-trajectories of the controlled piezoelectric composite are
presented, which go to zero.

Figure 3: Open-loop tip deflection $(w(1))$ of the piezoelectric composite.
The piezoelectric composite is actuated with 500 V for the first two seconds
and reaches a deflection of four cm at the tip. After two second, the
actuation is stopped and it can be seen that the tip of the composite moves
back to zero and continues to vibrate in a regular fashion. Figure 4: Closed-
loop tip deflection $(w(z))$ \- with $\kappa=10$ after being actuated with 500
V for the first two seconds. Then the asymptotic behaviour of the controlled
piezoelectric composite can be seen.

Figure 5: In (a) the asymptotic stabilizing behaviour of the voltage-controlled longitudinal $(v(z))$ and transverse displacement $(w(z)$ of the complete composite is shown, for $\kappa=10$. In (b), the state trajectories (including velocities) of the voltage-controlled piezoelectric composite are shown to go to zero, for $\kappa=10$. TABLE II: System parameters Geometry | Description | Value
---|---|---
$2g_{b}$ | Layer width | $0,1~{}m$
$h_{b}-h_{a}$ | Layer thicness | $0,01~{}m$
Piezo parameters |  |
$\rho_{p}$ | mass-density | $7950~{}\frac{\text{kg}}{\text{m}^{3}}$
$C_{p}$ | Stiffness | $66\times 10^{9}~{}\frac{\text{N}}{\text{m}^{2}}$
$\gamma$ | Coupling coefficient | $12.54~{}{\frac{\text{C}}{\text{m}^{2}}}$
$\beta$ | Impermittivity | $10^{-6}~{}\frac{m}{F}$
$\mu$ | Magnetic permeability | $1.2\times 10^{-6}~{}\frac{H}{m}$
Substrate parameters |  |
$\rho_{s}$ | mass-density | $8000v~{}\frac{\text{kg}}{\text{m}^{3}}$
$C_{s}$ | Stiffness | $193\times 10^{9}~{}\frac{\text{N}}{\text{m}^{2}}$

##  VIII Conclusion and future work

In this work, we propose a new well-posed voltage-actuated piezoelectric
actuator and composite model with a fully dynamic electromagnetic field and
show that the piezoelectric composite is asymptotically stabilizable for
certain system parameters. Furthermore, we show that for certain system
parameters, the composites under static and quasi-static electric field
assumptions are also asymptotically stabilizable. Therefore, we claim that the
voltage-controlled piezoelectric composites under any electromagnetic
assumption are stabilizable for certain system parameters. Furthermore, we
show that the non-linear current-through-the-boundary-controlled piezoelectric
composite proposed in [15] that exploits the boundary ports in the pH-
formalism is of the same family of the voltage-controlled piezoelectric
systems. The spatial discretization of this particular model satisfies
Brocket’s necessary condition for stabilizability [15], and we improved upon
this by showing the asymptotic stabilizability for the infinite-dimensional
model in the linear case for certain system parameters. Furthermore, we show
that the piezoelectric beam is a special case of the piezoelectric actuator by
neglecting the rotational inertia (by means of centroidal coordinates). We
show that the piezoelectric actuator is useful for developing piezoelectric
composite models, where a piezoelectric actuator is bonded to a mechanical
substrate, as opposed to the piezoelectric beam, which requires a redo of the
model derivation as the rotational inertias are neglected in the strain
expressions. Difficulties with the spatial discretization of (piezoelectric)
models using a mixed Finite-Element method [45], mentioned in [5], are solved
by the use of a trapezoidal spatial integration and time integration to
compute the longitudinal and transverse deflection. A full connection with the
various current controlled models, e.g. [39] and [15], remains and is part of
future research.

## Acknowlegdement

We would like to acknowledge Kirsten A. Morris for her insights and
suggestions during the development of this work.

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  *[PDE]: Partial Differential Equation
  *[TBT]: Timoshenko beam theory
  *[pH]: port-Hamiltonian
  *[PDEs]: Partial Differential Equations
  *[1D]: one-dimensional
  *[EBBT]: Euler-Bernoulli beam theory
  *[EB]: Euler-Bernoulli
  *[IBP]: integration by parts
  *[PZT]: lead zirconate titanate