# Passivity, Port-Hamiltonian Formulation and Solution Estimates for a Coupled
Magneto-Quasistatic System

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###### Abstract.

In this paper, we study a quasilinear coupled magneto-quasistatic model from a
systems theoretic perspective. First, by taking the injected voltages as input
and the associated currents as output, we prove that the magneto-quasistatic
system is passive. Moreover, by defining suitable Dirac and resistive
structures, we show that it admits a representation as a port-Hamiltonian
system. Thereafter, we consider dependence on initial and input data. We show
that the current and the magnetic vector potential can be estimated by means
of the initial magnetic vector potential and the voltage. We also analyse the
free dynamics of the system and study the asymptotic behavior of the solutions
for $t\to\infty$.

###### Key words and phrases:

magneto-quasistatic systems, abstract differential-algebraic equations,
magnetic energy, passivity, port-Hamiltonian systems.

###### 1991 Mathematics Subject Classification:

Primary: 12H20, 34A09, 37L05; Secondary: 35B45, 78A30,

Timo Reis∗

Institut für Mathematik

Technische Universität Ilmenau

Weimarer Str. 32

98693 Ilmenau, Germany

Tatjana Stykel

Institut für Mathematik & Centre for Advanced Analytics and Predictive
Sciences

Universität Augsburg

Universitätsstr. 12a

86159 Augsburg, Germany

(Communicated by the associate editor name)

##  1\. Introduction

A subject of this paper is the investigation of systems theoretic properties
of a quasulinear coupled magneto-quasistatic (MQS) model

$\displaystyle{\tfrac{\partial}{\partial
t}}\left(\sigma\bm{A}\right)+\nabla\times\left(\nu(\cdot,\|\nabla\times\bm{A}\|_{2})\nabla\times\bm{A}\right)=$
$\displaystyle\;\chi\,\bm{i}$ in $\displaystyle\mathit{\Omega}\times(0,T),$
(1a) $\displaystyle\tfrac{{\rm d}}{{\rm
d}t}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm d}\xi+R\,\bm{i}=$
$\displaystyle\;\bm{v}$ on $\displaystyle(0,T),$ (1b)
$\displaystyle\bm{A}\times\bm{n}_{o}=$ $\displaystyle\;0$ in
$\displaystyle\partial\mathit{\Omega}\times(0,T),$ (1c)
$\displaystyle\sigma\bm{A}(\cdot,0)=$ $\displaystyle\;\sigma\bm{A}_{0}$ in
$\displaystyle\mathit{\Omega},$ (1d)
$\displaystyle\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(\cdot,0)\,{\rm d}\xi=$
$\displaystyle\,\int_{\mathit{\Omega}}\chi^{\top}\bm{A}_{0}\,{\rm d}\xi,$ (1e)

where $\mathit{\Omega}\subset\mathbb{R}^{3}$ is a bounded domain with some
further properties specified later on,
$\bm{A}:\overline{\mathit{\Omega}}\times[0,T]\to\mathbb{R}^{3}$ is the
magnetic vector potential, $\nu:\mathit{\Omega}\times\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ is the magnetic reluctivity,
$\sigma:\mathit{\Omega}\to\mathbb{R}_{\geq 0}$ is the electric conductivity,
and $\bm{v},\bm{i}:{[0,T]}\to\mathbb{R}^{m}$ are, respectively, the voltage
and the electrical current through the electromagnetic conductive contacts.
Furthermore, $\chi:\mathit{\Omega}\to\mathbb{R}^{3\times m}$ is the winding
function, which expresses the geometry of $m$ windings,
$R\in\mathbb{R}^{m\times m}$ is the resistance of the winding,
$\bm{n}_{o}:\partial\mathit{\Omega}\to\mathbb{R}^{3}$ is the outward unit
normal vector to the boundary $\partial\mathit{\Omega}$, and
$\bm{A}_{0}:\mathit{\Omega}\to\mathbb{R}^{3}$ is the initial value for the
magnetic vector potential. The voltage $\bm{v}$ is considered as the input,
whereas the current $\bm{i}$ is the output of the system. The dynamics of the
magnetic vector potential is expressed by (1a), which arises from a
simplification of Maxwell’s equations [11], and (1c) means that the magnetic
flux through the boundary $\partial\mathit{\Omega}$ is zero. The term
$\chi\,\bm{i}$ stands for the external current induced by $m$ windings, and
(1b) expresses Kirchhoff’s voltage law including the applied voltage $\bm{v}$,
the resistive voltage $R\,\bm{i}$, and the term $\tfrac{{\rm d}}{{\rm
d}t}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm d}\xi$, which is the voltage
induced by the electromagnetic field. Note that $\sigma$ is positive and
constant on a subdomain $\mathit{\Omega}_{C}\subset\mathit{\Omega}$, whereas
it vanishes on the complement of $\mathit{\Omega}_{C}$. This implies that the
coupled MQS system (1) becomes of degenerate parabolic type.

Existence, uniqueness and regularity properties of solutions to (1) have been
studied in [5, 25, 26] for some special cases, while [11] presents well-
posedness and regularity results for more general setting. The purpose of this
paper is to investigate the dynamic behavior of the coupled MQS model (1) from
a systems theoretic point of view. In doing so, we analyse passivity of this
model by establishing the existence of a certain energy balance. Further, we
show that the MQS system (1) fits into the framework of port-Hamiltonian
systems. The findings on passivity will be the basis for solution estimates by
means of the initial value $\bm{A}_{0}$ and the input voltage $\bm{v}$.
Besides showing that the free dynamics of (1) (that is, the solution behavior
with $\bm{v}\equiv 0$) are bounded and that the $L^{2}$-norm of the curl of
the magnetic vector potential decays exponentially, we present estimates for
the $L^{2}$-norm of the output and the magnetic vector potential in terms of
the initial value, the input and the material parameters.

The paper is organized as follows. In Section 2, we collect our model
assumptions and review some results from [11] on existence, uniqueness and
regularity of solutions of the coupled MQS system (1). We also highlight that
this system admits a representation as a differenial-algebraic system in which
the state evolves in an infinite-dimensional Hilbert space. This section
closes with a brief introduction to the concept of magnetic energy. It plays
an essential role in Section 3, where we establish an energy balance of the
MQS system (1) and prove that this system is passive. In Section 4, we
introduce infinite-dimensional port-Hamiltonian systems via abstract Dirac and
resistive structures and show that the coupled MQS system (1) belongs to this
class. The energy balance from Section 3 is then used in Section 5 to
establish the solution estimates for (1). In Section 5.1, we derive a bound
for the $L^{2}$-norm of the output in terms of the $L^{2}$-norm of the input
and the initial value. This gives, in particular, an estimate for the
generalization of the $\mathcal{H}_{\infty}$-norm to the nonlinear infinite-
dimensional differential-algebraic case. In Section 5.2, we derive estimates
for the magnetic energy and the $L^{2}$-norm of the magnetic vector potential
at some given time $t\geq 0$ by means of the initial value and the
$L^{2}$-norm of the input. Finally, in Section 5.3, we consider the MQS system
(1) in which the zero voltage is applied. We show that the magnetic energy
decays exponentially, and we present estimates for the $L^{2}$-norm of the
magnetic vector potential at time $t\geq 0$.

##  2\. The MQS System: Assumptions, Solvability and Magnetic Energy

Throughout this paper, we use the notation and terminology from [11]. For ease
of reference, we recall here the assumptions on the spatial domain, the
material parameters, the initial value, and the winding function. Moreover, we
present the solution concept and recap the solvability results and some
essential properties of the magnetic energy established in [11] which will be
required in the following.

###  2.1. The model assumptions

First, we impose some assumptions on the spatial domain $\mathit{\Omega}$.

######  Assumption 2.1 (Spatial domain, geometry and topology).

The set $\mathit{\Omega}\subset\mathbb{R}^{3}$ is a simply connected bounded
Lipschitz domain, which is decomposed into two Lipschitz regular, open subsets
$\mathit{\Omega}_{C}$, $\mathit{\Omega}_{I}{\subset}\mathit{\Omega}$, called,
respectively, conducting and non-conducting subdomains, such that
$\overline{\mathit{\Omega}}_{C}{\subset}{\mathit{\Omega}}$ and
$\mathit{\Omega}_{I}=\mathit{\Omega}\setminus\overline{\mathit{\Omega}}_{C}$.
Furthermore, $\mathit{\Omega}_{C}$ is connected, and $\mathit{\Omega}_{I}$ has
finitely many connected internal subdomains
$\mathit{\Omega}_{I,1},\ldots,\mathit{\Omega}_{I,q}$ with single boundary
components $\Gamma_{1},\ldots,\Gamma_{q}$, respectively, and the external
subdomain $\mathit{\Omega}_{I,{\rm ext}}$ which has two boundary components
$\Gamma_{{\rm ext}}=\overline{\mathit{\Omega}}_{I,{\rm
ext}}\cap\overline{\mathit{\Omega}}_{C}$ and $\partial\mathit{\Omega}$.

Next, we state the assumptions on the electric conductivity $\sigma$, the
magnetic reluctivity $\nu$, and the resistance matrix $R$.

######  Assumption 2.2 (Material parameters).

  1. a)

The electric conductivity $\sigma:\mathit{\Omega}\to\mathbb{R}_{\geq 0}$ is of
the form $\sigma=\sigma_{C}\mathbbm{1}_{\mathit{\Omega}_{C}}$, where
$\sigma_{C}>0$ and $\mathbbm{1}_{\mathit{\Omega}_{C}}$ denotes the indicator
function of the subdomain $\mathit{\Omega}_{C}$.

  2. b)

The magnetic reluctivity $\nu:\mathit{\Omega}\times\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ has the following properties:

    1. (i)

$\nu$ is measurable;

    2. (ii)

the function $\zeta\mapsto\nu(\xi,\zeta)\zeta$ is strongly monotone with a
monotonicity constant $m_{\nu}>0$ independent of $\xi\in\mathit{\Omega}$. In
other words, there exists $m_{\nu}>0$ such that

$\bigl{(}\nu(\xi,\zeta)\zeta-\nu(\xi,\varsigma)\varsigma\bigr{)}(\zeta-\varsigma)\geq
m_{\nu}(\zeta-\varsigma)^{2}\enskip\text{ for all
}\xi\in\mathit{\Omega},\enskip\zeta,\varsigma\in\mathbb{R}_{\geq 0};$

    3. (iii)

the function $\zeta\mapsto\nu(\xi,\zeta)\zeta$ is Lipschitz continuous with a
Lipschitz constant $L_{\nu}>0$ independent of $\xi\in\mathit{\Omega}$. In
other words, there exists $L_{\nu}>0$ such that

$|\nu(\xi,\zeta)\zeta-\nu(\xi,\varsigma)\varsigma|\leq
L_{\nu}|\zeta-\varsigma|\quad\text{ for all
}\xi\in\mathit{\Omega},\enskip\zeta,\varsigma\in\mathbb{R}_{\geq 0}.$

  3. c)

The resistance matrix $R\in\mathbb{R}^{m\times m}$ is symmetric and positive
definite.

As the space in which the solutions of the coupled MQS system (1) evolve, we
consider the set $X(\mathit{\Omega},\mathit{\Omega}_{C})$ of all square
integrable functions which are $L^{2}$-orthogonal to all gradient fields of
functions from $H^{1}_{0}(\mathit{\Omega})$ being constant on each interface
component $\Gamma_{1},\ldots,\Gamma_{q}$ and $\Gamma_{\rm ext}$. The space
$X(\mathit{\Omega},\mathit{\Omega}_{C})$ is a Hilbert space equipped with the
standard inner product in ${L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$. We
further consider the space

$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})=H_{0}(\operatorname{curl},\mathit{\Omega})\cap
X(\mathit{\Omega},\mathit{\Omega}_{C}),$ (2)

which is again a Hilbert space, now provided with the inner product in
$H_{0}(\operatorname{curl},\mathit{\Omega})$. For any $\bm{A}\in
X(\mathit{\Omega},\mathit{\Omega}_{C})$, one has
$\mathbbm{1}_{\mathit{\Omega}_{C}}\bm{A}\in
X(\mathit{\Omega},\mathit{\Omega}_{C})$, see [11, Lemma 3.3]. Moreover, [11,
Lemma 3.4] states that
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ is dense in
$X(\mathit{\Omega},\mathit{\Omega}_{C})$.

The space $X(\mathit{\Omega},\mathit{\Omega}_{C})$ enables us to formulate the
assumptions on the initial magnetic vector potential $\bm{A}_{0}$ and the
winding function $\chi$. For the latter, we impose a condition for which it is
necessary that all components of the matrix-valued function
$\chi:\Omega\to\mathbb{R}^{3\times m}$ are square integrable, i.e., $\chi\in
L^{2}(\Omega;\mathbb{R}^{3\times m})$. Note that such a function can, loosely
speaking, be canonically identified with a $m$-tuple of elements of
$L^{2}(\Omega;\mathbb{R}^{3})$. More precisely, we will use the identification

$L^{2}(\Omega;\mathbb{R}^{3\times m})\cong
L^{2}(\Omega;\mathbb{R}^{3})^{1\times m},$ (3)

where elements of $L^{2}(\Omega;\mathbb{R}^{3})^{1\times m}$ are regarded as
operators from $\mathbb{R}^{m}$ to $L^{2}(\Omega;\mathbb{R}^{3})$. The
corresponding operator norm is given by

$\|\chi\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}=\sqrt{\lambda_{\max}\left(\int_{\mathit{\Omega}}\chi^{\top}\chi\,{\rm
d}\xi\right)},$ (4)

where $\lambda_{\max}(\mathcal{X})$ stands for the largest eigenvalue of a
symmetric matrix $\mathcal{X}\in\mathbb{R}^{m\times m}$. Note that the
resulting normed space $L^{2}(\mathit{\Omega};\mathbb{R}^{3\times m})$ is not
a Hilbert space unless $m=1$, since the above norm is not induced by an inner
product.

######  Assumption 2.3 (Initial condition and winding function).

  1. a)

The initial magnetic vector potential
$\bm{A}_{0}:\mathit{\Omega}\to\mathbb{R}^{3}$ belongs to
$X(\mathit{\Omega},\mathit{\Omega}_{C})$.

  2. b)

The winding function $\chi:\mathit{\Omega}\to\mathbb{R}^{3\times m}$ belongs,
by using the identification (3), to
$X(\mathit{\Omega},\mathit{\Omega}_{C})^{1\times m}$.

###  2.2. Solutions of the coupled MQS system

Before discussing the solution properties of the coupled MQS system (1), we
declare what we mean by solutions. Let $T>0$ and $\bm{v}\in
L^{2}([0,T];\mathbb{R}^{m})$. We call $(\bm{A},\bm{i})$ with
$\bm{A}\in\overline{\mathit{\Omega}}\times[0,T]\to\mathbb{R}^{3}$ and
$\bm{i}:[0,T]\to\mathbb{R}^{m}$ a weak solution (or just solution) of the
coupled MQS system (1), if

  1. a)

$\sigma\bm{A}\in C([0,T];X(\mathit{\Omega},\mathit{\Omega}_{C}))\cap H_{\rm
loc}^{1}((0,T];X(\mathit{\Omega},\mathit{\Omega}_{C}))$ and
$\sigma\bm{A}(0)=\sigma\bm{A}_{0}$,

  2. b)

$\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm d}\xi\in
C([0,T];\mathbb{R}^{m})\cap H_{\rm loc}^{1}((0,T];\mathbb{R}^{m})$ and
$\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(0)\,{\rm
d}\xi=\int_{\mathit{\Omega}}\chi^{\top}\bm{A}_{0}\,{\rm d}\xi$,

  3. c)

$\bm{A}\in
L^{2}([0,T];X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}))$
and $\bm{i}\in L_{\rm loc}^{2}((0,T];\mathbb{R}^{m})$,

  4. d)

for all $\bm{F}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ and almost all
$t\in[0,T]$,

$\begin{array}[]{rcl}\displaystyle{\tfrac{\rm d}{{\rm
d}t}\\!\int_{\mathit{\Omega}}\\!\sigma\bm{A}(t)\cdot\bm{F}\,{\rm
d}\xi+\\!\\!\int_{\mathit{\Omega}}\\!\nu(\cdot,\|\nabla\\!\times\\!\bm{A}(t)\|_{2})(\nabla\\!\times\\!\bm{A}(t))\cdot(\nabla\\!\times\\!\bm{F})\,{\rm
d}\xi}&\\!=&\\!\\!\displaystyle{\int_{\mathit{\Omega}}\\!\chi\,\bm{i}(t)\cdot\bm{F}\,{\rm
d}\xi,}\\\\[5.69054pt] \displaystyle{\tfrac{\rm d}{{\rm
d}t}\\!\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(t)\,{\rm
d}\xi+R\,\bm{i}(t)}&=&\\!\bm{v}(t).\end{array}$ (5)

It has been is proven in [11, Theorem 4.4] that, under Assumptions 2.1–2.3,
the solution the coupled MQS system (1) is unique. Existence and some
additional regularity properties of the solution have been established in [11,
Theorem 7.1]. In particular, it has been shown there that for almost all
$t\in[0,T]$,

$\begin{array}[]{rcl}\tfrac{\rm d}{{\rm
d}t}\left(\sigma\bm{A}(t)\right)+\nabla\times\left(\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)\right)&=&\chi\,\bm{i}(t),\\\\[2.84526pt]
\tfrac{\rm d}{{\rm
d}t}\displaystyle{\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(t)\,{\rm
d}\xi+R\,\bm{i}(t)}&=&\bm{v}(t),\end{array}$ (6)

and if, additionally, $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, then the
solution of (1) fulfills

$\displaystyle\nabla\times\bm{A}\in$
$\displaystyle\,L^{\infty}([0,T];X(\mathit{\Omega},\mathit{\Omega}_{C})),$
$\displaystyle\sigma\bm{A}\in$
$\displaystyle\,H^{1}([0,T];X(\mathit{\Omega},\mathit{\Omega}_{C})),$
$\displaystyle\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm d}\xi\in$
$\displaystyle\,H^{1}([0,T];\mathbb{R}^{m}),$ $\displaystyle\bm{i}\in$
$\displaystyle\,L^{2}([0,T];\mathbb{R}^{m}).$

######  Remark 1.

The above solution concept can easily be extended to the positive real axis
$\mathbb{R}_{\geq 0}$ by saying that, for $\bm{v}\in L_{\rm
loc}^{2}({\mathbb{R}_{\geq 0}};\mathbb{R}^{m})$, $(\bm{A},\bm{i})$ with
$\bm{A}:\overline{\mathit{\Omega}}\times{\mathbb{R}_{\geq
0}}\to\mathbb{R}^{3}$ and $\bm{i}:{\mathbb{R}_{\geq 0}}\to\mathbb{R}^{m}$ is a
solution of the coupled MQS system (1) if for any $T>0$, the restriction of
$(\bm{A},\bm{i})$ to $[0,T]$ is a solution of (1). In this case, existence and
uniqueness of solutions immediately follow from the results on finite time
intervals, and the above regularity results can be adapted straightforwardly.

The coupled MQS model (1) can be regarded as an abstract differential-
algebraic control system

$\begin{array}[]{rcl}\tfrac{\rm d}{{\rm
d}t}\scalebox{1.25}{$\mathpzc{E}$}x(t)&=&\scalebox{1.25}{$\mathpzc{A}$}(x(t))+\scalebox{1.25}{$\mathpzc{B}$}u(t),\quad\scalebox{1.25}{$\mathpzc{E}$}x(0)=\scalebox{1.25}{$\mathpzc{E}$}x_{0},\\\
y(t)&=&\scalebox{1.25}{$\mathpzc{C}$}x(t)\end{array}$ (7)

with the input $u(t)=\bm{v}(t)$, the state $x(t)=(\bm{A}(t),\bm{i}(t))$, the
output $y(t)=\bm{i}(t)$, and the initial value $x_{0}=(\bm{A}_{0},0)$.
Further, the linear operators  $\mathpzc{E}$ ,  $\mathpzc{B}$ ,  $\mathpzc{C}$
and the nonlinear operator  $\mathpzc{A}$ are given by

$\displaystyle\scalebox{1.25}{$\mathpzc{E}$}:\;$ $\displaystyle
X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}$
$\displaystyle\,\to\,X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m},$
(8a) $\displaystyle(\bm{A},\bm{i})$
$\displaystyle\,\mapsto\,\Bigl{(}\sigma\bm{A},\int_{\mathit{\Omega}}\chi^{T}\bm{A}\,{\rm
d}\xi\Bigr{)},$ $\displaystyle\scalebox{1.25}{$\mathpzc{A}$}:\;$
$\displaystyle
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}$
$\displaystyle\,\to\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}\times\mathbb{R}^{m},$
(8b) $\displaystyle(\bm{A},\bm{i})$
$\displaystyle\,\mapsto\,\bigl{(}-\scalebox{1.25}{$\mathpzc{A}$}_{11}(\bm{A})+\chi\,\bm{i},-R\,\bm{i}\bigr{)},\qquad\qquad$
$\displaystyle\scalebox{1.25}{$\mathpzc{B}$}:\;$ $\displaystyle\mathbb{R}^{m}$
$\displaystyle\,\to\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}\times\mathbb{R}^{m},$
(8c) $\displaystyle\bm{v}$ $\displaystyle\,\mapsto\,(0,\bm{v}),$
$\displaystyle\scalebox{1.25}{$\mathpzc{C}$}:\;$ $\displaystyle
X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}$
$\displaystyle\,\to\,\mathbb{R}^{m},$ (8d) $\displaystyle(\bm{A},\bm{i})$
$\displaystyle\,\mapsto\,(0,\bm{i}),$

where

$\displaystyle\qquad\scalebox{1.25}{$\mathpzc{A}$}_{11}:\;$ $\displaystyle
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$
$\displaystyle\,\to\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime},$
$\displaystyle\bm{A}$
$\displaystyle\,\mapsto\,\displaystyle{\biggl{(}\bm{F}\mapsto\int_{\mathit{\Omega}}\nu(\cdot,\|\nabla\times\bm{A}\|_{2})(\nabla\times\bm{A})\cdot(\nabla\times\bm{F})\,{\rm
d}\xi\biggr{)}.}$

Hereby, the first “block row” in $\tfrac{\rm d}{{\rm
d}t}\scalebox{1.25}{$\mathpzc{E}$}x(t)=\scalebox{1.25}{$\mathpzc{A}$}(x(t))+\scalebox{1.25}{$\mathpzc{B}$}u(t)$
corresponds to (1a) and also includes the boundary condition (1d), as this is
included in the domain of $\scalebox{1.25}{$\mathpzc{A}$}_{11}$. The second
“block row” expresses the coupling relation (1b). The initial condition
$\scalebox{1.25}{$\mathpzc{E}$}x(0)=\scalebox{1.25}{$\mathpzc{E}$}x_{0}$
comprises (1d) and (1e), and the output equation
$y(t)=\scalebox{1.25}{$\mathpzc{C}$}x(t)$ states that the output $y$ is given
by the current $\bm{i}$.

###  2.3. Magnetic energy

The magnetic energy plays an important role in our forthcoming discussions.
For the magnetic reluctivity $\nu$ as in Assumption 2.2 b), consider the
function $\vartheta:\mathit{\Omega}\times\mathbb{R}_{\geq
0}\to\mathbb{R}_{\geq 0}$ given by

$\vartheta(\xi,\varrho)=\frac{1}{2}\int_{0}^{\varrho}\nu(\xi,\sqrt{\zeta})\,{\rm
d}\zeta=\int_{0}^{\sqrt{\varrho}}\nu(\xi,\zeta)\zeta\,{\rm d}\zeta.$ (9)

Further, define a functional

$\displaystyle E:$ $\displaystyle X(\mathit{\Omega},\mathit{\Omega}_{C})\to$
$\displaystyle\,\mathbb{R}\cup\\{\infty\\},$ (10) $\displaystyle\bm{A}\mapsto$
$\displaystyle\,\begin{cases}\displaystyle{\int_{\mathit{\Omega}}\vartheta\bigl{(}\xi,\|\nabla\times\bm{A}(\xi)\|_{2}^{2}\bigr{)}\,{\rm
d}\xi}\;&\text{if }\bm{A}{}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}),\\\
\infty\;&\text{else}.\end{cases}$

Given a magnetic vector potential $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, the scalar
function
$\xi\mapsto\vartheta\bigl{(}\xi,\|\nabla\times\bm{A}(\xi,t)\|_{2}^{2}\bigr{)}$
is the magnetic energy density, and $E(\bm{A}(t))$ is the magnetic energy.
Many properties of the magnetic energy are collected in [11, Proposition 5.2].
In particular, it has been shown there that $E$ is convex, and for all
$\bm{A}\in X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, it
fulfills the estimates

$\frac{m_{\nu}}{2}\|\nabla\times\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\leq
E(\bm{A})\leq\frac{L_{\nu}}{2}\|\nabla\times\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2},$
(11)

where $m_{\nu},L_{\nu}>0$ are the constants as in Assumption 2.2 b).

##  3\. Passivity

In this section, we investigate passivity of the coupled MQS system (1).
Passive systems form a special class of dissipative dynamical systems which
have extensively been studied in [14, 30]. They are of particular interest in
circuit simulation [3] and controller design [10]. Roughly speaking, a system
is passive if it does not generate energy or, equivalently, the energy is
dissipated. Mathematically, passivity can be defined in terms of a storage
function. An important property of passive systems is that an interconnection
of passive subsystems often provides a new passive system.

Consider a general abstract differential-algebraic control system

$\begin{array}[]{rcl}\tfrac{\rm d}{{\rm
d}t}\scalebox{1.25}{$\mathpzc{E}$}(x(t))&=&\scalebox{1.25}{$\mathpzc{A}$}(x(t))+\scalebox{1.25}{$\mathpzc{B}$}(u(t)),\\\
y(t)&=&\scalebox{1.25}{$\mathpzc{C}$}(x(t))\end{array}$ (12)

with (possibly nonlinear) operators $\scalebox{1.25}{$\mathpzc{E}$}:X\to Z$,
$\scalebox{1.25}{$\mathpzc{A}$}:D(\scalebox{1.25}{$\mathpzc{A}$})\subset X\to
Z$, $\scalebox{1.25}{$\mathpzc{B}$}:U\to Z$ and
$\scalebox{1.25}{$\mathpzc{C}$}:X\to U^{\prime}$ for some Hilbert spaces $X$,
$Z$ and $U$. The input $u\in L^{2}([0,T];U)$ is called admissible with the
initial condition
$\scalebox{1.25}{$\mathpzc{E}$}(x(0))=\scalebox{1.25}{$\mathpzc{E}$}(x_{0})$,
if (12) has a solution $x:{[0,T]}\to X$ with
$\scalebox{1.25}{$\mathpzc{E}$}(x(0))=\scalebox{1.25}{$\mathpzc{E}$}(x_{0})$
and $y\in L^{2}([0,T];U^{\prime})$. Similarly to the finite-dimensional case
[30], we define passivity for the infinite-dimensional system (12) as follows.

######  Definition 3.1 (Passivity).

A function $\mathcal{S}:{X}\to\mathbb{R}_{\geq 0}\cup\\{\infty\\}$ is called a
storage function for passivity of system (12), if for all $T>0$, $x_{0}\in{X}$
with $\mathcal{S}(x_{0})<\infty$ and all inputs $u\in L^{2}([0,T];U)$
admissible with the initial condition
$\scalebox{1.25}{$\mathpzc{E}$}(x(0))=\scalebox{1.25}{$\mathpzc{E}$}(x_{0})$,
the following conditions are fulfilled:

  1. a)

$t\mapsto\mathcal{S}(x(t))$ is continuous as a function from $[0,T]$ to
$\mathbb{R}_{\geq 0}\cup\\{\infty\\}$;

  2. b)

for all $0\leq t_{0}\leq t_{1}\leq T$, it holds the dissipation inequality

$\mathcal{S}(x(t_{1}))-\mathcal{S}(x(t_{0}))\leq\int_{t_{0}}^{t_{1}}\langle
u(\tau),y(\tau)\rangle_{2}\,{\rm d}\tau,$ (13)

where $\langle\cdot,\cdot\rangle_{2}$ stands for the standard Euclidean inner
product in $\mathbb{R}^{m}$. System (12) is called passive, if there exists a
storage function for passivity.

The dissipation inequality (13) typically has the interpretation of an energy
balance. Namely, $\mathcal{S}(x(t))$ expresses the energy of the state $x(t)$,
whereas the energy extracted from the system is given by
$\int_{t_{0}}^{t_{1}}\langle u(\tau),y(\tau)\rangle_{2}\,{\rm d}\tau$. The
nonnegative term

$\int_{t_{0}}^{t_{1}}\langle u(\tau),y(\tau)\rangle_{2}\,{\rm
d}\tau-\mathcal{S}(x(t_{1}))+\mathcal{S}(x(t_{0}))$

is the energy which is dissipated by the system on the time interval
$[t_{0},t_{1}]$.

######  Remark 2.

  1. a)

Assume that the initial value $x_{0}$ fulfills $\mathcal{S}(x_{0})<\infty$ and
$u\in L^{2}([0,T];U)$ is admissible with the initial condition
$\scalebox{1.25}{$\mathpzc{E}$}(x(0))=\scalebox{1.25}{$\mathpzc{E}$}(x_{0})$.
Then it immediately follows from the dissipation inequality (13) that
$\mathcal{S}(x(t))<\infty$ for all $t\in[0,T]$.

  2. b)

If, additionally, $\mathcal{S}(0)=0$, then for system (12) initialized with
$\scalebox{1.25}{$\mathpzc{E}$}(x(0))=\scalebox{1.25}{$\mathpzc{E}$}(0)$ and
for all inputs $u\in L^{2}([0,T];U)$ admissible with this initial condition,
the dissipation inequality (13) implies

$\mathcal{S}(x(t))\leq\int_{0}^{t}\langle u(\tau),y(\tau)\rangle_{2}\,{\rm
d}\tau\quad\text{ for all }t\in[0,T].$

In this case, by the nonnegativity of $\mathcal{S}$, we have

$0\leq\int_{0}^{t}\langle u(\tau),y(\tau)\rangle_{2}\,{\rm d}\tau.$

Systems with this property are called input-output passive. It has been shown
in [14] that reachable and stabilizable input-output passive finite-
dimensional standard state space systems possess a storage function. In
particular, such systems are passive. Passivity of infinite-dimensional linear
systems has been studied in [16, 28].

We now return to the coupled MQS system (1). We have seen in Section 2.2 that
it can be written as an abstract differential-algebraic control system (7)
with the operators as in (8) and, in particular,
$U=U^{\prime}=\mathbb{R}^{m}$. Then the existence result in [11, Theorem 7.1]
consequences that for $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, any input
$u=\bm{v}\in L^{2}([0,T];\mathbb{R}^{m})$ is admissible with the initial
condition

$\scalebox{1.25}{$\mathpzc{E}$}(\bm{A}(0),\bm{i}(0))=\scalebox{1.25}{$\mathpzc{E}$}(\bm{A}_{0},0).$

Next, we show that the function $\mathcal{S}_{\rm
MQS}:X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}\to\mathbb{R}_{\geq
0}\cup\\{\infty\\}$ defined by the magnetic energy as in (10), i.e.,

$\mathcal{S}_{\rm MQS}(\bm{A}(t),\bm{i}(t))=E(\bm{A}(t))$ (14)

is a storage function for passivity of the coupled MQS system (1). A formal
consideration by invoking the chain rule for $\vartheta$ being as in (9), and
the integration by parts formula with the weak curl operator, see [11, eq.
(2.1)], yields that the solution of system (1) fulfills

$\displaystyle{\tfrac{\rm d}{{\rm d}t}\mathcal{S}_{\rm
MQS}(\bm{A}(t),\bm{i}(t))}$ $\displaystyle=$
$\displaystyle\displaystyle{\int_{\mathit{\Omega}}\tfrac{\partial}{\partial\varrho}\vartheta\bigl{(}\cdot,\|\nabla\times\bm{A}(t)\|_{2}^{2}\bigr{)}\tfrac{{\rm
d}}{{\rm d}t}\|\nabla\times\bm{A}(t)\|_{2}^{2}\,{\rm d}\xi}$ $\displaystyle=$
$\displaystyle\displaystyle{-\int_{\mathit{\Omega}}\bigl{\|}\tfrac{{\rm
d}}{{\rm d}t}\sqrt{\sigma}\bm{A}(t)\bigr{\|}_{2}^{2}\,{\rm
d}\xi-\langle\bm{i}(t),R\,\bm{i}(t)\rangle_{2}+\langle\bm{v}(t),\bm{i}(t)\rangle_{2}}.$

Integrating this equation on ${[t_{0},t_{1}]}$ and using that $u=\bm{v}$,
$y=\bm{i}$, and $R$ is a positive definite matrix, we obtain

$\displaystyle\mathcal{S}_{\rm MQS}$
$\displaystyle(\bm{A}({t_{1}}),\bm{i}({t_{1}}))-\mathcal{S}_{\rm
MQS}(\bm{A}{(t_{0})},\bm{i}{(t_{0})})$ $\displaystyle=$
$\displaystyle-\int_{{t_{0}}}^{{t_{1}}}\int_{\mathit{\Omega}}\bigl{\|}\tfrac{{\rm
d}}{{\rm d}\tau}\sqrt{\sigma}\bm{A}(\tau)\bigr{\|}_{2}^{2}\,{\rm d}\xi\,{\rm
d}\tau-\int_{{t_{0}}}^{{t_{1}}}\langle\bm{i}(\tau),R\,\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau+\int_{{t_{0}}}^{{t_{1}}}\langle\bm{v}(\tau),\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau$ $\displaystyle\leq$ $\displaystyle\int_{{t_{0}}}^{{t_{1}}}\langle
u(\tau),y(\tau)\rangle_{2}\,{\rm d}\tau.$

In particular, the non-negative expression

$\int_{{t_{0}}}^{{t_{1}}}\int_{\mathit{\Omega}}\bigl{\|}\tfrac{{\rm d}}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\bigr{\|}_{2}^{2}\,{\rm d}\xi\,{\rm
d}\tau+\int_{{t_{0}}}^{{t_{1}}}\langle\bm{i}(\tau),R\,\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau\geq 0$

stands for the energy dissipated by the system on the time interval
${[t_{0},t_{1}]}$. Next, we show that a rigorous analysis indeed leads to the
above dissipation inequality.

######  Theorem 3.2 (Energy balance for the coupled MQS system).

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, $T>0$, $\bm{v}\in L^{2}([0,T];\mathbb{R}^{m})$, and let
$(\bm{A},\bm{i})$ be a solution of the coupled MQS system (1). Then the
magnetic energy function $E$ as defined in (10) has the following properties:

$\displaystyle\bigl{(}t\mapsto E(\bm{A}(t))\bigr{)}\in$
$\displaystyle\,L^{1}({[0,T]})\cap W^{1,1}_{\rm loc}((0,T]),$ (15)
$\displaystyle\bigl{(}t\mapsto t\,E(\bm{A}(t))\bigr{)}\in$
$\displaystyle\,L^{\infty}([0,T]).$ (16)

Further, for all $0<t_{0}\leq t_{1}\leq T$, it holds

$E(\bm{A}(t_{1}))-E(\bm{A}(t_{0}))=-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau\\\
-\int_{t_{0}}^{t_{1}}\langle\bm{i}(\tau),R\,\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau+\int_{t_{0}}^{t_{1}}\langle\bm{v}(\tau),\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau.$ (17)

If, additionally, $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, then

$\left(t\mapsto E(\bm{A}(t))\right)\in\,W^{1,1}([0,T]),$ (18)

and the identity (17) holds for all $0\leq t_{0}\leq t_{1}\leq T$.

###### Proof.

It has been shown in the proof of [11, Theorem 7.1] that the coupled MQS
system (1) can be formulated as an abstract differential-algebraic gradient
system with a subgradient of $E$. This system meets the assumptions of [11,
Corollary 6.4], which leads to the existence of a unique solution
$(\bm{A},\bm{i})$ satisfying (15), (16) and (17). If, further, $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, then (18) is
fulfilled, and the identity (17) even holds for all $0\leq t_{0}\leq t_{1}\leq
T$. ∎

As a consequence of Theorem 3.2, we obtain the following result establishing
the passivity of the coupled MQS system (1).

######  Corollary 1.

Under the assumptions of Theorem 3.2, the function $\mathcal{S}_{\rm MQS}$ as
in (14) is a storage function for passivity of the coupled MQS system (1), and
thus, this system is passive.

###### Proof.

Since the expression

$\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\,{\rm
d}\tau+\int_{t_{0}}^{t_{1}}\langle\bm{i}(\tau),R\,\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau$

is positive, Theorem 3.2 implies that for all $0<t_{0}\leq t_{1}\leq T$, the
solution $(\bm{A},\bm{i})$ of (1) fulfills the dissipation inequality

$\mathcal{S}_{\rm MQS}(\bm{A}(t_{1}),\bm{i}(t_{1}))-\mathcal{S}_{\rm
MQS}(\bm{A}(t_{0}),\bm{i}(t_{0}))\leq\int_{t_{0}}^{t_{1}}\langle\bm{v}(\tau),\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau.$

Thus, $\mathcal{S}_{\rm MQS}$ is a storage function for passivity of the
coupled MQS system (1) and this system is indeed passive. ∎

######  Remark 3.

Theorem 3.2 implies that for $\bm{v}\\!\in\\!L^{2}([0,T];\mathbb{R}^{m})$
and $\bm{A}_{0}\\!\in\\!X(\mathit{\Omega},\mathit{\Omega}_{C})$, we have
$\mathcal{S}_{\rm MQS}(\bm{A}(t),\bm{i}(t))<\infty$ for all $t\in(0,T]$. In
other words, the storage function takes finite values after an arbitrary short
time, even if it is infinite at time zero.

##  4\. The Coupled MQS System in Port-Hamiltonian Formulation

Port-Hamiltonian systems have meanwhile become rather popular as a modelling
tool especially for coupled (multi-)physical systems [17]. In the past few
years, this theory has successfully been extended to (finite-dimensional)
differential-algebraic systems [20, 21, 29, 13]. The aim of this section is to
show that the coupled MQS system (1) also fits into the framework of port-
Hamiltonian systems.

We first introduce some basics of port-Hamiltonian systems which are heavily
inspired by [17]. Since infinite-dimensional port-Hamiltonian systems have
been so far treated mainly from a differential geometric rather than from a
functional analytic perspective [23, 24, 19], the authors of this paper have a
certain sovereignty of definition. Note that, though a functional analytic
approach to infinite-dimensional port-Hamiltonian systems has been discussed
in [15], the theory presented therein, however, restricts to the very limited
class of coupled systems of linear transport equations.

An important concept is the Dirac structure which describes the power
preserving energy-routing of the system. Dirac structures on Hilbert spaces
have been considered in [8], and their structure has been analysed by using
the theory of Krein spaces.

######  Definition 4.1 (Dirac structure).

Let $X$ be a Banach space. A subspace $\mathcal{D}\subset X\times X^{\prime}$
is called a  _Dirac structure_ , if for all $f\in X$, $e\in X^{\prime}$, it
holds

$(f,e)\in\mathcal{D}\;\Longleftrightarrow\;\left(\langle{f},{\hat{e}}\rangle+\langle{\hat{f}},{{e}}\rangle=0\,\text{
for all }\,(\hat{f},\hat{e})\in\mathcal{D}\right).$

Hereby, $X$ is called the space of flows, whereas $X^{\prime}$ is called the
space of efforts.

Another concept needed for port-Hamiltonian systems is the resistive relation
which represents the internal energy dissipation of the system [17, Section
2.4].

######  Definition 4.2 (Resistive relation).

Let $X$ be a Banach space. A relation $\mathcal{R}\subset X\times X^{\prime}$
is called _resistive_ , if

$\left\langle f,e\right\rangle\leq 0\,\text{ for all
}\,(f,e)\in{\mathcal{R}}.$

Having defined the Dirac structure and the resistive relation, we are now
ready to introduce port-Hamiltonian systems. The subsequent definition uses
the concepts of Gâteaux differentiability and Gâteaux derivative, for which we
refer to [31, Definition 4.5].

######  Definition 4.3 (Port-Hamiltonian system).

Let $X_{\mathcal{S}}$, $X_{\mathcal{R}}$ and $X_{\mathcal{P}}$ be Banach
spaces. A _port-Hamiltonian system_ is a triple
$(\mathcal{D},\mathcal{H},\mathcal{R})$, where

$\mathcal{D}\subset(X_{\mathcal{S}}\times X_{\mathcal{R}}\times
X_{\mathcal{P}})\times(X_{\mathcal{S}}^{\prime}\times
X_{\mathcal{R}}^{\prime}\times X_{\mathcal{P}}^{\prime})$

is a Dirac structure, $\mathcal{H}:X_{\mathcal{S}}\to\mathbb{R}$ is Lipschitz
continuous on bounded sets and Gâteaux differentiable, and $\mathcal{R}\subset
X_{\mathcal{R}}\times X_{\mathcal{R}}^{\prime}$ is a resistive relation. The
_dynamics_ of the port-Hamiltonian system on an interval
$\mathbb{I}\subset\mathbb{R}$ are specified by the differential inclusions

$\begin{array}[]{rcl}\bigl{(}-\tfrac{\rm d}{{\rm
d}t}x(t),f_{\mathcal{R}}(t),f_{\mathcal{P}}(t),{\rm
D}\mathcal{H}(x(t)),e_{\mathcal{R}}(t),e_{\mathcal{P}}(t))&\in&\mathcal{D},\\\
(f_{\mathcal{R}}(t),e_{\mathcal{R}}(t)\bigr{)}&\in&\mathcal{R},\quad
t\in\mathbb{I},\end{array}$ (19)

where ${\rm D}\mathcal{H}:X_{\mathcal{S}}\to X_{\mathcal{S}}^{\prime}$ is the
Gâteaux derivative of $\mathcal{H}$. The function $\mathcal{H}$ is called
Hamiltonian, whereas, for some time $t$ in which the system is driven, $x(t)$
is called state. The spaces $X_{\mathcal{S}}\times X_{\mathcal{S}}^{\prime}$,
$X_{\mathcal{R}}\times X_{\mathcal{R}}^{\prime}$ and $X_{\mathcal{P}}\times
X_{\mathcal{P}}^{\prime}$ are referred to as energy storage port, resistive
port and external port, respectively.

######  Remark 4.

  1. a)

Though the above definition of a port-Hamiltonian system includes the case of
infinite-dimensional Banach spaces and therefore has a certain generality,
there are quite a lot of opportunities for even further generalizations. For
instance, in a very general setting, a Dirac structure is defined on a
manifold $\mathcal{M}$ by a certain subbundle of $\mathcal{D}\subset
T\mathcal{M}\oplus T^{*}\mathcal{M}$, where $T\mathcal{M}$ is the tangent
bundle and $T^{*}\mathcal{M}$ is the co-tangent bundle of $\mathcal{M}$ [12,
Definition 2.2.1]. In [20, 21], the energy storage port has been determined by
so-called Lagrange manifolds, which generalize the above use of Hamiltonians.

  2. b)

Since the Hamiltonian is assumed to be Lipschitz continuous on bounded sets
and is Gâteaux differentiable, [4, Theorem 4.2] can be applied to obtain that
for any interval $\mathbb{I}\subset\mathbb{R}$ and any $x\in
W^{1,p}(\mathbb{I};X_{\mathcal{S}})$, it holds that
$(t\mapsto\mathcal{H}(x(t))\in W^{1,p}(\mathbb{I})$, and the weak derivative
fulfills the generalized chain rule

$\tfrac{\rm d}{{\rm d}t}\mathcal{H}(x)=\langle{\tfrac{\rm d}{{\rm d}t}}x,{\rm
D}\mathcal{H}(x)\rangle.$ (20)

Combining this with the properties of the Dirac structure and resistive
structure, we obtain from (19) that for almost all $t\in\mathbb{I}$, it holds
the energy balance

$\displaystyle 0=$ $\displaystyle-\langle\tfrac{\rm d}{{\rm d}t}x(t),{\rm
D}\mathcal{H}(x(t))\rangle+\langle
f_{\mathcal{R}}(t),e_{\mathcal{R}}(t)\rangle+\langle
f_{\mathcal{P}}(t),e_{\mathcal{P}}(t)\rangle$ $\displaystyle=$
$\displaystyle-\tfrac{\rm d}{{\rm d}t}\mathcal{H}(x(t))+\langle
f_{\mathcal{R}}(t),e_{\mathcal{R}}(t)\rangle+\langle
f_{\mathcal{P}}(t),e_{\mathcal{P}}(t)\rangle$ $\displaystyle\leq$
$\displaystyle-\tfrac{\rm d}{{\rm d}t}\mathcal{H}(x(t))+\langle
f_{\mathcal{P}}(t),e_{\mathcal{P}}(t)\rangle.$

and, hence, an integration on $[t_{0},t_{1}]\subset\mathbb{I}$ yields

$\mathcal{H}(x(t_{1}))-\mathcal{H}(x(t_{0}))\leq\,\int_{t_{0}}^{t_{1}}\langle
f_{\mathcal{P}}(t),e_{\mathcal{P}}(t)\rangle\,{\rm d}t.$ (21)

If, for instance, the flows at the external ports form the input, and the
efforts at the external ports form the output (or vice-versa), then we indeed
obtain the dissipation inequality (13).

  3. c)

A generalization of the Gâteaux derivative is given by the subdifferential as
considered in [6] from the perspective of nonlinear evolution equations. This
approach is applicable to a class of Hamiltonians which are further allowed to
map to $\mathbb{R}_{\geq 0}\cup\\{\infty\\}$, and typically results into a
subdifferential which is set-valued and only defined on some subset of
$X_{\mathcal{P}}$. Under the additional assumption that $X_{\mathcal{P}}$ is a
Hilbert space, it is shown in [6, Lemma 4.4] that the generalized chain rule
(20) also holds, if the Gâteaux derivative is replaced by a subdifferential.
Since this is the essential ingredient used in (21), the incorporation of
subdifferentials is a further possible generalization of our approach to
infinite-dimensional port-Hamiltonian systems.

We now show that the coupled MQS system (1) admits a formulation as a port-
Hamiltonian system. For this purpose, we introduce the function

$\displaystyle\mathcal{H}_{\rm
MQS}:X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$
$\displaystyle\to\,\mathbb{R},$ $\displaystyle\bm{A}$
$\displaystyle\,\mapsto\int_{\mathit{\Omega}}\vartheta\bigl{(}\xi,\|\nabla\times\bm{A}(\xi)\|_{2}^{2}\bigr{)}\,{\rm
d}\xi,$

with the magnetic energy density $\vartheta$ defined in (9). Then it follows
from [11, Proposition 5.2 a)] that $\mathcal{H}_{\rm MQS}$ is Lipschitz
continuous on bounded sets, whereas [11, Proposition 5.2 c)] shows that
$\mathcal{H}_{\rm MQS}$ is Gâteaux differentiable, and the Gâteaux derivative
fulfills for all $\bm{A},\bm{F}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$,

$\langle\bm{F},{\rm D}\mathcal{H}_{\rm
MQS}(\bm{A})\rangle=\int_{\mathit{\Omega}}\nu(\cdot,\|\nabla\times\bm{A}\|_{2})\,(\nabla\times\bm{A})\cdot(\nabla\times\bm{F})\,{\rm
d}\xi.$ (22)

Next, consider the Hilbert spaces

$\displaystyle X_{\mathcal{S}}=$
$\displaystyle\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}),\qquad
X_{\mathcal{R}}=X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}\times\mathbb{R}^{m},\qquad
X_{\mathcal{P}}=\mathbb{R}^{m},$ (23) $\displaystyle X\,=$
$\displaystyle\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})\times
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}\times\mathbb{R}^{m}\times\mathbb{R}^{m}.$

Since Hilbert spaces are reflexive, and the dual space of $\mathbb{R}^{m}$ can
be canonically identified with itself, we have

$\displaystyle\\!\\!X_{\mathcal{S}}^{\prime}=$
$\displaystyle\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime},\qquad
X_{\mathcal{R}}^{\prime}=X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m},\qquad
X_{\mathcal{P}}^{\prime}=\mathbb{R}^{m},$ $\displaystyle X^{\prime}\,=$
$\displaystyle\,X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}\times
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}\times\mathbb{R}^{m}.$

We introduce the relations

$\displaystyle\mathcal{D}_{\rm MQS}=$
$\displaystyle\Biggl{\\{}\left(\begin{pmatrix}\bm{f}_{\bm{A}}\\\ \bm{J}_{\rm
int}\\\ \bm{v}_{\rm w}\\\ \bm{i}\end{pmatrix},\begin{pmatrix}{\bm{J}}\\\
\bm{E}\\\ \bm{i}_{\rm w}\\\ \bm{v}\end{pmatrix}\right)\in X\times
X^{\prime}\;:\;\bm{f}_{\bm{A}}=\bm{E},\;\;\bm{J}_{\rm
int}=-\bm{J}+\chi\,\bm{i}_{\rm w},\Biggr{.}$ (24) $\displaystyle\hskip
136.57323pt\Biggl{.}\bm{v}_{\rm
w}=-\int_{\mathit{\Omega}}\chi^{\top}\bm{E}\,{\rm
d}\xi-\bm{v},\enskip\bm{i}=\bm{i}_{\rm w}\Biggr{\\}},$
$\displaystyle{\mathcal{R}}_{\rm MQS}=$
$\displaystyle\left\\{\left(\begin{pmatrix}\bm{J}_{\rm int}\\\ \bm{v}_{\rm
w}\end{pmatrix},\begin{pmatrix}\bm{E}\\\ \bm{i}_{\rm w}\end{pmatrix}\right)\in
X_{\mathcal{R}}\times X_{\mathcal{R}}^{\prime}\;:\;\bm{J}_{\rm
int}=-\sigma\bm{E},\enskip\bm{v}_{\rm w}=-R\,\bm{i}_{\rm w}\right\\}.$ (25)

It immediately follows from Assumption 2.2 a) and c) that ${\mathcal{R}}_{\rm
MQS}$ is a resistive relation. To prove that ${\mathcal{D}}_{\rm MQS}$ is a
Dirac structure, we advance the following lemma, which is a straightforward
generalization of the well-known statement that the graphs of a skew-symmetric
matrices define Dirac structures [20, 13].

######  Lemma 4.4.

Let $X$ be a Banach space, and let $\mathcal{J}\in\mathcal{L}(X^{\prime},X)$
be a skew-dual operator in the sense that it fulfills
$\left\langle\mathcal{J}v,w\right\rangle=-\left\langle\mathcal{J}w,v\right\rangle$
for all $v,w\in X^{\prime}$. Then
$\mathcal{D}=\\{(\mathcal{J}e,e)\enskip:\enskip e\in X^{\prime}\\}$ is a Dirac
structure.

###### Proof.

Assume that $(f,e)\in\mathcal{D}$. Then $f=\mathcal{J}e$. This implies that
for all $(\hat{f},\hat{e})\in\mathcal{D}$,

$\langle{f},{\hat{e}}\rangle+\langle{\hat{f}},{{e}}\rangle=\langle{\mathcal{J}e},{\hat{e}}\rangle+\langle{\mathcal{J}\hat{e}},{{e}}\rangle=-\langle{\mathcal{J}e},{\hat{e}}\rangle+\langle{\mathcal{J}\hat{e}},{{e}}\rangle=0.$

On the other hand, if $(f,e)\in X\times X^{\prime}$ fulfills
$\langle{f},{\hat{e}}\rangle+\langle{\hat{f}},{{e}}\rangle=0$ for all
$(\hat{f},\hat{e})\in\mathcal{D}$, then we obtain for all $\hat{e}\in
X^{\prime}$ that

$0=\langle{f},{\hat{e}}\rangle+\langle{\mathcal{J}\hat{e}},{{e}}\rangle=\langle{f},{\hat{e}}\rangle-\langle{\mathcal{J}e},{\hat{e}}\rangle=\langle{f-\mathcal{J}e},{\hat{e}}\rangle.$

Hence, [1, Corollary 6.17 (2)] (which is a direct consequence of the famous
Hahn-Banach theorem) leads to $f=\mathcal{J}e$, which in turn gives
$(f,e)\in\mathcal{D}$. ∎

We can now easily verify that $\mathcal{D}_{\rm MQS}$ as in (24) is a Dirac
structure.

######  Proposition 1.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1, and let
$\chi:\mathit{\Omega}\to\mathbb{R}^{3\times m}$ satisfy Assumption 2.3 b).
Then $\mathcal{D}_{\rm MQS}$ as in (24) is a Dirac structure.

###### Proof.

For $X$ as in (23), consider the bounded operator $\mathcal{J}_{\rm
MQS}:X^{\prime}\to X$ with

$\mathcal{J}_{\rm MQS}\begin{pmatrix}\bm{J}\\\ \bm{E}\\\ \bm{i}_{\rm w}\\\
\bm{v}\end{pmatrix}=\begin{pmatrix}\bm{E}\\\ -\bm{J}+\chi\,\bm{i}_{\rm w}\\\
-\int_{\mathit{\Omega}}\chi^{\top}\bm{E}\,{\rm d}\xi-\bm{v}\\\ \bm{i}_{\rm
w}\end{pmatrix}.$

Then it can be immediately seen that $\mathcal{J}_{\rm MQS}$ is skew-dual,
whence Lemma 4.4 implies that $\mathcal{D}_{\rm MQS}$ is a Dirac structure. ∎

Next, we show that the coupled MQS system (1) represents the dynamics of the
port-Hamiltonian system $(\mathcal{D}_{\rm MQS},\mathcal{H}_{\rm
MQS},\mathcal{R}_{\rm MQS})$ in some sense. Denote the state $x(t)$ by
$\bm{A}(t)$ and the flow and effort at the external port by $\bm{i}(t)$ and
$\bm{v}(t)$, respectively. Then

$\displaystyle\left(-\tfrac{\rm d}{{\rm
d}t}\bm{A}(t),\begin{pmatrix}\bm{J}_{\rm int}(t)\\\ \bm{v}_{\rm
w}(t)\end{pmatrix},{\bm{i}}(t),{\rm D}\mathcal{H}_{\rm
MQS}(\bm{A}(t)),\begin{pmatrix}\bm{E}(t)\\\ \bm{i}_{\rm
w}(t)\end{pmatrix},{\bm{v}}(t)\right)$ $\displaystyle\in{\mathcal{D}}_{\rm
MQS},$ (26) $\displaystyle\left(\begin{pmatrix}\bm{J}_{\rm int}(t)\\\
\bm{v}_{\rm w}(t)\end{pmatrix},\begin{pmatrix}\bm{E}(t)\\\ \bm{i}_{\rm
w}(t)\end{pmatrix}\right)$ $\displaystyle\in{\mathcal{R}}_{\rm MQS}.$

By using the definition of $\mathcal{D}_{\rm MQS}$ in (24) and
$\mathcal{R}_{\rm MQS}$ in (25), we obtain

$\displaystyle-\tfrac{\rm d}{{\rm d}t}\bm{A}(t)=$ $\displaystyle\,\bm{E}(t),$
$\displaystyle-\sigma\bm{E}(t)=$ $\displaystyle\,-{\rm D}\mathcal{H}_{\rm
MQS}(\bm{A}(t))+\chi\,\bm{i}_{\rm w}(t),$ $\displaystyle-R\,\bm{i}_{\rm
w}(t)=$ $\displaystyle\,-\int_{\mathit{\Omega}}\chi^{\top}\bm{E}(t)\,{\rm
d}\xi-\bm{v}(t),$ $\displaystyle\bm{i}(t)=$ $\displaystyle\,\bm{i}_{\rm
w}(t),$

and, thus,

$\displaystyle\tfrac{\rm d}{{\rm d}t}\sigma\bm{A}(t)=$ $\displaystyle\,-{\rm
D}\mathcal{H}_{\rm MQS}(\bm{A}(t))+\chi\,\bm{i}(t),$ $\displaystyle\tfrac{\rm
d}{{\rm d}t}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(t)\,{\rm d}\xi=$
$\displaystyle\,-R\,\bm{i}(t)+\bm{v}(t).$

Since the first equation takes place in
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})^{\prime}$ and
${\rm D}\mathcal{H}_{\rm MQS}$ reads as in (22), we obtain the weak
formulation (5) of the coupled MQS system (1). Moreover, the external port is
formed by the voltage $\bm{v}$ and the current $\bm{i}$ at the conductive
contacts, which are, respectively, the input and the output of (1). The
variable $\bm{J}_{\rm int}$ in (26) stands for the current density in
$\mathit{\Omega}$ induced by the electromagnetic field, whereas $\bm{E}$ is
the electric field intensity. Further, $\bm{v}_{\rm w}$ is the part of the
voltage at the winding which is caused by the resistive effect, and
$\bm{i}_{\rm w}$ is the corresponding current.

######  Remark 5.

Note that the port-Hamiltonian model (26) requires weak differentiability with
respect to time of the magnetic vector potential $\bm{A}$, whereas the
solution concept in Section 2.2 only requires $\sigma\bm{A}$ to be weakly
differentiable.

The port-Hamiltonian model (26) can at least formally be represented in a
compact form

$\displaystyle\tfrac{\rm d}{{\rm d}t}\begin{bmatrix}I\;&0\;&0\;\\\ 0&0&0\\\
0&0&0\end{bmatrix}\\!\begin{bmatrix}\bm{A}(t)\\\ \bm{E}(t)\\\
\bm{i}(t)\end{bmatrix}=$ $\displaystyle\begin{bmatrix}0\enskip\;&-I&\;0\\\
I\enskip\;&-\sigma&-\chi\\\
0\enskip\;&\int_{\mathit{\Omega}}\chi^{\top}\cdot\,{\rm
d}\xi\enskip\;&-R\;\end{bmatrix}\\!\begin{bmatrix}{\rm D}\mathcal{H}_{\rm
MQS}(\bm{A}(t))\\\ \bm{E}(t)\\\ \bm{i}(t)\end{bmatrix}+\begin{bmatrix}0\\\
0\\\ I\end{bmatrix}\bm{v}(t),$ $\displaystyle\bm{i}(t)=$
$\displaystyle\begin{bmatrix}0\,&\,0\,&\,I\end{bmatrix}\begin{bmatrix}{\rm
D}\mathcal{H}_{\rm MQS}(\bm{A}(t))\\\ \bm{E}(t)\\\ \bm{i}(t)\end{bmatrix},$

whose structure amazingly resembles the class of finite-dimensional port-
Hamiltonian differential-algebraic systems studied in [22, 7].

##  5\. Solution Estimates

In this section, we study the quantitative properties of the coupled MQS
system (1) by considering its input-to-output and input-to-state behavior. We
present estimates for the current, the magnetic vector potential and the
magnetic energy upon the initial magnetic vector potential and the voltage. We
further show that, under some additional assumption on the initial value, the
free dynamics of the MQS system decay exponentially. Note that our estimates
also cover the case of infinite intervals. To this end, we refer to Remark 1
for the definition and existence of solutions on the whole positive real axis
$\mathbb{R}_{\geq 0}$.

###  5.1. Input-to-output behavior

First, we derive estimates for the current $\bm{i}$, which is considered as
the output of the coupled MQS system (1), by means of the initial value
$\bm{A}_{0}$ and the voltage $\bm{v}$, which forms the input. In most cases,
we will stick to the case $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Note that by
[11, Theorem 7.1] (see also Section 2.2), the solution $(\bm{A},\bm{i})$ has
the property that $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ for almost all
$t>0$ even if the weak curl of $\bm{A}_{0}\in
X(\mathit{\Omega},\mathit{\Omega}_{C})$ is not in
$L^{2}(\mathit{\Omega};\mathbb{R}^{3})$.

Our forthcoming considerations rely on the following auxiliary result, which
is solely based on the energy balance (17) developed in Theorem 3.2.

######  Lemma 5.1.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and let $T\in\mathbb{R}_{\geq 0}\cup\\{\infty\\}$ and
$\bm{v}\in L^{2}_{\rm loc}([0,T);\mathbb{R}^{m})$. Moreover, let $E$ be the
magnetic energy as defined in (10), and let $(\bm{A},\bm{i})$ be the solution
of the coupled MQS system (1). Then for all $\varepsilon>0$ and all
$0<t_{0}\leq t_{1}<T$, it holds

$\displaystyle E(\bm{A}(t_{1}))$ $\displaystyle-E(\bm{A}(t_{0}))$
$\displaystyle\leq-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau-\Bigl{(}1-\frac{\varepsilon}{2}\Bigr{)}\int_{t_{0}}^{t_{1}}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm
d}\tau$
$\displaystyle\qquad+\frac{1}{2\varepsilon}\int_{t_{0}}^{t_{1}}\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}\,{\rm
d}\tau.$

If, additionally, $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, then the
above inequality holds even for all $0\leq t_{0}\leq t_{1}<T$.

###### Proof.

Assume that $0<t_{0}\leq t_{1}<T$, and let $(\bm{A},\bm{i})$ be a solution of
(1). A combination of the Cauchy-Schwarz with Young’s inequality [1, p. 53]
yields that for almost all $\tau\in[t_{0},t_{1}]$,

$\displaystyle\langle\bm{i}(\tau),\bm{v}(\tau)\rangle_{2}$
$\displaystyle\leq\|R^{1/2}\bm{i}(\tau)\|_{2}\,\|R^{-1/2}\bm{v}(\tau)\|_{2}$
$\displaystyle\leq\frac{\varepsilon}{2}\,\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}+\frac{1}{2\varepsilon}\,\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}.$

Then we obtain from the energy balance (17) that

$\displaystyle E(\bm{A}(t_{1}))-E(\bm{A}(t_{0}))$
$\displaystyle\quad\leq-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau-\int_{t_{0}}^{t_{1}}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm d}\tau$
$\displaystyle\qquad+\int_{t_{0}}^{t_{1}}\frac{\varepsilon}{2}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}+\frac{1}{2\varepsilon}\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}\,{\rm
d}\tau$ $\displaystyle\quad=-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau-\Bigl{(}1-\frac{\varepsilon}{2}\Bigr{)}\int_{t_{0}}^{t_{1}}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm
d}\tau$
$\displaystyle\qquad+\frac{1}{2\varepsilon}\int_{t_{0}}^{t_{1}}\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}\,{\rm
d}\tau.$

As Theorem 3.2 states that (17) holds for all $0\leq t_{0}\leq t_{1}<T$, if,
additionally, $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, the above
estimate is then as well fulfilled in the remaining case $t_{0}=0$. ∎

In the following theorem, we establish an estimate for the $L^{2}$-norm of the
output $y=\bm{i}$ in terms of the $L^{2}$-norm of the input $u=\bm{v}$. In
particular, we show that $\bm{i}\in L^{2}(\mathbb{R}_{\geq
0};\mathbb{R}^{m})$, if $\bm{v}\in L^{2}(\mathbb{R}_{\geq 0};\mathbb{R}^{m})$
and $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$.

######  Theorem 5.2.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and let $T\in\mathbb{R}_{\geq 0}\cup\\{\infty\\}$,
$\bm{v}\in L^{2}([0,T);\mathbb{R}^{m})$, and $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Moreover, let
$E$ be the magnetic energy as defined in (10), and let $(\bm{A},\bm{i})$ be a
solution of the coupled MQS system (1). Then $\bm{i}\in
L^{2}([0,T);\mathbb{R}^{m})$, and for all $\varepsilon\in(0,{2})$,

$\displaystyle\\!\\!\|R^{1/2}\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}^{2}\\!$
$\displaystyle\leq{\tfrac{2}{{2-\varepsilon}}}\,E(\bm{A}_{0})+{\tfrac{1}{{\varepsilon(2-\varepsilon)}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}^{2},$
(27) $\displaystyle\\!\\!\|R^{1/2}\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}\\!$
$\displaystyle\leq\sqrt{{\tfrac{{L_{\nu}}}{{2-\varepsilon}}}}\|\nabla\\!\times\\!\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\\!+\\!\tfrac{1}{\sqrt{{\varepsilon(2-\varepsilon)}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})},\\!$
(28)

where $L_{\nu}$ is the Lipschitz constant as in Assumption 2.2 b). If,
moreover, the initial value is zero, i.e., $\bm{A}_{0}=0$, then

$\|R^{1/2}\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}\leq\|R^{-1/2}\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}.$
(29)

###### Proof.

By [11, Theorem 7.1], we have $\bm{i}\in L^{2}([0,T);\mathbb{R}^{m})$.
Further, let $t\in[0,T)$ and $\varepsilon\in(0,2)$. Then using Lemma 5.1, we
obtain

$\displaystyle 0\leq\,E(\bm{A}(t))\leq$
$\displaystyle\,E(\bm{A}_{0})-\int_{0}^{t}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau$
$\displaystyle\qquad-\Bigl{(}1-\frac{\varepsilon}{2}\Bigr{)}\int_{{0}}^{{t}}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm
d}\tau+\frac{1}{2\varepsilon}\int_{{0}}^{{t}}\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}\,{\rm
d}\tau$ $\displaystyle\leq$
$\displaystyle\,E(\bm{A}_{0})-\Bigl{(}1-\frac{\varepsilon}{2}\Bigr{)}\int_{{0}}^{{t}}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm
d}\tau+\frac{1}{2\varepsilon}\int_{{0}}^{{t}}\|R^{-1/2}\bm{v}(\tau)\|_{2}^{2}\,{\rm
d}\tau.$

Adding the expression
$\bigl{(}1-\frac{\varepsilon}{2}\bigr{)}\int_{0}^{t}\|R^{1/2}\bm{i}(\tau)\|_{2}^{2}\,{\rm
d}\tau$ to both sides, dividing by $1-\tfrac{\varepsilon}{2}$, and
subsequently taking the limit for $t\nearrow T$, we obtain (27). The estimate
(28) immediately follows by a combination of (27) with (11). Finally, if
$\bm{A}_{0}=0$, then (29) follows from (28) by setting $\varepsilon={1}$. ∎

######  Remark 6.

  1. a)

Let $\lambda_{\min}(R)$ be the minimal eigenvalue of $R$. Then, under the
assumptions of Theorem 5.2, we obtain from (27) and (28) that

$\displaystyle\|\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}^{2}$
$\displaystyle\leq{\tfrac{2}{(2-\varepsilon){\lambda_{\min}(R)}}}\,E(\bm{A}_{0})+{\tfrac{1}{{\varepsilon(2-\varepsilon)}{\lambda_{\min}^{2}(R)}}}\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}^{2},$
$\displaystyle\|\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}$
$\displaystyle\leq{\sqrt{\tfrac{L_{\nu}}{(2-\varepsilon)\lambda_{\min}(R)}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\qquad+{\tfrac{1}{\sqrt{\varepsilon(2-\varepsilon)}\,\lambda_{\min}(R)}}\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}.$

If, further, $\bm{A}_{0}=0$, then (29) implies

$\|\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}\leq\|R^{-1}\|\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}=\tfrac{{1}}{\lambda_{\min}(R)}\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}.$
(30)

Note that in the above estimates, we can further replace $\lambda_{\min}(R)$
by $R$, if $m=1$.

  2. b)

In systems theory and model reduction, estimates of the $L^{2}$-norm of the
output by means of the $L^{2}$-norm of the input play a crucial role. In the
linear and time-invariant case, such estimates can be obtained by using the
so-called $\mathcal{H}_{\infty}$-norm of the transfer function. In particular,
it has been shown in [18] that discretizing the linear coupled MQS system (1)
using the finite element method and regularizing the resulting system, one
obtains a differential-algebraic system whose transfer function has the
$\mathcal{H}_{\infty}$-norm coinciding with $\|R^{-1}\|$ independent of the
fineness of the discretization. In this case, we have

$\|\bm{i}\|_{L^{2}([0,T);\mathbb{R}^{m})}\leq\|R^{-1}\|\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})},$

and, further, in the case where $m=1$, this estimate is sharp in the sense
that there exists a sequence of inputs $(\bm{v}_{k})_{k}$ in
$L^{2}(\mathbb{R}_{\geq 0})$ such that for the corresponding sequence of
outputs $(\bm{i}_{k})_{k}$ of the system with $\bm{A}_{0}=0$, the sequence of
quotients $\|\bm{i}_{k}\|_{L^{2}(\mathbb{R}_{\geq
0})}/\|\bm{v}_{k}\|_{L^{2}(\mathbb{R}_{\geq 0})}$ tends to $R^{-1}$. Thus, the
estimate (30) is an extension of this result to the quasilinear infinite-
dimensional case, and it is presumably sharp.

###  5.2. Input-to-state behavior

Next, we present a quantitative estimate for the magnetic energy
$E(\bm{A}(t))$ in terms of the $L^{2}$-norm of the input $\bm{v}$, if the
coupled MQS system (1) is initialized with $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$.

######  Proposition 2.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and let $T\in\mathbb{R}_{\geq 0}\cup\\{\infty\\}$,
$\bm{v}\in L^{2}_{\rm loc}([0,T);\mathbb{R}^{m})$, and $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Moreover, let
$E$ be the magnetic energy as defined in (10), and let $(\bm{A},\bm{i})$ be a
solution of the coupled MQS system (1). Then for all $0\leq t<T$,

$\displaystyle E(\bm{A}(t))$ $\displaystyle\leq
E(\bm{A}_{0})+\tfrac{1}{4}\,{\|R^{-1/2}\bm{v}\|^{2}_{L^{2}([0,t);\mathbb{R}^{m})}},$
(31)
$\displaystyle\|\nabla\\!\times\\!\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\leq\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\\!\times\\!\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{1}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})},$
(32)

where $m_{\nu},L_{\nu}$ are the monotonicity and Lipschitz constants as in
Assumption 2.2 b). In particular, if $\bm{v}\in L^{2}([0,T);\mathbb{R}^{m})$,
then $\nabla\times\bm{A}\in
L^{\infty}([0,T);L^{2}(\mathit{\Omega};\mathbb{R}^{3}))$ with

$\|\nabla\\!\times\\!\bm{A}\|_{L^{\infty}([0,T);L^{2}(\mathit{\Omega};\mathbb{R}^{3}))}\leq\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\\!\times\\!\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{1}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}.$

###### Proof.

The estimate (31) follows from Lemma 5.1 by choosing $\varepsilon=2$. Further,
by invoking (11), we obtain from (31) that

$\displaystyle\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\leq$
$\displaystyle\,\tfrac{2}{m_{\nu}}\,E(\bm{A}(t))$ $\displaystyle\leq$
$\displaystyle\,\tfrac{2}{m_{\nu}}\,E(\bm{A}_{0})+\tfrac{1}{2m_{\nu}}{\|R^{-1/2}\bm{v}\|^{2}_{L^{2}([0,t);\mathbb{R}^{m})}}$
$\displaystyle\leq$
$\displaystyle\,\tfrac{L_{\nu}}{m_{\nu}}\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}+\tfrac{1}{2m_{\nu}}{\|R^{-1/2}\bm{v}\|^{2}_{L^{2}([0,t);\mathbb{R}^{m})}}$
$\displaystyle\leq$
$\displaystyle\,{\Bigl{(}\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{1}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}\Bigr{)}^{2}}.$

Thus, (32) follows by taking the square root. ∎

######  Remark 7.

Let $\lambda_{\min}(R)$ be the minimal eigenvalue of $R$. Then, under the
assumptions of Proposition 2, we obtain that

$\displaystyle E(\bm{A}(t))$ $\displaystyle\,\leq
E(\bm{A}_{0})+\tfrac{1}{4\,{\lambda_{\min}(R)}}\,{\|\bm{v}(\tau)\|^{2}_{L^{2}([0,t);\mathbb{R}^{m})}},$
$\displaystyle\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\,\leq\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{1}{\sqrt{2m_{\nu}{\lambda_{\min}(R)}}}\,\|\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})},$
$\displaystyle\|\nabla\times\bm{A}\|_{L^{\infty}([0,T);L^{2}(\mathit{\Omega};\mathbb{R}^{3}))}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$
$\displaystyle\,\leq\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{1}{\sqrt{2m_{\nu}\lambda_{\min}(R)}}\,\|\bm{v}\|_{L^{2}([0,T);\mathbb{R}^{m})}.$

We can again replace $\lambda_{\min}(R)$ by $R$ in the single-input single-
output case, i.e., $m=1$.

Finally, we analyze the dependence of
$\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$ upon the input
$\bm{v}$ and the initial value $\bm{A}_{0}$. Hereby, we make essential use of
[11, Lemma 3.4] which states that there exists a constant $L_{C}>0$ such that
for all $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$,

$\|\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\leq
L_{C}\,\left(\|\bm{A}\|_{L^{2}(\mathit{\Omega_{C}};\mathbb{R}^{3})}^{2}+\|\nabla\times\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\right).$
(33)

As a preliminary thought, we use the fact that for the electric conductivity
$\sigma$ as in Assumption 2.2 a), one has
$\langle\sigma\bm{A}_{1},\bm{A}_{2}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}=\sigma_{C}\langle\bm{A}_{1},\bm{A}_{2}\rangle_{L^{2}(\mathit{\Omega}_{C};\mathbb{R}^{3})}$.
Then (33) implies that

$\langle\bm{A}_{1},\bm{A}_{2}\rangle_{X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})}=\langle\sigma\bm{A}_{1},\bm{A}_{2}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\langle\nabla\times\bm{A}_{1},\nabla\times\bm{A}_{2}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\\\
+\int_{\mathit{\Omega}}\chi^{\top}\bm{A}_{1}\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}_{2}\,{\rm
d}\xi\qquad\qquad\quad\quad$ (34)

defines an inner product in
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ whose induced
norm is equivalent to the standard norm in
$H_{0}(\operatorname{curl},\mathit{\Omega})$. Consider now the space

$X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})=\\{\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})\enskip:\enskip\nabla\times\bm{A}=0\\}$
(35)

which is a closed subspace of both $X(\mathit{\Omega},\mathit{\Omega}_{C})$
and $X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ with
respect to the respective norms.

The following lemma is essential for our further analysis.

######  Lemma 5.3.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Let
$P\in\mathcal{L}(X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}))$
be the orthogonal projector onto
$X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})$ with
respect to the inner product (34). Then the following statements hold:

  1. a)

The orthogonal projector $P$ can be uniquely extended to a bounded projector
$\widetilde{P}\in\mathcal{L}(X(\mathit{\Omega},\mathit{\Omega}_{C}))$, where
$X(\mathit{\Omega},\mathit{\Omega}_{C})$ is provided with the norm in
$L^{2}(\mathit{\Omega};\mathbb{R}^{3})$. The operator norm of $\widetilde{P}$
fulfills $\|\widetilde{P}\|\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}$ with
$\sigma_{C}$ as in Assumption 2.2 a), $L_{C}$ as in (33), and

$\gamma={\sigma_{C}}+\|\chi\,R^{-1/2}\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}.$ (36)

  2. b)

There exists a constant ${L_{1}}>0$ such that for all $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$,

$\|(I-P)\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq
L_{1}\|\nabla\times\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$ (37)

  3. c)

The operator

$\displaystyle\mathcal{T}:$
$\displaystyle\\!\\!\operatorname{im}\widetilde{P}$
$\displaystyle\;\to\operatorname{im}\widetilde{P}^{*},$
$\displaystyle\\!\\!\bm{A}$
$\displaystyle\;\mapsto\widetilde{P}^{*}\Bigl{(}\sigma\bm{A}+\chi\,R^{-1}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm
d}\xi\Bigr{)}$

has a bounded inverse. The operator norm of this inverse fulfills
$\|\mathcal{T}^{-1}\|\leq\tfrac{L_{C}}{\sigma_{C}}$.

###### Proof.

a) Let $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Then
$\nabla\times(P\bm{A})=0$, and we obtain by using the definition of the inner
product (34) that

$\displaystyle 0$ $\displaystyle\,=\langle
P\bm{A},(I-P)\bm{A}\rangle_{X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})}$
$\displaystyle\,=\int_{\mathit{\Omega}}\sigma(P\bm{A})\cdot((I-P)\bm{A})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}((I-P)\bm{A})\,{\rm d}\xi.$

This relation leads to

$\displaystyle\|P\bm{A}$
$\displaystyle\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\overset{\eqref{eq:curlest}}{\leq}L_{C}\,\|P\bm{A}\|_{L^{2}(\mathit{\Omega}_{C};\mathbb{R}^{3})}^{2}=\tfrac{L_{C}}{\sigma_{C}}\,\int_{\mathit{\Omega}}\sigma\|P\bm{A}\|_{2}^{2}\,{\rm
d}\xi$ $\displaystyle\leq$
$\displaystyle\;\tfrac{L_{C}}{\sigma_{C}}\,\biggl{(}\int_{\mathit{\Omega}}\sigma\|P\bm{A}\|_{2}^{2}\,{\rm
d}\xi{+\int_{\mathit{\Omega}}\\!\sigma\|(I-P)\bm{A}\|_{2}^{2}\,{\rm
d}\xi}\biggr{.}$
$\displaystyle+{2}\\!\int_{\mathit{\Omega}}\sigma(P\bm{A})\cdot((I-P)\bm{A})\,{\rm
d}\xi+{2}\\!\int_{\mathit{\Omega}}\\!\chi^{\top}(P\bm{A})\,{\rm d}\xi\cdot
R^{-1}\\!\int_{\mathit{\Omega}}\\!\chi^{\top}((I-P)\bm{A})\,{\rm d}\xi$
$\displaystyle\biggl{.}\;{+\Bigl{\|}R^{-1/2}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm
d}\xi\Bigr{\|}_{2}^{2}}+\Bigl{\|}R^{-1/2}\int_{\mathit{\Omega}}\chi^{\top}((I-P)\bm{A})\,{\rm
d}\xi\Bigr{\|}_{2}^{2}\biggr{)}$ $\displaystyle=$
$\displaystyle\;\tfrac{L_{C}}{\sigma_{C}}\,\biggl{(}\int_{\mathit{\Omega}_{C}}{\sigma_{C}}\|\bm{A}\|_{2}^{2}\,{\rm
d}\xi+\Bigl{\|}R^{-1/2}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}\,{\rm
d}\xi\Bigr{\|}_{2}^{2}\biggr{)}\leq\tfrac{\gamma\,L_{C}}{\sigma_{C}}\,\|\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}.$

Since $X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ is
dense in $X(\mathit{\Omega},\mathit{\Omega}_{C})$, we can make use of [1,
Theorem E5.3] to see that the projector $P$ uniquely extends to an operator
$\widetilde{P}\in\mathcal{L}(X(\mathit{\Omega},\mathit{\Omega}_{C}))$ with
$\|\widetilde{P}\|^{2}\leq{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}$. As the
operator
$\widetilde{P}^{2}-\widetilde{P}\in\mathcal{L}(X(\mathit{\Omega},\mathit{\Omega}_{C}))$
vanishes on the dense subspace
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, it has to
vanish everywhere. Consequently, $\widetilde{P}$ is a projector.

b) Step 1: First, we show that the mapping

$\displaystyle\Psi:\quad
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$
$\displaystyle\,\to\,L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3}),$
$\displaystyle\bm{A}$ $\displaystyle\,\mapsto\,\nabla\times\bm{A}$

is surjective. Let $\bm{F}\in
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})$. Then, by
[2, Theorem 3.17], there exists some $\mathbf{C}\in
H_{0}(\operatorname{curl},\mathit{\Omega})$ such that
$\bm{F}=\nabla\times\mathbf{C}$. By definition of
$X(\mathit{\Omega},\mathit{\Omega}_{C})$, we may consider an orthogonal
decomposition $\mathbf{C}=\bm{A}+\nabla{\psi}$ with $\bm{A}\in
X(\mathit{\Omega},\mathit{\Omega}_{C})$ and
$\psi\in{H_{0}^{1}}(\mathit{\Omega})$ which is constant on each boundary
component of $\mathit{\Omega}_{C}$. Consequently, the tangential boundary
trace of $\nabla\psi$ vanishes, whence this also holds for
$\bm{A}=\mathbf{C}-\nabla{\psi}$. Then we obtain
$\nabla\times\bm{A}=\nabla\times\mathbf{C}-\nabla\times\nabla\psi=\nabla\times\mathbf{C}=\bm{F}$
and $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, i.e.,
$\bm{F}\in\operatorname{im}\Psi$.

Step 2: Next, we show that the restriction $\left.\Psi\right|_{\ker P}$ of
$\Psi$ to $\ker P$ is bijective. By definition of $P$, we have
$\ker\Psi=\operatorname{im}P$, which is the orthogonal complement of $\ker P$
with respect to the inner product (34). Therefore, $\left.\Psi\right|_{\ker
P}$ is injective. To prove surjectivity, let $\bm{F}\in
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})$. Then, by
Step 1, there exists some $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ with
$\bm{F}=\nabla\times\bm{A}$, and thus

$\nabla\times((I-P)\bm{A})=\nabla\times((I-P)\bm{A})+\nabla\times(P\bm{A})=\nabla\times\bm{A}=\bm{F}.$

Step 3: Finally, we show that there exists $L_{1}>0$ such that (37) holds. We
have seen in Step 2 that $\left.\Psi\right|_{\ker P}:\ker P\to
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})$ is
bijective. It can be further shown that this mapping is bounded. Then the
inverse mapping theorem [1, Theorem 7.8] yields that $\left.\Psi\right|_{\ker
P}$ has a bounded inverse, which implies that there exists a constant
$c_{1}>0$ such that for all $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$,

$\|(I-P)\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}+\|\nabla\times((I-P)\bm{A})\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\leq
c_{1}\|\nabla\times(I-P)\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}.$

Then, clearly, $c_{1}>1$, and we obtain that (37) holds with
$L_{1}=c_{1}-1>0$.

c) Let $\bm{A}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Then

$\displaystyle\langle\widetilde{P}\bm{A},$
$\displaystyle\,\mathcal{T}\widetilde{P}\bm{A}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}=\langle\widetilde{P}\bm{A},\mathcal{T}{P}\bm{A}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;=\Bigl{\langle}\widetilde{P}\bm{A},\widetilde{P}^{*}\bigl{(}\sigma
P\bm{A}+\chi R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm
d}\xi\bigr{)}\Bigr{\rangle}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;=\Bigl{\langle}\widetilde{P}\widetilde{P}\bm{A},\sigma
P\bm{A}+\chi R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm
d}\xi\Bigr{\rangle}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;=\Bigl{\langle}P\bm{A},\sigma P\bm{A}+\chi
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}({P}\bm{A})\,{\rm
d}\xi\Bigr{\rangle}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;=\int_{\mathit{\Omega}}\sigma(P\bm{A})\cdot(P\bm{A})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A})\,{\rm d}\xi$
$\displaystyle\;\geq\int_{\mathit{\Omega}}\sigma(P\bm{A})\cdot(P\bm{A})\,{\rm
d}\xi$
$\displaystyle\;=\sigma_{C}\Bigl{(}\|P\bm{A}\|_{L^{2}(\mathit{\Omega}_{C};\mathbb{R}^{3})}^{2}+\|\nabla\times(P\bm{A})\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\Bigr{)}$
$\displaystyle\;\overset{\eqref{eq:curlest}}{\geq}\,\tfrac{\sigma_{C}}{L_{C}}\|P\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}.$

Since $X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ is
dense in $X(\mathit{\Omega},\mathit{\Omega}_{C})$, we have

$\langle\widetilde{P}\bm{A},\mathcal{T}\widetilde{P}\bm{A}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\geq\tfrac{\sigma_{C}}{L_{C}}\|\widetilde{P}\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}^{2}\,\text{
for all }\,\bm{A}\in X(\mathit{\Omega},\mathit{\Omega}_{C}).$

Consequently, $\mathcal{T}$ has a bounded inverse with
$\|\mathcal{T}^{-1}\|\leq\frac{L_{C}}{\sigma_{C}}$. ∎

For the next result on the dependence of the $L^{2}$-norm of $\bm{A}(t)$ upon
the input and the initial value, we recall that we use the identification (3)
and the norm (4) on $L^{2}(\mathit{\Omega};\mathbb{R}^{3\times m})$.

######  Theorem 5.4.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and let $T\in\mathbb{R}_{\geq 0}\cup\\{\infty\\}$,
$\bm{v}\in L^{2}_{\rm loc}([0,T);\mathbb{R}^{m})$, and $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$. Moreover, let
$E$ be the magnetic energy as defined in (10), and let $(\bm{A},\bm{i})$ be a
solution of the coupled MQS system (1). Let $m_{\nu}$ and $L_{\nu}$ be as in
Assumption 2.2 b), $L_{C}$ as in (33), $L_{1}$ as in Lemma 5.3 b), and
$\gamma$ as in (36).

  1. a)

For all $0<t\leq T$,

$\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq
L_{1}\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{1}}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}\\\
+\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\left(\\!\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\\!+\tfrac{L_{C}}{\sigma_{C}}\|\chi
R^{-1/2}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}\left\|\int_{0}^{t}\\!\\!R^{-1/2}\bm{v}(\tau){\rm
d}\tau\right\|_{2}\right).$

  2. b)

If $\chi\in
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})^{1\times m}$,
then

$\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\;\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\\\
+L_{1}\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{1}}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}.$

###### Proof.

a) Let $(\bm{A},\bm{i})$ be a solution of (1), and let
$P\in\mathcal{L}(X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}))$
be the orthogonal projector onto
$X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})$ as
defined in (35), where
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ is provided
with the inner product (34). Then the definition of the solution, see Section
2.2, yields that for all
$\bm{F}\\!\\!\in\\!X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$,

$\displaystyle\tfrac{\rm d}{{\rm d}t}\int_{\mathit{\Omega}}$
$\displaystyle\sigma\bm{A}(t)\cdot(P\bm{F})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})(\nabla\times\bm{A}(t))\cdot(\nabla\times(P\bm{F}))\,{\rm
d}\xi$ $\displaystyle=$
$\displaystyle\;\int_{\mathit{\Omega}}\chi\,\bm{i}(t)\cdot(P\bm{F})\,{\rm
d}\xi$ $\displaystyle=$ $\displaystyle\;\bm{v}(t)\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi-\tfrac{\rm
d}{{\rm d}t}\\!\int_{\mathit{\Omega}}\chi^{\top}{\bm{A}(t)}\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi.$

Since $\nabla\times(P\bm{F})=0$, this equation reduces to

$\displaystyle\tfrac{\rm d}{{\rm d}t}$
$\displaystyle\left(\int_{\mathit{\Omega}}\sigma\bm{A}(t)\cdot(P\bm{F})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}{\bm{A}(t)}\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi\right)$ (38)
$\displaystyle\quad=\bm{v}(t)\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi.$

By using that
$P\in\mathcal{L}(X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C}))$
is an orthogonal projector onto the space
$X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})$ with
respect to the inner product (34) and, by [11, Theorem 7.1], $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ for almost all
$t\in[0,T)$, we obtain

$\displaystyle\int_{\mathit{\Omega}}\sigma$
$\displaystyle\bm{A}(t)\cdot(P\bm{F})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(t)\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi$
$\displaystyle\enskip=\int_{\mathit{\Omega}}\sigma\bm{A}(t)\cdot(P\bm{F})\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(t)\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi$
$\displaystyle\qquad+\int_{\mathit{\Omega}}(\nabla\times\bm{A})\cdot(\underbrace{\nabla\times(P\bm{F}))}_{=0}\,{\rm
d}\xi$
$\displaystyle\overset{\eqref{eq:Xinner}}{=}\;\langle\bm{A}(t),P\bm{F}\rangle_{X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})}=\langle
P\bm{A}(t),\bm{F}\rangle_{X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})}$
$\displaystyle\enskip=\int_{\mathit{\Omega}}\sigma(P\bm{A}(t))\cdot\bm{F}\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A}(t))\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}{\bm{F}}\,{\rm d}\xi$
$\displaystyle\qquad+\int_{\mathit{\Omega}}\underbrace{(\nabla\times(P\bm{A}(t)))}_{=0}\cdot(\nabla\times\bm{F})\,{\rm
d}\xi$
$\displaystyle\enskip=\int_{\mathit{\Omega}}\sigma(P\bm{A}(t))\cdot\bm{F}\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A}(t))\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}\bm{F}\,{\rm d}\xi.$

Since by Lemma 5.3 a), $P$ extends to a projector
$\widetilde{P}\in\mathcal{L}(X(\mathit{\Omega},\mathit{\Omega}_{C}))$, we
further have

$\displaystyle\bm{v}(t)\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{F})\,{\rm d}\xi=$
$\displaystyle\int_{\mathit{\Omega}}(\chi\,R^{-1}\bm{v}(t))\cdot(P\bm{F})\,{\rm
d}\xi=\langle\chi\,R^{-1}\bm{v}(t),P\bm{F}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle=$
$\displaystyle\,\langle\chi\,R^{-1}\bm{v}(t),\widetilde{P}\bm{F}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}=\langle\widetilde{P}^{*}\chi\,R^{-1}\bm{v}(t),\bm{F}\rangle_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})},$

where $\widetilde{P}^{*}$ is the adjoint of $\widetilde{P}$. By using the
density of the space
$X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ in
$X(\mathit{\Omega},\mathit{\Omega}_{C})$ and the latter two equations, the
integration of (38) implies that

$t\mapsto\sigma(P\bm{A}(t))+\chi\,R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A}(t))\,{\rm
d}\xi$

is a continuous mapping from $[0,T)$ to
$X(\mathit{\Omega},\mathit{\Omega}_{C})$ with

$\left(\sigma(P\bm{A}(t))+\chi\,R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A}(t))\,{\rm
d}\xi\right)-\left(\sigma(P\bm{A}_{0})+\chi\,R^{-1}\int_{\mathit{\Omega}}\chi^{\top}(P\bm{A}_{0})\,{\rm
d}\xi\right)\\\ =\widetilde{P}^{*}\chi\,R^{-1}\int_{0}^{t}\bm{v}(\tau)\,{\rm
d}\tau\quad\text{for all }t\in[0,T).$

An application of $\widetilde{P}^{*}$ to both sides of this equation yields

$\mathcal{T}(P\bm{A}(t)-P\bm{A}_{0})=\widetilde{P}^{*}\chi\,R^{-1}\int_{0}^{t}\bm{v}(\tau)\,{\rm
d}\tau\quad\text{for all }t\in[0,T),$

where $\mathcal{T}$ is the operator as in Lemma 5.3 c). Since $\mathcal{T}$ is
invertible, we obtain

$P\bm{A}(t)-P\bm{A}_{0}=\mathcal{T}^{-1}\left(\widetilde{P}^{*}\chi\,R^{-1}\int_{0}^{t}\bm{v}(\tau)\,{\rm
d}\tau\right)\quad\text{for all }t\in[0,T).$ (39)

Hence, using Lemma 5.3 a) and c), we obtain for all $t\in[0,T)$,

$\displaystyle\|P\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\quad\;\leq\|P\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\|\mathcal{T}^{-1}\|\|\widetilde{P}^{*}\|\|\chi
R^{-1/2}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}\left\|\int_{0}^{t}R^{-1/2}\bm{v}(\tau)\,{\rm d}\tau\right\|_{2}$
$\displaystyle\quad\;\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\left(\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{C}}{\sigma_{C}}\|\chi
R^{-1/2}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}\left\|\int_{0}^{t}R^{-1/2}\bm{v}(\tau)\,{\rm d}\tau\right\|_{2}\right).$

Further, Lemma 5.3 b) and Proposition 2 imply for all $t\in[0,T)$,

$\|(I-P)\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq
L_{1}\,\|\nabla\times\bm{A}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\\\ \leq
L_{1}\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{1}}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}$

and, thus,

$\displaystyle\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq\|P\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\|(I-P)\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\qquad\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\left(\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{C}}{\sigma_{C}}\|\chi
R^{-1/2}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3\times
m})}\left\|\int_{0}^{t}R^{-1/2}\bm{v}(\tau)\,{\rm d}\tau\right\|_{2}\right)$
$\displaystyle\qquad\qquad+L_{1}\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{1}}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}.$

b) If $\chi\in
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})^{1\times m}$,
then it follows from [2, Theorem 3.17] that there exists some $\bm{F}\in
H_{0}(\operatorname{curl},\mathit{\Omega)}^{1\times m}$ with
$\chi=\nabla\times F$. Using the integration by parts formula for the curl
operator, see [11, eq. (2.1)], we obtain that the columns of $\chi$ are
orthogonal with respect to the inner product in
$L^{2}(\mathit{\Omega};\mathbb{R}^{3})$ to all
$\bm{C}\in{X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})}$.
In other words,

$\chi\in((\operatorname{im}\widetilde{P})^{\bot})^{1\times
m}=(\ker\widetilde{P}^{*})^{1\times m},$

which gives $\widetilde{P}^{*}\chi=0$. Then (39) reduces to
$P\bm{A}(t)=P\bm{A}_{0}$ for all $t\in[0,T)$. Now proceeding as in the
previous case, we obtain

$\displaystyle\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq$
$\displaystyle\;\|P\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\|(I-P)\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\leq$
$\displaystyle\;\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\,\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;+L_{1}\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\tfrac{L_{1}}{\sqrt{2m_{\nu}}}\,\|R^{-1/2}\bm{v}\|_{L^{2}([0,t);\mathbb{R}^{m})}.$

∎

######  Remark 8.

Note that the divergence-freeness condition in Theorem 5.4 b) is guaranteed,
if the support of $\chi$ does not meet the interface between the conducting
and non-conducting subdomains, i.e.,
$\mbox{supp}(\chi)\subset\mathit{\Omega}_{C}\cup\mathit{\Omega}_{I}$. In this
case, the condition $\chi\in X(\mathit{\Omega},\mathit{\Omega}_{C})^{1\times
m}$ is even equivalent to $\chi\in
L^{2}(\operatorname{div}\\!=\\!0,\mathit{\Omega};\mathbb{R}^{3})^{1\times m}$.

###  5.3. The free dynamics

By free dynamics, we mean the solution behavior of the coupled MQS system (1)
in which (1) no voltage is applied, i.e., $\bm{v}=0$. Our goal is now to
analyse their asymptotic behavior for $t\to\infty$.

The following theorem shows that the magnetic energy $E(\bm{A}(t))$ as well as
the $L^{2}$-norm of $\nabla\times\bm{A}(t)$ can be bounded from above by an
exponentially decaying function provided $\bm{v}\equiv 0$.

######  Theorem 5.5.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and
$\bm{A}_{0}\\!\in\\!X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$.
Moreover, let $E$ be the magnetic energy as defined in (10), and let
$(\bm{A},\bm{i})$ be a solution of the coupled MQS system (1) with
$\bm{v}\equiv 0$. Then for $\sigma_{C}$, $m_{\nu}$ and $L_{\nu}$ as in
Assumption 2.2, $\gamma$ as in (36), and

$\omega:=\tfrac{m_{\nu}^{2}}{\gamma\,L_{\nu}},$ (40)

it holds for all $t\geq 0$ that

$\displaystyle E(\bm{A}(t))\leq$ $\displaystyle\,e^{-2\,\omega
t}E(\bm{A}_{0}),$ (41)
$\displaystyle\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq$
$\displaystyle\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,e^{-\omega
t}\,\|\nabla\times{\bm{A}_{0}}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$
(42)

###### Proof.

Consider the operator

$\displaystyle\mathcal{F}:\;\;X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}$
$\displaystyle\,\to\,X(\mathit{\Omega},\mathit{\Omega}_{C}),$
$\displaystyle(\bm{A},\bm{i})$
$\displaystyle\,\mapsto\,\sqrt{\sigma}\bm{A}+\chi\,R^{-1/2}\,\bm{i}.$

Then $\mathcal{F}$ is linear and bounded with
$\|\mathcal{F}\|\leq{\sqrt{\gamma}}$. Further, $\operatorname{im}\mathcal{F}$
is closed, as it is the sum of the closed space
$L^{2}(\mathit{\Omega_{C}};\mathbb{R}^{3})$ and a finite-dimensional subspace
of $X(\mathit{\Omega},\mathit{\Omega}_{C})$. Therefore, we conclude from [9,
Theorem 9.3.3] that $\mathcal{F}$ has a bounded Moore-Penrose inverse
$\mathcal{F}^{+}:X(\mathit{\Omega},\mathit{\Omega}_{C})\to
X(\mathit{\Omega},\mathit{\Omega}_{C})\times\mathbb{R}^{m}$ such that
$\mathcal{F}\mathcal{F}^{+}$ and $\mathcal{F}^{+}\mathcal{F}$ are the
orthogonal projectors onto $\operatorname{im}\mathcal{F}$ and
$\operatorname{im}\mathcal{F}^{*}$, respectively. Using
$\|\mathcal{F}\|\leq{\sqrt{\gamma}}$, we obtain for all
$\bm{F}\in\operatorname{im}\mathcal{F}$,

$\|\bm{F}\|_{{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3})}}=\|\mathcal{F}\mathcal{F}^{+}\bm{F}\|_{{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3})}}\leq{\sqrt{\gamma}}\,\|\mathcal{F}^{+}\bm{F}\|_{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3}){\times\mathbb{R}^{m}}}.$
(43)

Let $(\bm{A},\bm{i})$ be a solution of (1) with $\bm{v}\equiv 0$. Then it
follows from (6) that for almost all $t\geq 0$,

$\displaystyle\nabla\times\left(\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)\right)$
$\displaystyle=-\tfrac{\rm d}{{\rm
d}t}\left(\sigma\bm{A}(t)\right)+\chi\,\bm{i}(t)$ (44)
$\displaystyle=-\mathcal{F}\tfrac{\rm d}{{\rm
d}t}\mathcal{F}^{*}\bm{A}(t)\in\operatorname{im}\mathcal{F},$

and Assumption 2.2 b) gives

$\langle\nabla\times\bm{A}(t),\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)\rangle_{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3})}\geq
m_{\nu}\|\nabla\times{\bm{A}}(t)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$

Further, using the integration by parts formula for the weak curl operator
together with the fact that $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ for almost all
$t\geq 0$, we obtain

$\|\nabla\times\left(\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)\right)\|_{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3})}\geq
m_{\nu}\|\nabla\times{\bm{A}}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$
(45)

Then it follows from (43) and (45) that for almost all $t\geq 0$,

$\displaystyle\left\|\mathcal{F}^{+}(\nabla\right.$
$\displaystyle\left.\times(\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)))\right\|_{L^{2}(\mathit{{\Omega}};\mathbb{R}^{3})\times\mathbb{R}^{m}}$
(46)
$\displaystyle\geq\tfrac{1}{{\sqrt{\gamma}}}\,\left\|\nabla\times\left(\nu(\cdot,\|\nabla\times\bm{A}(t)\|_{2})\nabla\times\bm{A}(t)\right)\right\|_{L^{2}(\mathit{\Omega}_{C};\mathbb{R}^{3})}$
$\displaystyle\geq\tfrac{m_{\nu}}{{\sqrt{\gamma}}}\,\left\|\nabla\times\bm{A}(t)\right\|_{L^{2}(\mathit{\Omega}_{C};\mathbb{R}^{3})}.$

Let $\omega$ be as in (40). Using the energy balance (17) and the relation
$\mathcal{F}^{*}=\mathcal{F}^{+}\mathcal{F}\mathcal{F}^{*}$, we obtain, by
invoking $\bm{v}\equiv 0$, that for all $0\leq t_{0}<t<\infty$,

$\displaystyle E(\bm{A}$ $\displaystyle(t_{1}))-E(\bm{A}(t_{0}))$
$\displaystyle\;=\;-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau))\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau-\int_{t_{0}}^{t_{1}}\langle\bm{i}(\tau),R\,\bm{i}(\tau)\rangle_{2}\,{\rm
d}\tau$
$\displaystyle\\!\overset{\eqref{eq:MQSsol}}{=}-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm
d}{{\rm
d}\tau}\sqrt{\sigma}\bm{A}(\tau))\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau-{\int_{t_{0}}^{t_{1}}}\left\|\tfrac{\rm d}{{\rm
d}\tau}R^{-1/2}\int_{\mathit{\Omega}}\chi^{\top}\bm{A}(\tau)\,{\rm
d}\xi\right\|_{2}^{2}\,{\rm d}\tau$
$\displaystyle\;=\;-\int_{t_{0}}^{t_{1}}\|\tfrac{\rm d}{{\rm
d}\tau}\mathcal{F}^{*}\bm{A}(\tau)\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})\times\mathbb{R}^{m}}{\rm
d}\tau$
$\displaystyle\;=\;-\int_{t_{0}}^{t_{1}}\|\mathcal{F}^{+}\mathcal{F}\tfrac{\rm
d}{{\rm
d}\tau}\mathcal{F}^{*}\bm{A}(\tau))\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})\times\mathbb{R}^{m}}{\rm
d}\tau$
$\displaystyle\\!\overset{\eqref{eq:pSAimE}}{=}-\int_{t_{0}}^{t_{1}}\left\|\mathcal{F}^{+}\left(\nabla\times\left(\nu(\|\nabla\times\bm{A}(\tau)\|_{2})\nabla\times\bm{A}(\tau)\right)\right)\right\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})\times\mathbb{R}^{m}}{\rm
d}\tau$
$\displaystyle\\!\overset{\eqref{eq:E-est}}{\leq}-\frac{m_{\nu}^{2}}{\gamma}\,\int_{t_{0}}^{t_{1}}\left\|\nabla\times\bm{A}(\tau)\right\|^{2}_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}{\rm
d}\tau$
$\displaystyle\\!\overset{\eqref{eq:equivalence}}{\leq}-2\,\omega\,\int_{t_{0}}^{t_{1}}E(\bm{A}(\tau)){\rm
d}\tau$

with $\omega$ as in (40). By a division of the above inequality by
$t_{1}-t_{0}$ and then taking the limit $t_{1}\to t_{0}$, we obtain that the
weak derivative of $t\mapsto E(\bm{A}(t))$ fulfills the differential
inequality

$\tfrac{\rm d}{{\rm d}t}E(\bm{A}(t))\leq-2\,\omega\,E(\bm{A}(t)).$ (47)

Then Grönwall’s inequality for the weak derivative [27, Lemma IV.4.1] gives
rise to (41). The estimate (42) can be concluded from (41) by further using
the inequalities in (11), and subsequently taking the square root. ∎

######  Remark 9.

  1. a)

The inequality (47) shows that the scalar function $t\mapsto E(\bm{A}(t))$ is
strictly decaying unless $\nabla\times\bm{A}(t){=0}$. This is not surprising
from a physical point of view, as, by $\bm{v}\equiv 0$, no external energy is
put into the system.

  2. b)

Let us briefly consider the case where the free MQS system is initialized with
$\bm{A}_{0}\in X(\mathit{\Omega},\mathit{\Omega}_{C})$, which is not
necessarily in $H_{0}(\operatorname{curl},\mathit{\Omega})$. By [11, Theorem
7.1] on the existence and regularity properties of the solutions of the
coupled MQS system (1), we have $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ for almost all
$t>0$. Further, for each finite interval $[0,T]$, the functions $t\mapsto
E(\bm{A}(t))$ and
$t\mapsto\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
can be bounded by a constant times $\tfrac{1}{t}$ and $\tfrac{1}{\sqrt{t}}$,
respectively. As a consequence, there exists some constants $M_{1},M_{2}>0$
such that the solution $(\bm{A},\bm{i})$ of (1) with $\bm{v}\equiv 0$
satisfies for all $t>0$,

$E(\bm{A}(t))\leq M_{1}\,(1+\tfrac{1}{t})\,e^{-2\omega
t},\quad\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq
M_{2}\,(1+\tfrac{1}{\sqrt{t}})\,e^{-\omega t}.$

Finally, we derive the estimates for the $L^{2}$-norm of the magnetic vector
potential $\bm{A}(t)$ of the uncontrolled coupled MQS system (1).

######  Theorem 5.6.

Assume that $\mathit{\Omega}\subset\mathbb{R}^{3}$ with a subdomain
$\mathit{\Omega}_{C}$ satisfies Assumption 2.1. Further, let Assumptions 2.2
and 2.3 be fulfilled, and $\bm{A}_{0}\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$ and
$\bm{v}\equiv 0$. Moreover, let $E$ be the magnetic energy as defined in (10),
and let $(\bm{A},\bm{i})$ be a solution of the coupled MQS system (1). Then
for $\sigma_{C}$, $L_{\nu}$, $m_{\nu}$ as in Assumption 2.2, $\gamma$ as in
(36), and $\omega$ as in (40), it holds for all $t\geq 0$,

$\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}\,\|\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+L_{1}\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,e^{-\omega
t}\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$ (48)

If, additionally, the initial value fulfills for all $\bm{F}\in
X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})$,

$\int_{\mathit{\Omega}}\sigma\bm{A}_{0}\cdot\bm{F}\,{\rm
d}\xi+\int_{\mathit{\Omega}}\chi^{\top}\bm{A}_{0}\,{\rm d}\xi\cdot
R^{-1}\int_{\mathit{\Omega}}\chi^{\top}\bm{F}\,{\rm d}\xi=0,$ (49)

then for all $t\geq 0$,

$\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}\leq
L_{1}\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,e^{-\omega
t}\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$ (50)

###### Proof.

Let $\widetilde{P}\in\mathcal{L}(X(\mathit{\Omega},\mathit{\Omega}_{C}))$ be
the projector as in Lemma 5.3 a). Then by using the argumentation as in the
proof of Theorem 5.4 a) and invoking $\bm{v}\equiv 0$, we obtain that
$\widetilde{P}\bm{A}(t)=\widetilde{P}\bm{A}_{0}$ for all $t\geq 0$. Thus, by
further using that $\bm{A}(t)\in
X_{0}(\operatorname{curl},\mathit{\Omega},\mathit{\Omega}_{C})$, we obtain for
almost all $t\geq 0$,

$\displaystyle\|\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\;\leq\;\|\widetilde{P}\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\|(I-\widetilde{P})\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
(51)
$\displaystyle\;=\;\|\widetilde{P}\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+\|(I-{P})\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\\!\overset{\eqref{eq:L1const}}{\leq}\|\widetilde{P}\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+L_{1}\|\nabla\times\bm{A}(t)\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}$
$\displaystyle\\!\overset{\eqref{eq:MQSexpstab2}}{\leq}\|\widetilde{P}\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}+L_{1}\,\sqrt{\tfrac{L_{\nu}}{m_{\nu}}}\,e^{-\omega
t}\|\nabla\times\bm{A}_{0}\|_{L^{2}(\mathit{\Omega};\mathbb{R}^{3})}.$

Then (48) follows by using the bound
$\|\widetilde{P}\|\leq\sqrt{\tfrac{\gamma\,L_{C}}{\sigma_{C}}}$ from Lemma 5.3
a).

On the other hand, if $\bm{A}_{0}$ satisfies (49), then $\bm{A}_{0}$ is
orthogonal to all elements of
$X_{0}(\operatorname{curl}\\!=\\!0,\mathit{\Omega},\mathit{\Omega}_{C})$ with
respect to the inner product (34). Since $P$ is an orthogonal projector with
respect to that inner product, we have
$\widetilde{P}\bm{A}_{0}={P}\bm{A}_{0}=0$, and, hence, (51) immediatelly
implies (50). ∎

##  6\. Conclusion

We have considered a quasilinear magneto-quasistatic approximation of
Maxwell’s equations, which is furthermore coupled with an integral equation.
By employing the magnetic energy, we have shown that this system is passive
and admits a representation as a port-Hamiltonian system. Further, we have
derived estimates of the state and output of the system by means of the
initial value and the input. A special emphasis in the solution estimates is
placed on the free system with the zero input voltage, where we have shown
that the magnetic energy decays exponentially.

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