remarkRemark hypothesisHypothesis claimClaim Infinite-dimensional pH systems
with a stationary interfaceA. Kilian, B. M. Maschke, A. Mironchenko, and F. R.
Wirth

# Infinite-dimensional port-Hamiltonian systems with a stationary
interface††thanks: Submitted to the editors DATE. A. Mironchenko is supported
by the German Research Foundation (DFG) via the grant MI 1886/2-2.

A. Kilian Faculty of Computer Science and Mathematics, University of Passau,
Germany ( (corresponding author<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>B. M. Maschke
LAGEP, UCB Lyon 1 - CNRS UMR 5007, CPE Lyon - Bâtiment 308 G, Université
Claude Bernard Lyon-1, 43, bd du 11 Novembre 1918, F-69622 Villeurbanne cedex,
France<EMAIL_ADDRESS>A. Mironchenko22footnotemark: 2 F.
R. Wirth22footnotemark: 2

###### Abstract

We consider two systems of two conservation laws that are defined on
complementary spatial intervals and coupled by an interface as a single port-
Hamiltonian system. In case of a fixed interface position, we characterize the
boundary and interface conditions for which the associated port-Hamiltonian
operator generates a contraction semigroup. Furthermore, we present sufficient
conditions for the exponential stability of the generated $C_{0}$-semigroup.
The results are illustrated by the example of two acoustic waveguides coupled
by an interface consisting of some membrane.

###### keywords:

port-Hamiltonian systems, strongly continuous semigroups, stationary
interface, exponential stability

35L02, 35Q93, 37L15, 35B35, 93C05

##  1 Introduction

Linear infinite-dimensional systems theory is a well-established subject [5,
26, 15], constituting a basis for analysis of broad classes of distributed
parameter systems. However, _physical systems_ belonging to mechanical,
electric, hydraulic, thermal, and other domains share common specific
structures that can be used to develop methods for modeling, analysis and
control, which are more natural and easy-to-use than those available for
general linear systems. The port-Hamiltonian systems theory [7, 15, 28] is a
systematic framework that develops such methods. This modeling framework is an
extension of the _Hamiltonian systems_ , stemming from mechanics [1], to open
systems in a control systems’ perspective where the interaction with the
environment is given by the definition of a pair of external variables called
_port variables_ [27, 11].

Following the semigroup approach, consider the abstract Cauchy problem of the
form

$\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t),\quad t>0,$ $\displaystyle
x(0)$ $\displaystyle=x_{0}\in X,$

where $A\colon D(A)\subset X\to X$ is a linear unbounded operator on some
Hilbert space $X$. In the port-Hamiltonian framework [29, Chapter 2], [11,
Section 3], [7, pp. 226-228], [3, Chapter 3], [24] the operator is assumed to
be the composition

$\displaystyle A=\mathcal{J}(\mathcal{Q}\cdot)$

where $\mathcal{J}$ is a Hamiltonian operator [22], which is a formally skew-
symmetric constant coefficient matrix differential operator, modeling the
interdomain coupling in physical systems, and $\mathcal{Q}$ is a symmetric
coercive and bounded multiplication operator defining the energy of the
physical system. Furthermore, the system is augmented with a pair of _boundary
port variables_ defined by a boundary port operator $R_{\operatorname{ext}}$.
This operator is associated with the Hamiltonian operator $\mathcal{J}$ [11,
Def. 3.5], in a very similar way as for boundary triplets [18]:

$\displaystyle\begin{bmatrix}f_{\partial}\\\
e_{\partial}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}\left(\mathcal{Q}x\right),$

where $\operatorname{tr}$ denotes the trace operator. The well-documented
theory of strongly continuous semigroups [5, 8, 23] may then be used to derive
results on well-posedness, stability and stabilization by boundary feedback
[24, section 4.4]. The most important class of semigroups in the framework of
infinite-dimensional port-Hamiltonian systems are contraction semigroups,
which are closely related to energy dissipation. Typically, dissipation
phenomena are incorporated into the domain of the operator $A$ by means of the
port variables, and this operator is the infinitesimal generator of a
contraction semigroup. For control purposes, control and conjugated
observation variables may be defined as a linear combination of the port
variables, leading to the definition of (port-Hamiltonian) boundary control
systems [9], [26, Chapter 10]. Such systems have been extensively studied in
[3, 29].

An important property of port-Hamiltonian systems is that the power-conserving
interconnection of port-Hamiltonian systems again defines a port-Hamiltonian
system. This interconnection is defined by the composition of the underlying
(Stokes-)Dirac structures [28, Section 2.2, Section 14.1] through a number of
mutually shared port variables. The internally stored energy of the
interconnected system, described by the Hamiltonian on the product space, is
simply the sum of the Hamiltonians of the constituting subsystems. We refer to
[28, Chapter 6] for the finite-dimensional case as well as to [13], [29,
Chapter 7], and the references therein for the infinite-dimensional case. In
[6, Section 2.2], it has been depicted how port-Hamiltonian systems are
coupled through their (fixed) boundaries.

This paper is motivated by the models of systems with an interface within
their spatial domain and which may be moving as the result of non-equilibrium
conditions at the interface. Such systems arise, for instance, in varying
causality systems, such as extrusion processes where some parts are completely
filled and some other not [20], phase transition phenomena in multiphase
systems such as evaporators or condensers [31] or the celebrated Stefan
problem [12].

Here we follow the frame suggested in [6] where the authors proposed the port-
Hamiltonian formulation of a physical system containing a moving interface,
not considering coupling two systems by a Dirac structure or a port-
Hamiltonian system [19, 14] but by introducing the interface using additional
variables, the _color functions_ [10, 4]. These color functions are actually
the characteristic functions of spatial domains separated by the interface and
correspond to the so-called thin interface assumption. A motivation for this
perspective is also provided by the approach by the phase field theory for
multiphase systems, where the interface separating the different phases has
actually a certain thickness and is characterized by a distributed variable
called _phase field_ [30].

Contribution. In this paper, we propose a mathematically rigorous definition
of port-Hamiltonian systems with an interface suggested in [6] where the
Hamiltonian operator depends on the color functions. Then we model such
systems as abstract evolution equations and use the methods of semigroup
theory to provide easy-to-check criteria for boundary and interface conditions
that guarantee that the system with a _fixed_ interface induces a contraction
semigroup. Under somewhat stronger conditions, we prove the exponential
stability of such a system. We apply our findings to the interface coupling of
two acoustic waveguides.

Structure of the paper. Section 2 introduces port-Hamiltonian systems with an
interface. It is split into three parts. In Subsection 2.1, we consider two
systems of conservation laws defined on two complementary spatial domains that
we want to interconnect at the interface position, which is assumed to be
fixed. The notions of interface port variables and color functions are
introduced, which allows us to formalize the state variables of the aggregate
system on the composed domain. In Subsection 2.2, we briefly describe the
derivation of the conservation laws of the aggregate system having regard to
the interface conditions. Subsection 2.3 is devoted to the port-Hamiltonian
formulation of the system dynamics. We introduce the formally skew-symmetric
differential operator associated with the aggregate system. Furthermore, we
define the boundary and interface port variables in case of a fixed interface
position, and state an energy balance equation for this class of systems. In
Section 3, we characterize boundary and interface conditions for which the
port-Hamiltonian operator associated with the considered port-Hamiltonian
system generates a contraction semigroup on the energy space. Under stronger
assumptions concerning the boundary conditions, we guarantee exponential
stability of the semigroup, as we work out in Section 4. In Section 5, we
apply our findings to the interface coupling of two acoustic waveguides.
Finally, Section 6 concludes the paper.

#### Notation

Throughout the paper $X$ is a Hilbert space and $\mathcal{L}(X)$ is the space
of linear bounded operators on $X$. We denote by $\mathcal{C}([a,b],X)$ and
$\mathcal{C}^{k}([a,b],X)$ the vector spaces of $X$-valued, continuous and
$k$-times continuously differentiable functions $f\colon[a,b]\to X$;
$L^{2}([a,b],\mathbb{R}^{n})$ denotes the vector space of
$\mathbb{R}^{n}$-valued square-integrable functions on the interval $[a,b]$,
and $L^{\infty}([a,b],\mathbb{R}^{n})$ is the vector space of
$\mathbb{R}^{n}$-valued essentially bounded functions on $[a,b]$. Moreover,
$H^{1}([a,b],\mathbb{R}^{n})$ denotes the Sobolev space of order $1$ on
$[a,b]$. Further, the vector space of infinitely differentiable functions with
compact support in $(a,b)\subset\mathbb{R}$, that is, the set of test
functions on $(a,b)$, is denoted by $\mathcal{D}(a,b)$. The Euclidean norm in
$\mathbb{R}^{n}$ is denoted by $|\cdot|$ and $I_{n}$ is the identity matrix of
size $n\times n$.

##  2 Port-Hamiltonian Systems with Interface

In this section, we propose a port-Hamiltonian formulation for a system
composed of two conservation laws, coupled by means of the interface,
satisfying certain interface relations. For a detailed discussion, we refer to
[6], [16, Section 5.2].

###  2.1 Model Description

The situation is as follows: Let $a<0<b$, and let $l\in(a,b)$ be the fixed
position of the interface. Consider two systems of conservation laws of the
form

(1) $\displaystyle\begin{split}\frac{\partial}{\partial
t}x^{-}(z,t)&=P_{1}\frac{\partial}{\partial
z}\big{(}\mathcal{Q}^{-}(z)x^{-}(z,t)\big{)},\quad z\in[a,l),\quad t>0,\\\
\frac{\partial}{\partial t}x^{+}(z,t)&=P_{1}\frac{\partial}{\partial
z}\big{(}\mathcal{Q}^{+}(z)x^{+}(z,t)\big{)},\quad z\in(l,b],\quad
t>0,\end{split}$

defined on the respective interval $[a,l)$ or $(l,b]$, where

(2) $P_{1}=\begin{bmatrix}0&-1\\\ -1&0\end{bmatrix}\in\mathbb{R}^{2\times 2},$

and $\mathcal{Q}^{\pm}\in L^{\infty}([a,b],\mathbb{R}^{2\times 2})$ are
symmetric and satisfy $mI_{2}\leq\mathcal{Q}^{\pm}(z)\leq MI_{2}$ for almost
all $z\in[a,b]$ and for some $0<m\leq M$.

Let us define the _flux variables_

$\displaystyle\mathcal{N}^{-}(x^{-})$
$\displaystyle=-P_{1}(\mathcal{Q}^{-}x^{-}),\quad x^{-}\in
X^{-}=L^{2}((a,l),\mathbb{R}^{2}),$ $\displaystyle\mathcal{N}^{+}(x^{+})$
$\displaystyle=-P_{1}(\mathcal{Q}^{+}x^{+}),\quad x^{+}\in
X^{+}=L^{2}((l,b),\mathbb{R}^{2}).$

Our objective is to interconnect these two systems to a single port-
Hamiltonian system on the composed spatial domain $[a,b]$, where the position
of the interface is assumed to be at the fixed position $l\in(a,b)$. To do so,
we need to introduce suitable state variables defined on the composed domain,
and we need to define some _interface relations_ at $z=l$. The interface
relations simultaneously constitute the _interface port variables_
$(e_{I},f_{I})\in\mathbb{R}^{2}$. These are given by

(3) $\displaystyle f_{I}$
$\displaystyle=\mathcal{N}_{1}^{+}(x^{+})(l^{+})=\mathcal{N}_{1}^{-}(x^{-})(l^{-}),$
(4) $\displaystyle-e_{I}$
$\displaystyle=\mathcal{N}_{2}^{+}(x^{+})(l^{+})-\mathcal{N}_{2}^{-}(x^{-})(l^{-}).$

Equation (3) is a continuity equation of the flux variables
$\mathcal{N}_{1}^{\pm}(x^{\pm})$, and (4) is a balance equation of the flux
variables $\mathcal{N}_{2}^{\pm}(x^{\pm})$. These interface relations are
commonly considered, cf. [4, 10].

Next, we define the _color functions_. They are defined as the characteristic
functions of the respective complementary subdomain of the considered spatial
domain $[a,b]$, and are used to keep track of the interface position:

(5) $c_{l}^{-}(z)=\begin{cases}1,&z\in[a,l),\\\
0,&z\in[l,b],\end{cases}\quad\text{and}\quad
c_{l}^{+}(z)=\begin{cases}0,&z\in[a,l],\\\ 1,&z\in(l,b].\end{cases}$

Since, in this paper, we assume that the interface position is fixed, the
color functions do not depend on time. However, they allow to write the state
variables of the coupled system as the sum of prolongations of the variables
of each subsystem to the composed domain $[a,b]$. By abuse of notation, for
all $t\geq 0$ and $x(\cdot,t)\in X=L^{2}([a,b],\mathbb{R})$, we have

$\displaystyle x(z,t)$ $\displaystyle=x^{-}(z,t)+x^{+}(z,t)$
$\displaystyle=c_{l}^{-}(z)x(z,t)+c_{l}^{+}(z)x(z,t).$

Furthermore, the flux variable of the coupled system becomes

$\displaystyle\mathcal{N}(x)=c_{l}^{-}\mathcal{N}^{-}(x)+c_{l}^{+}\mathcal{N}^{+}(x),\quad
x\in X,$

which obviously satisfies

$\displaystyle
c_{l}^{-}\mathcal{N}(x)=c_{l}^{-}\mathcal{N}^{-}(x)\quad\text{and}\quad
c_{l}^{+}\mathcal{N}(x)=c_{l}^{+}\mathcal{N}^{+}(x).$

For the sake of brevity, for $i=1,2,$ we will sometimes write
$\mathcal{N}_{i}^{\pm}(z,t)$ instead of $\mathcal{N}_{i}^{\pm}(x)(z,t)$, and
$\mathcal{N}_{i}(z,t)$ instead of
$\mathcal{N}_{i}(x)(z,t)=\mathcal{N}_{i}(x(\cdot,t))(z)$.

###  2.2 Derivation of the Balance Equations

To define the coupled system as a port-Hamiltonian system on the state space
$X$, we derive next the balance equations of the variables $x_{1}$ and $x_{2}$
having regard to the interface relations (3)-(4).

We will show this only for the state variable $x_{1}$. For further details and
for the balance equation regarding $x_{2}$, we refer to [6], [16, Section
5.2].

Assume that the individual conservation laws (1) are satisfied in a classical
sense for some $x^{-}\colon[0,\infty)\to X^{-}$, $x^{+}\colon[0,\infty)\to
X^{+}$. This means that both $x^{-}\in\mathcal{C}^{1}([0,\infty),X^{-})$ and
$x^{+}\in\mathcal{C}^{1}([0,\infty),X^{+})$ solve the respective conservation
law in (1). Then the conservation law for $x_{1}$ of the aggregate system may
be formally written as

(6) $\displaystyle\begin{split}\frac{\partial}{\partial
t}x_{1}&=-\frac{\partial}{\partial z}\mathcal{N}_{1}(x)\\\
&=-\frac{\partial}{\partial
z}\big{(}c_{l}^{-}\mathcal{N}_{1}^{-}(x)+c_{l}^{+}\mathcal{N}_{1}^{+}(x)\big{)}\\\
&=-\frac{\partial}{\partial
z}\left(c_{l}^{-}\mathcal{N}_{1}(x)+c_{l}^{+}\mathcal{N}_{1}(x)\right)\\\
&=\underbrace{-\left[\frac{\partial}{\partial
z}(c_{l}^{-}\cdot)+\frac{\partial}{\partial
z}(c_{l}^{+}\cdot)\right]}_{=:\mathbf{d}_{l}}\mathcal{N}_{1}(x).\end{split}$

Next we are going to justify the above computations.

######  Proposition 2.1.

The equation (6) holds in a distributional sense for all $t>0$, that is, for
every test function $\varphi\in\mathcal{D}(a,b)$ it holds that

$\displaystyle\int_{a}^{b}\left[\mathbf{d}_{l}\mathcal{N}_{1}(x)(z,t)\right]\varphi(z)\,dz=\int_{a}^{b}\frac{\partial}{\partial
t}x_{1}(z,t)\varphi(z)\,dz.$

######  Proof 2.2.

So let $\varphi\in\mathcal{D}(a,b)$ be some test function. Due to (1) and the
continuity condition (3) of $\mathcal{N}_{1}(x)$, the following holds for all
$t>0$:

$\displaystyle\allowdisplaybreaks\int_{a}^{b}\left[\mathbf{d}_{l}\mathcal{N}_{1}(x)(z,t)\right]\varphi(z)dz$
$\displaystyle=\int_{a}^{b}-\left[\frac{\partial}{\partial
z}(c_{l}^{-}(z)\mathcal{N}_{1}(z,t))+\frac{\partial}{\partial
z}(c_{l}^{+}(z)\mathcal{N}_{1}(z,t))\right]\varphi(z)\,dz$ (part. int.)
$\displaystyle=\int_{a}^{b}c_{l}^{-}(z)\mathcal{N}_{1}(z,t)\frac{d}{dz}\varphi(z)\,dz+\int_{a}^{b}c_{l}^{+}(z)\mathcal{N}_{1}(z,t)\frac{d}{dz}\varphi(z)\,dz$
$\displaystyle=\int_{a}^{l}\mathcal{N}_{1}^{-}(z,t)\frac{d}{dz}\varphi(z)\,dz+\int_{l}^{b}\mathcal{N}_{1}^{+}(z,t)\frac{d}{dz}\varphi(z)\,dz$
(part. int.)
$\displaystyle=\mathcal{N}_{1}^{-}(l^{-},t)\varphi(l)+\int_{a}^{l}-\frac{\partial}{\partial
z}\mathcal{N}_{1}^{-}(z,t)\varphi(z)\,dz$
$\displaystyle\quad-\mathcal{N}_{1}^{+}(l^{+},t)\varphi(l)+\int_{l}^{b}-\frac{\partial}{\partial
z}\mathcal{N}_{1}^{+}(z,t)\varphi(z)\,dz$
$\displaystyle\stackrel{{\scriptstyle\eqref{Systems of Conservation
Laws}}}{{=}}{}\mathcal{N}_{1}^{-}(l^{-},t)\varphi(l)+\int_{a}^{l}\frac{\partial}{\partial
t}x_{1}^{-}(z,t)\varphi(z)\,dz$
$\displaystyle\quad-\mathcal{N}_{1}^{+}(l^{+},t)\varphi(l)+\int_{l}^{b}\frac{\partial}{\partial
t}x_{1}^{+}(z,t)\varphi(z)\,dz$
$\displaystyle\stackrel{{\scriptstyle\eqref{Interface Variable
fI}}}{{=}}{}\int_{a}^{b}\frac{\partial}{\partial t}x_{1}(z,t)\varphi(z)\,dz.$

This proves the claim.

Note that for all $t>0$, $\varphi\in\mathcal{D}(a,b)$, one computes

$\displaystyle-\int_{a}^{b}\frac{d}{dz}c_{l}^{-}(z)\mathcal{N}_{1}(z,t)\varphi(z)\,dz=\int_{a}^{l}\frac{\partial}{\partial
z}\big{(}\mathcal{N}_{1}^{-}(z,t)\varphi(z)\big{)}\,dz=\mathcal{N}_{1}^{-}(l^{-},t)\varphi(l),$

and

$\displaystyle-\int_{a}^{b}\frac{d}{dz}c_{l}^{+}(z)\mathcal{N}_{1}(z,t)\varphi(z)\,dz=\int_{l}^{b}\frac{\partial}{\partial
z}\big{(}\mathcal{N}_{1}^{+}(z,t)\varphi(z)\big{)}\,dz=-\mathcal{N}_{1}^{+}(l^{+},t)\varphi(l).$

This implies that, due to the continuity condition (3), the evaluation of the
flux variables of the respective subdomain at the interface cancel each other
out, and equation (6) becomes (when understood in a distributional sense)

(7) $\displaystyle\begin{split}\frac{\partial}{\partial
t}x_{1}(\cdot,t)&=\mathbf{d}_{l}\mathcal{N}_{1}(x)(\cdot,t)\\\
&=-c_{l}^{-}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{-}(x)(\cdot,t)-\frac{d}{dz}c_{l}^{-}(\cdot)\mathcal{N}_{1}(x)(\cdot,t)\\\
&\quad-c_{l}^{+}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{+}(x)(\cdot,t)-\frac{d}{dz}c_{l}^{+}(\cdot)\mathcal{N}_{1}(x)(\cdot,t)\\\
&=-c_{l}^{-}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{-}(x)(\cdot,t)-c_{l}^{+}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{+}(x)(\cdot,t).\end{split}$

Note that $x_{1}\in\mathcal{C}^{1}([0,\infty),L^{2}([a,b],\mathbb{R}))$ and,
under certain regularity conditions on $\mathcal{N}_{1}(x)$, we have for all
$t>0$,

$\displaystyle\mathbf{d}_{l}\mathcal{N}_{1}(x)(\cdot,t)=-c_{l}^{-}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{-}(x)(\cdot,t)-c_{l}^{+}(\cdot)\frac{\partial}{\partial
z}\mathcal{N}_{1}^{+}(x)(\cdot,t)\in L^{2}([a,b],\mathbb{R}).$

As a consequence, we have the following result:

######  Proposition 2.3.

Equation (7) holds in the a.e. sense.

This motivates the definition of the operator $\mathbf{d}_{l}\colon
D(\mathbf{d}_{l})\subset L^{2}([a,b],\mathbb{R})\to L^{2}([a,b],\mathbb{R})$
in the following way:

$\displaystyle D(\mathbf{d}_{l})$ $\displaystyle=\left\\{x\in
L^{2}([a,b],\mathbb{R})\mid x_{|(a,l)}\in
H^{1}((a,l),\mathbb{R}),\,x_{|(l,b)}\in
H^{1}((l,b),\mathbb{R}),\,x\in\mathcal{C}([a,b],\mathbb{R})\right\\}$ (8)
$\displaystyle=H^{1}([a,b],\mathbb{R}),$ $\displaystyle\mathbf{d}_{l}x$
$\displaystyle=-\left[\frac{d}{dz}(c_{l}^{-}x)+\frac{d}{dz}(c_{l}^{+}x)\right],\quad
x\in D(\mathbf{d}_{l}).$

For a function $x=c_{l}^{-}x^{-}+c_{l}^{+}x^{+}\in D(\mathbf{d}_{l})$ we have
in particular that

$\displaystyle\mathbf{d}_{l}x=-c_{l}^{-}\frac{d}{dz}x^{-}-c_{l}^{+}\frac{d}{dz}x^{+}.$

This way, the continuity equation (3) of the flux variable
$\mathcal{N}_{1}(x)$ is incorporated into the domain of the operator
$\mathbf{d}_{l}$.

Taking the balance equation (4) into consideration, the conservation law of
the state variable $x_{2}$ can be defined with respect to an operator closely
related to $\mathbf{d}_{l}$. Once again, we refer to [6], [16, Section 5.2]
for a detailed discussion. This admits a port-Hamiltonian formulation of these
conservation laws, as we will see in the following section.

###  2.3 Port-Hamiltonian Formulation

Let $X=L^{2}([a,b],\mathbb{R}^{2})$ be the state space and $l\in(a,b)$ be the
position of the interface. Consider the system

(9) $\frac{\partial}{\partial
t}x(z,t)=\mathcal{J}_{l}\big{(}\mathcal{Q}_{l}(z)x(z,t)\big{)},\quad
z\in[a,b]\setminus\\{l\\},\quad t>0,$

where $\mathcal{J}_{l}$ is defined below, and
$\mathcal{Q}_{l}=c_{l}^{-}\mathcal{Q}^{-}+c_{l}^{+}\mathcal{Q}^{+}\in\mathcal{L}(X)$
is a coercive matrix multiplication operator. In particular, the matrix
functions $\mathcal{Q}^{\pm}\in L^{\infty}([a,b],\mathbb{R}^{2\times 2})$ are
point-wise symmetric a.e., and there exist constants $0<m\leq M$ such that
$mI_{2}\leq\mathcal{Q}^{\pm}(z)\leq MI_{2}$ for a.e. $z\in[a,b]$. The
functions $c_{l}^{-}$ and $c_{l}^{+}$ are the color functions defined in (5).

We endow $X$ with the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{Q}_{l}}=\frac{1}{2}\langle\cdot,\mathcal{Q}_{l}\cdot\rangle_{L^{2}}$,
i.e.,

(10) $\langle
x,y\rangle_{\mathcal{Q}_{l}}=\frac{1}{2}\int_{a}^{b}y^{\top}(z)\mathcal{Q}_{l}(z)x(z)\,dz,\quad
x,y\in X,$

with associated norm $\|\cdot\|_{\mathcal{Q}_{l}}$. Note that
$\langle\cdot,\cdot\rangle_{\mathcal{Q}_{l}}$ is equivalent to the standard
inner product $\langle\cdot,\cdot\rangle_{L^{2}}$ on $X$. The Hamiltonian
$H\colon X\to\mathbb{R}$ on the energy space $(X,\|\cdot\|_{\mathcal{Q}_{l}})$
is defined as

(11)
$H(x)=\frac{1}{2}\int_{a}^{b}x^{\top}(z)\mathcal{Q}_{l}(z)x(z)\,dz=\|x\|_{\mathcal{Q}_{l}}^{2},\quad
x\in X.$

The operator $\mathcal{J}_{l}\colon D(\mathcal{J}_{l})\subset X\to X$ is given
by

(12) $\displaystyle\begin{split}D(\mathcal{J}_{l})&=\left\\{x=(x_{1},x_{2})\in
X\mid x_{1}\in D(\mathbf{d}_{l}^{\ast}),\,x_{2}\in
D(\mathbf{d}_{l})\right\\},\\\
\mathcal{J}_{l}x&=\begin{bmatrix}0&\mathbf{d}_{l}\\\
-\mathbf{d}_{l}^{\ast}&0\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix}=\begin{bmatrix}\mathbf{d}_{l}x_{2}\\\
-\mathbf{d}_{l}^{\ast}x_{1}\end{bmatrix},\quad x\in
D(\mathcal{J}_{l}),\end{split}$

where the operator $\mathbf{d}_{l}$ has been defined in (8), and its formal
adjoint111For a definition of formally adjoint operators, see [2, Chapter 13].
$\mathbf{d}_{l}^{\ast}$ is given by

$\displaystyle D(\mathbf{d}_{l}^{\ast})$ $\displaystyle=\left\\{y\in
L^{2}([a,b],\mathbb{R})\mid y_{|(a,l)}\in
H^{1}((a,l),\mathbb{R}),\,y_{|(l,b)}\in H^{1}((l,b),\mathbb{R})\right\\},$
$\displaystyle\mathbf{d}_{l}^{\ast}y$
$\displaystyle=\left(-\mathbf{d}_{l}-\left[\frac{d}{dz}c_{l}^{-}+\frac{d}{dz}c_{l}^{+}\right]\right)y,\quad
y\in D(\mathbf{d}_{l}^{\ast}).$

Similar to the operator $\mathbf{d}_{l}$, for a function
$y=c_{l}^{-}y^{-}+c_{l}^{+}y^{+}\in D(\mathbf{d}_{l}^{\ast})$ it holds that

$\mathbf{d}_{l}^{\ast}y=c_{l}^{-}\frac{d}{dz}y^{-}+c_{l}^{+}\frac{d}{dz}y^{+}.$

For all $x\in D(\mathbf{d}_{l}),\,y\in D(\mathbf{d}_{l}^{\ast})$, we have the
following relation between the operators $\mathbf{d}_{l}$ and
$\mathbf{d}_{l}^{\ast}$:

(13)
$\langle\mathbf{d}_{l}x,y\rangle_{L^{2}}=-\big{[}x(z)y(z)\big{]}_{a}^{b}+x(l)\left[y(l^{+})-y(l^{-})\right]+\langle
x,\mathbf{d}_{l}^{\ast}y\rangle_{L^{2}}.$

In particular, equation (13) describes the formal adjointness of
$\mathbf{d}_{l}$ and $\mathbf{d}_{l}^{\ast}$, that is, adjointness up to
boundary and interface evaluations.

Note that on any subinterval of $[a,b]$ not containing the interface position
$l\in(a,b)$, the operator $\mathcal{J}_{l}$ simply acts as the matrix
differential operator $P_{1}\frac{d}{dz}$, where $P_{1}$ is defined in (2).
Thus, (9) is a reformulation of (1). Next we reveal the port-Hamiltonian
structure of (9).

For $n\in\mathbb{N}$ we introduce the matrix $\Sigma_{2n}$, which will be used
extensively in the following sections:

(14) $\Sigma_{2n}:=\begin{bmatrix}0&I_{n}\\\
I_{n}&0\end{bmatrix}\in\mathbb{R}^{2n\times 2n}.$

In the framework of port-Hamiltonian systems, it is crucial that the
associated differential operator is formally skew-symmetric. It describes the
power-conserving interconnection of the system components, see, e.g., [11]. As
a first step, we show that the operator $\mathcal{J}_{l}$ defined in (12) is
_formally skew-symmetric_ , i.e.,

$\langle\mathcal{J}_{l}x,y\rangle_{L^{2}}=-\langle
x,\mathcal{J}_{l}y\rangle_{L^{2}}$

for all $x,y\in D(\mathcal{J}_{l})$ vanishing both at the boundary and at the
interface position.

######  Lemma 2.4.

The operator $\mathcal{J}_{l}\colon D(\mathcal{J}_{l})\subset X\to X$ defined
in (12) is formally skew-symmetric.

######  Proof 2.5.

Applying the relation (13) between the operators $\mathbf{d}_{l}$ and
$\mathbf{d}_{l}^{\ast}$, for $x=(x_{1},x_{2})$, $y=(y_{1},y_{2})\in
D(\mathcal{J}_{l})$ we have

(15)
$\displaystyle\begin{split}&\quad\langle\mathcal{J}_{l}x,y\rangle_{L^{2}}+\langle
x,\mathcal{J}_{l}y\rangle_{L^{2}}\\\
&=\langle\mathbf{d}_{l}x_{2},y_{1}\rangle_{L^{2}}-\langle\mathbf{d}_{l}^{\ast}x_{1},y_{2}\rangle_{L^{2}}+\langle
x_{1},\mathbf{d}_{l}y_{2}\rangle_{L^{2}}-\langle
x_{2},\mathbf{d}_{l}^{\ast}y_{1}\rangle_{L^{2}}\\\
&=-\big{[}y_{1}(z)x_{2}(z)\big{]}_{a}^{b}+x_{2}(l)\left[y_{1}(l^{+})-y_{1}(l^{-})\right]\\\
&\quad-\big{[}x_{1}(z)y_{2}(z)\big{]}_{a}^{b}+y_{2}(l)\left[x_{1}(l^{+})-x_{1}(l^{-})\right]\\\
&=\big{[}x^{\top}(z)P_{1}y(z)\big{]}_{a}^{b}+x_{2}(l)\left[y_{1}(l^{+})-y_{1}(l^{-})\right]+y_{2}(l)\left[x_{1}(l^{+})-x_{1}(l^{-})\right],\end{split}$

with $P_{1}$ defined in (2). This proves the claim.

Next, we augment the system (9) with boundary and interface port variables.
Port variables are used to express the previously mentioned power-conserving
interconnection between the system components ([28, Chapter 2], [29, Chapter
2]). Ports consist of physical quantities whose (inner) product yields power.
As the operator $\mathcal{J}_{l}$ is skew-symmetric up to boundary and
interface evaluations, see (15), we allow for a power flow both at the spatial
boundary (hence, an interaction with the system’s physical environment), and
at the interface position. Thus, by introducing boundary and interface port
variables, we can express the power flow of the system at these positions.

Since the interface position lies in the interior of the spatial domain
$[a,b]$, we may define the trace operator $\operatorname{tr}_{l}\colon
D(\mathcal{J}_{l})\to\mathbb{R}^{4}$ as follows:

$\operatorname{tr}_{l}(x)=\begin{bmatrix}x(b)\\\ x(a)\end{bmatrix},\quad x\in
D(\mathcal{J}_{l}).$

For all $x\in X$ such that $\mathcal{Q}_{l}x\in D(\mathcal{J}_{l})$, the
boundary flow $f_{\partial}=f_{\partial,\mathcal{Q}_{l}x}\in\mathbb{R}^{2}$
and the boundary effort
$e_{\partial}=e_{\partial,\mathcal{Q}_{l}x}\in\mathbb{R}^{2}$ are defined as

(16) $\begin{bmatrix}f_{\partial,\mathcal{Q}_{l}x}\\\
e_{\partial,\mathcal{Q}_{l}x}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}_{l}(\mathcal{Q}_{l}x)=\begin{bmatrix}\frac{1}{\sqrt{2}}(P_{1}(\mathcal{Q}_{l}x)(b)-P_{1}(\mathcal{Q}_{l}x)(a))\\\
\frac{1}{\sqrt{2}}((\mathcal{Q}_{l}x)(b)+(\mathcal{Q}_{l}x)(a))\end{bmatrix},$

where

(17) $R_{\operatorname{ext}}=\frac{1}{\sqrt{2}}\begin{bmatrix}P_{1}&-P_{1}\\\
I_{2}&I_{2}\end{bmatrix}\in\mathbb{R}^{4\times 4}.$

Now, we define the interface port variables $e_{I}=e_{I,\mathcal{Q}_{l}x}$,
$f_{I}=f_{I,\mathcal{Q}_{l}x}\in\mathbb{R}$. For all $x\in X$ such that
$\mathcal{Q}_{l}x\in D(\mathcal{J}_{l})$, the interface flow and the interface
effort are given as follows:

(18) $\displaystyle f_{I,\mathcal{Q}_{l}x}$
$\displaystyle=\left(\mathcal{Q}_{l}x\right)_{2}(l^{+})=\left(\mathcal{Q}_{l}x\right)_{2}(l^{-}),$
(19) $\displaystyle-e_{I,\mathcal{Q}_{l}x}$
$\displaystyle=\left(\mathcal{Q}_{l}x\right)_{1}(l^{+})-\left(\mathcal{Q}_{l}x\right)_{1}(l^{-}).$

For all $\mathcal{Q}_{l}x,\mathcal{Q}_{l}y\in D(\mathcal{J}_{l})$, we may
write the last line of (15) with respect to the boundary and interface port
variables as follows:

(20)
$\displaystyle\begin{split}&\quad\big{[}(\mathcal{Q}_{l}x)^{\top}(z)P_{1}(\mathcal{Q}_{l}y)(z)\big{]}_{a}^{b}+\left(\mathcal{Q}_{l}x\right)_{2}(l)\left[\left(\mathcal{Q}_{l}y\right)_{1}(l^{+})-\left(\mathcal{Q}_{l}y\right)_{1}(l^{-})\right]\\\
&\quad+\left(\mathcal{Q}_{l}y\right)_{2}(l)\left[\left(\mathcal{Q}_{l}x\right)_{1}(l^{+})-\left(\mathcal{Q}_{l}x\right)_{1}(l^{-})\right]\\\
&=\langle
e_{\partial,\mathcal{Q}_{l}y},f_{\partial,\mathcal{Q}_{l}x}\rangle+\langle
e_{\partial,\mathcal{Q}_{l}x},f_{\partial,\mathcal{Q}_{l}y}\rangle-
f_{I,\mathcal{Q}_{l}x}e_{I,\mathcal{Q}_{l}y}-f_{I,\mathcal{Q}_{l}y}e_{I,\mathcal{Q}_{l}x}.\end{split}$

Next, we simply substitute the relation (20) into equation (15). This yields
for all $\mathcal{Q}_{l}x,\mathcal{Q}_{l}y\in D(\mathcal{J}_{l})$,

(21)
$\displaystyle\begin{split}&\quad\langle\mathcal{J}_{l}(\mathcal{Q}_{l}x),\mathcal{Q}_{l}y\rangle_{L^{2}}+\langle\mathcal{Q}_{l}x,\mathcal{J}_{l}(\mathcal{Q}_{l}y)\rangle_{L^{2}}\\\
&=\langle
e_{\partial,\mathcal{Q}_{l}y},f_{\partial,\mathcal{Q}_{l}x}\rangle+\langle
e_{\partial,\mathcal{Q}_{l}x},f_{\partial,\mathcal{Q}_{l}y}\rangle-
f_{I,\mathcal{Q}_{l}x}e_{I,\mathcal{Q}_{l}y}-f_{I,\mathcal{Q}_{l}y}e_{I,\mathcal{Q}_{l}x}.\end{split}$

This allows us to state a (power) balance equation for classical solutions
$x\in C^{1}([0,\infty),X)$ of the first-order system (9) by means of the
boundary and interface port variables.

######  Lemma 2.6.

Let $x\in C^{1}([0,\infty),X)$ be a classical solution of the system (9) with
Hamiltonian (11). Then we have for all $t\geq 0$,

(22) $\frac{d}{dt}\|x(t)\|_{\mathcal{Q}_{l}}^{2}=\frac{d}{dt}H(x(t))=\langle
e_{\partial}(t),f_{\partial}(t)\rangle-e_{I}(t)f_{I}(t),$

where $e_{\partial}(t)=e_{\partial,\mathcal{Q}_{l}x(t)}$,
$f_{\partial}(t)=f_{\partial,\mathcal{Q}_{l}x(t)}$,
$e_{I}(t)=e_{I,\mathcal{Q}_{l}x(t)}$, and
$f_{I}(t)=f_{I,\mathcal{Q}_{l}x(t)}$.

######  Proof 2.7.

By exploiting the fact that $x$ satisfies (9) and by using (21), we get for
all $t\geq 0$,

$\displaystyle\frac{d}{dt}H(x(t))$
$\displaystyle=\frac{1}{2}\frac{d}{dt}\langle
x(t),\mathcal{Q}_{l}x(t)\rangle_{L^{2}}$
$\displaystyle=\frac{1}{2}\langle\mathcal{J}_{l}(\mathcal{Q}_{l}x(t)),\mathcal{Q}_{l}x(t)\rangle_{L^{2}}+\frac{1}{2}\langle\mathcal{Q}_{l}x(t),\mathcal{J}_{l}(\mathcal{Q}_{l}x(t))\rangle_{L^{2}}$
$\displaystyle=\langle e_{\partial}(t),f_{\partial}(t)\rangle-
e_{I}(t)f_{I}(t).$

Equation (22) states that the change of energy is equal to the power flow both
at the boundary and at the interface position.

######  Remark 2.8 (Dirac structure of (9)).

Note that the existence of an underlying, power-conserving interconnection
structure, called a (Stokes-)Dirac structure, is constitutive for port-
Hamiltonian systems [11], [28, Chapter 2], [29, Chapter 2]. It allows us to
geometrically specify the dynamics of the associated system by means of its
constituting port variables. However, in this work, we will not address this
topic. The underlying Dirac structure of the system (9) has been defined in
[16, Subsection 5.4.1].

Now that we have established the port-Hamiltonian formulation of the system
(9), we want to discuss under which conditions this system is well-posed. In
the subsequent sections, we will assume (without loss of generality) that the
fixed interface position is given by $l=0$.

##  3 Generation of a Contraction Semigroup

In this section, we characterize the boundary and interface conditions, under
which the system (9) generates a contraction semigroup. In particular, this
guarantees well-posedness in the sense of [8, Definition II.6.8]. To this end,
we investigate the port-Hamiltonian operator associated with the system (9)
with Hamiltonian (11). In [11, Section 4], it has been shown that suitable
boundary conditions in terms of the boundary port variables can be used to
characterize those boundary conditions for which the associated port-
Hamiltonian operator generates a contraction semigroup. With this approach in
mind, we will introduce interface conditions and boundary condition, which are
specified by the interface respectively boundary port variables. These
conditions will be incorporated into the definition of the domain of the port-
Hamiltonian operator. It will then turn out that the problem admits a
reformulation as a pure boundary port-Hamiltonian system of the type studied
in [15]. Using this reformulation, we show in Theorem 3.6 that the port-
Hamiltonian operator from (9) generates a contraction semigroup if and only if
certain passivity conditions both at the spatial boundary and at the interface
position hold. Furthermore, criteria are presented that guarantee the
generation of an isometric semigroup by the port-Hamiltonian operator.

###  3.1 Boundary and Interface Conditions

We begin by introducing boundary conditions for the system (9). Following [11,
Section 4], we impose linear boundary conditions of the form

(23) $\tilde{W}_{B}\begin{bmatrix}(\mathcal{Q}_{0}x)(b)\\\
(\mathcal{Q}_{0}x)(a)\end{bmatrix}=0$

for some $\tilde{W}_{B}\in\mathbb{R}^{2\times 4}$. Since the matrix
$R_{\operatorname{ext}}$ defined in (17) is invertible, we may write (23)
equivalently in terms of the boundary port variables (16) as

(24) $W_{B}\begin{bmatrix}f_{\partial,\mathcal{Q}_{0}x}\\\
e_{\partial,\mathcal{Q}_{0}x}\end{bmatrix}=0,\qquad
W_{B}=\tilde{W}_{B}R_{\operatorname{ext}}^{-1}\in\mathbb{R}^{2\times 4}.$

Next, we impose an _interface condition_ upon the interface port variables
(18)-(19). For a fixed $r\in\mathbb{R}$, it is given by

(25) $f_{I,\mathcal{Q}_{0}x}=re_{I,\mathcal{Q}_{0}x}.$

Recall that the interface port variables are defined by

(26) $\displaystyle f_{I,\mathcal{Q}_{0}x}$
$\displaystyle=\left(\mathcal{Q}^{+}x\right)_{2}(0^{+})=\left(\mathcal{Q}^{-}x\right)_{2}(0^{-}),$
$\displaystyle-e_{I,\mathcal{Q}_{0}x}$
$\displaystyle=\left(\mathcal{Q}^{+}x\right)_{1}(0^{+})-\left(\mathcal{Q}^{-}x\right)_{1}(0^{-}).$

With these boundary and interface conditions, we denote by
$A_{\mathcal{Q}_{0}}\colon D(A_{\mathcal{Q}_{0}})\subset X\to X$ the port-
Hamiltonian operator associated with the system (9) with Hamiltonian (11). It
is given by

(27) $\displaystyle\begin{split}D(A_{\mathcal{Q}_{0}})&=\left\\{x\in
X\,\big{|}\,\mathcal{Q}_{0}x\in
D(\mathcal{J}_{0}),\,f_{I,\mathcal{Q}_{0}x}=re_{I,\mathcal{Q}_{0}x},\,W_{B}\begin{bmatrix}f_{\partial,\mathcal{Q}_{0}x}\\\
e_{\partial,\mathcal{Q}_{0}x}\end{bmatrix}=0\right\\},\\\
A_{\mathcal{Q}_{0}}x&=\mathcal{J}_{0}(\mathcal{Q}_{0}x),\quad x\in
D(A_{\mathcal{Q}_{0}}).\end{split}$

We wish to characterize the boundary and interface conditions (i.e., those
$W_{B}$ and $r$) for which $A_{\mathcal{Q}_{0}}$ generates a contraction
semigroup $(T(t))_{t\geq 0}$ on the energy space
$(X,\|\cdot\|_{\mathcal{Q}_{0}})$. In particular, such conditions guarantee
the well-posedness of the abstract Cauchy problem

(28) $\displaystyle\begin{split}\dot{x}(t)&=A_{\mathcal{Q}_{0}}x(t),\quad
t>0,\\\ x(0)&=x_{0}\in D(A_{\mathcal{Q}_{0}}),\end{split}$

associated with the operator $A_{\mathcal{Q}_{0}}$.

###  3.2 A Reformulation as a Boundary Port-Hamiltonian System

To characterize well-posedness of our port-Hamiltonian system with interface,
we reformulate it in terms of a system without interface in the framework of
[15]. This allows to use the well-posedness criteria for port-Hamiltonian
systems developed in [15] for the analysis of our model. We continue to assume
$l=0$.

Let $\eta:=-a/b>0$ and consider the transformation
$w_{1}(\zeta,t):=x^{-}(\eta\zeta+a,t)$, $\zeta\in[0,b]$. With the
transformation $z=\eta\zeta+a$, the differential equation for $x^{-}$ on
$[a,0]$ given by

(29) $\frac{\partial}{\partial t}x^{-}(z,t)=P_{1}\frac{\partial}{\partial
z}\left(\mathcal{Q}^{-}(z)x^{-}(z,t)\right),\quad z\in(a,0),\quad t>0$

transforms to the equation on $(0,b)$ given by

(30) $\frac{\partial}{\partial
t}w_{1}(\zeta,t)=P_{1}\frac{\partial}{\partial\zeta}\left(\frac{1}{\eta}\mathcal{Q}^{-}(\eta\zeta+a)w_{1}(\zeta,t)\right),\quad\zeta\in(0,b),\quad
t>0.$

We now introduce a system of four first-order PDEs on $[0,b]$ that consists of
copies of the original system on $[a,0]$ (in a transformed form) and $[0,b]$.
The system is given by

(31) $\frac{\partial}{\partial
t}w(\zeta,t)=\hat{P}_{1}\frac{\partial}{\partial\zeta}\left(\mathcal{H}_{0}w\right)(\zeta,t),\quad
t>0,\quad\zeta\in(0,b),$

where $\hat{P}_{1}:=\operatorname{diag}(P_{1},P_{1})\in\mathbb{R}^{4\times
4}$, and

$\mathcal{H}_{0}(\zeta):=\operatorname{diag}(\eta^{-1}\mathcal{Q^{-}}(\eta\zeta+a),\mathcal{Q}^{+}(\zeta)),\quad\zeta\in[0,b].$

Let $\hat{X}:=L^{2}([0,b],\mathbb{R}^{4})$ be endowed with the scalar product
$\langle\cdot,\cdot\rangle_{\mathcal{H}_{0}}$ (defined analogously to (10))
and consider the map

(32a) $\displaystyle S:\hat{X}\to X,\quad S:w$
$\displaystyle=\begin{bmatrix}w_{1}\\\ w_{2}\end{bmatrix}\mapsto
x:=\frac{1}{\eta}{}c^{-}w_{1}(\eta^{-1}\cdot+b)+c^{+}w_{2},$ (32b)
$\displaystyle S^{-1}:X\to\hat{X},\quad S^{-1}:x$
$\displaystyle=c^{-}x^{-}+c^{+}x^{+}\mapsto w:=\begin{bmatrix}\eta
x^{-}(\eta\cdot+a)\\\ x^{+}\end{bmatrix}.$

The relation between $X$ and $\hat{X}$ is established via the following lemma:

######  Lemma 3.1.

$S$ induces an isometric isomorphism of the Hilbert spaces
$(X,\|\cdot\|_{\mathcal{Q}_{0}})$ and $(\hat{X},\|\cdot\|_{\mathcal{H}_{0}})$.

######  Proof 3.2.

For $x,y\in X$, denote $w=S^{-1}y,v=S^{-1}x$, where $S^{-1}$ is given by
(32b). We have

$\displaystyle 2\langle x,y\rangle_{\mathcal{Q}_{0}}$
$\displaystyle=\int_{a}^{b}y^{\top}(z)\mathcal{Q}_{0}(z)x(z)\,dz$
$\displaystyle=\int_{a}^{0}y^{-,\top}(z)\mathcal{Q}^{-}(z)x^{-}(z)\,dz+\int_{0}^{b}y^{+,\top}(z)\mathcal{Q}^{+}(z)x^{+}(z)\,dz.$

Substituting $z=\eta\zeta+a$, $dz=\eta\,d\zeta$ in the first integral and
writing $\zeta:=z$ in the second integral, we see that

$\displaystyle 2\langle x,y\rangle_{\mathcal{Q}_{0}}$
$\displaystyle=\int_{0}^{b}y^{-,\top}(\eta\zeta+a)\mathcal{Q}^{-}(\eta\zeta+a)x^{-}(\eta\zeta+a)\eta\,d\zeta+\int_{0}^{b}y^{+,\top}(\zeta)\mathcal{Q}^{+}(\zeta)x^{+}(\zeta)\,d\zeta$
$\displaystyle=\int_{0}^{b}\begin{bmatrix}y^{-,\top}(\eta\zeta+a)&y^{+,\top}(\zeta)\end{bmatrix}\begin{bmatrix}\eta\mathcal{Q}^{-}(\eta\zeta+a)&0\\\
0&\mathcal{Q}^{+}(\zeta)\end{bmatrix}\begin{bmatrix}x^{-}(\eta\zeta+a)\\\
x^{+}(\zeta)\end{bmatrix}d\zeta$
$\displaystyle=\int_{0}^{b}\begin{bmatrix}w_{1}(\zeta)\\\
w_{2}(\zeta)\end{bmatrix}^{\top}\begin{bmatrix}\eta^{-1}\mathcal{Q}^{-}(\eta\zeta+a)&0\\\
0&\mathcal{Q}^{+}(\zeta)\end{bmatrix}\begin{bmatrix}v_{1}(\zeta)\\\
v_{2}(\zeta)\end{bmatrix}d\zeta=2\left\langle\begin{bmatrix}w_{1}\\\
w_{2}\end{bmatrix},\begin{bmatrix}v_{1}\\\
v_{2}\end{bmatrix}\right\rangle_{\mathcal{H}_{0}}.$

Recall $r$ from the interface condition (25) and define the auxiliary matrices

(33) $U_{1}:=\begin{bmatrix}-r&1\\\ 0&1\end{bmatrix},\qquad
U_{2}:=\begin{bmatrix}r&0\\\ 0&-1\end{bmatrix}.$

With (26), it is easy to see that the conditions (18) and (19) together with
the interface condition $f_{I}=re_{I}$ are equivalent to

$U_{1}(\mathcal{Q}^{-}x^{-})(0^{-})+U_{2}(\mathcal{Q}^{+}x^{+})(0^{+})=0.$

For $\tilde{W}_{B}\in\mathbb{R}^{2\times 4}$ and $r\in\mathbb{R}$, introduce
the boundary condition matrix $\tilde{W}_{B,c}$ by

(34) $\tilde{W}_{B,c}=\begin{bmatrix}U_{1}&0_{2\times 4}&U_{2}\\\ 0_{2\times
2}&\tilde{W}_{B}&0_{2\times 2}\end{bmatrix}\in\mathbb{R}^{4\times 8}.$

Note that with this definition and the relation $w=S^{-1}x$, we have that

$\displaystyle\tilde{W}_{B,c}\begin{bmatrix}(\mathcal{H}_{0}w)(b)\\\
(\mathcal{H}_{0}w)(0)\end{bmatrix}=\tilde{W}_{B,c}\begin{bmatrix}(\mathcal{H}_{0}w)_{1}(b)\\\
(\mathcal{H}_{0}w)_{2}(b)\\\ (\mathcal{H}_{0}w)_{1}(0)\\\
(\mathcal{H}_{0}w)_{2}(0)\end{bmatrix}$
$\displaystyle=\tilde{W}_{B,c}\begin{bmatrix}(\mathcal{Q}^{-}x^{-})(0^{-})\\\
(\mathcal{Q}^{+}x^{+})(b)\\\ (\mathcal{Q}^{-}x^{-})(a)\\\
(\mathcal{Q}^{+}x^{+})(0^{+})\end{bmatrix}$
$\displaystyle=\begin{bmatrix}U_{1}(\mathcal{Q}^{-}x^{-})(0^{-})+U_{2}(\mathcal{Q}^{+}x^{+})(0^{+})\\\
\tilde{W}_{B}\begin{bmatrix}(\mathcal{Q}^{+}x^{+})(b)\\\
(\mathcal{Q}^{-}x^{-})(a)\end{bmatrix}\end{bmatrix}.$

In particular, the boundary condition
$\tilde{W}_{B,c}\begin{bmatrix}(\mathcal{H}_{0}w)(b)\\\
(\mathcal{H}_{0}w)(0)\end{bmatrix}=0$ is equivalent to the conditions (18),
(19), $f_{I}=re_{I}$, and
$\tilde{W}_{B}\begin{bmatrix}(\mathcal{Q}^{+}x^{+})(b)\\\
(\mathcal{Q^{-}}x^{-})(a)\end{bmatrix}=0$ under the transformation $S$.

The transformation $\hat{R}_{\operatorname{ext}}$ associated to $\hat{P}_{1}$
is given as before by

(35)
$\hat{R}_{\operatorname{ext}}=\frac{1}{\sqrt{2}}\begin{bmatrix}\hat{P}_{1}&-\hat{P}_{1}\\\
I_{4}&I_{4}\end{bmatrix}\in\mathbb{R}^{8\times 8}.$

Again, we define $W_{B,c}:=\tilde{W}_{B.c}\hat{R}_{\operatorname{ext}}^{-1}$.
A direct computation shows that
$\hat{R}_{\operatorname{ext}}^{-1}=\hat{R}_{\operatorname{ext}}^{\top}$. Also,
in view of [15, Lemma 7.2.2], it holds that

(36)
$\displaystyle\hat{R}_{\operatorname{ext}}^{\top}\Sigma_{8}\hat{R}_{\operatorname{ext}}=\begin{bmatrix}\hat{P}_{1}&0\\\
0&-\hat{P}_{1}\end{bmatrix}.$

The following observation is required for the final result.

######  Lemma 3.3.

The following two statements are equivalent:

  1. (i)

$W_{B}\Sigma_{4}W_{B}^{\top}\geq 0$ and $r\geq 0$.

  2. (ii)

$W_{B,c}\Sigma_{8}W_{B,c}^{\top}\geq 0$.

######  Proof 3.4.

Using (36), we obtain that

$W_{B,c}\Sigma_{8}W_{B,c}^{\top}=\tilde{W}_{B,c}\hat{R}_{\operatorname{ext}}^{-1}\Sigma_{8}(\hat{R}_{\operatorname{ext}}^{-1})^{\top}\tilde{W}^{\top}_{B,c}=\tilde{W}_{B,c}\hat{R}_{\operatorname{ext}}^{\top}\Sigma_{8}\hat{R}_{\operatorname{ext}}\tilde{W}^{\top}_{B,c}\\\
=\tilde{W}_{B,c}\operatorname{diag}(P_{1},P_{1},-P_{1},-P_{1})\tilde{W}^{\top}_{B,c}\operatorname{diag}(U_{1}P_{1}U_{1}^{\top}-U_{2}P_{1}U_{2}^{\top},W_{B}\Sigma_{4}W_{B}^{\top}).$

The assertion thus depends on the definiteness of

$\displaystyle U_{1}P_{1}U_{1}^{\top}-U_{2}P_{1}U_{2}^{\top}$
$\displaystyle=\begin{bmatrix}-r&1\\\ 0&1\end{bmatrix}\begin{bmatrix}0&-1\\\
-1&0\end{bmatrix}\begin{bmatrix}-r&0\\\ 1&1\end{bmatrix}+\begin{bmatrix}r&0\\\
0&-1\end{bmatrix}\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\begin{bmatrix}r&0\\\
0&-1\end{bmatrix}$ $\displaystyle=\begin{bmatrix}2r&r\\\
r&0\end{bmatrix}+\begin{bmatrix}0&-r\\\
-r&0\end{bmatrix}=\begin{bmatrix}2r&0\\\ 0&0\end{bmatrix}.$

The latter matrix is clearly positive semidefinite if and only if $r\geq 0$.

We will now study the $C_{0}$-semigroups associated to the operators
$A_{\mathcal{Q}_{0}}$ introduced in (27):

(37) $\displaystyle\begin{split}D(A_{\mathcal{Q}_{0}})&=\left\\{x\in
X\,\big{|}\,\mathcal{Q}_{0}x\in
D(\mathcal{J}_{0}),\,f_{I,\mathcal{Q}_{0}x}=re_{I,\mathcal{Q}_{0}x},\,W_{B}\begin{bmatrix}f_{\partial,\mathcal{Q}_{0}x}\\\
e_{\partial,\mathcal{Q}_{0}x}\end{bmatrix}=0\right\\},\\\
A_{\mathcal{Q}_{0}}x&=\mathcal{J}_{0}(\mathcal{Q}_{0}x),\quad x\in
D(A_{\mathcal{Q}_{0}}),\end{split}$

and $A_{\mathcal{H}_{0}}$ defined next:

(38)
$\displaystyle\begin{split}D(A_{\mathcal{H}_{0}})&=\left\\{x\in\hat{X}\,\big{|}\,\mathcal{H}_{0}x\in
H^{1}([0,b],\mathbb{R}^{4}),\,W_{B.c}\begin{bmatrix}f_{\partial,\mathcal{H}_{0}x}\\\
e_{\partial,\mathcal{H}_{0}x}\end{bmatrix}=0\right\\},\\\
A_{\mathcal{H}_{0}}x&=P_{1}\frac{d}{d\zeta}(\mathcal{H}_{0}x),\quad x\in
D(A_{\mathcal{H}_{0}}).\end{split}$

We note the following simple observation, which follows directly from the
previous arguments in this section:

######  Lemma 3.5.

The operators $A_{\mathcal{Q}_{0}}$ and $A_{\mathcal{H}_{0}}$ are similar via
the isometric isomorphism $S$, i.e.,
$SD(A_{\mathcal{H}_{0}})=D(A_{\mathcal{Q}_{0}})$ and for all $x\in
D(A_{\mathcal{H}_{0}})$ we have
$S^{-1}A_{\mathcal{Q}_{0}}Sx=A_{\mathcal{H}_{0}}x$.

With this in mind, we can show the following result.

######  Theorem 3.6.

Consider the port-Hamiltonian operator $A_{\mathcal{Q}_{0}}\colon
D(A_{\mathcal{Q}_{0}})\subset X\to X$ defined in (37) with a full rank matrix
$W_{B}\in\mathbb{R}^{2\times 4}$. Then the following assertions are
equivalent:

  1. 1.

$A_{\mathcal{Q}_{0}}$ is the infinitesimal generator of a contraction
semigroup on $X$ endowed with the scalar product
$\langle\cdot,\cdot\rangle_{\mathcal{Q}_{0}}=\frac{1}{2}\langle\cdot,\mathcal{Q}_{0}\cdot\rangle_{L^{2}}$.

  2. 2.

$A_{\mathcal{Q}_{0}}$ is dissipative.

  3. 3.

$r\geq 0$ and with $\Sigma_{4}$ defined in (14) we have

(39) $W_{B}\Sigma_{4}W_{B}^{\top}\geq 0.$

######  Proof 3.7.

The assumptions imply that $\hat{P}_{1}$ is Hermitian and invertible.
Furthermore, we have that $\mathcal{H}_{0}\in
L^{\infty}([0,b],\mathbb{R}^{4\times 4})$ is pointwise a.e. symmetric and
positive definite, bounded and bounded away from $0$. Also, the matrix
$W_{B,c}$ has full rank $4$, as $W_{B}$ has full rank. Thus,
$A_{\mathcal{H}_{0}}$ meets the assumption of Theorem 7.2.4 in [15] and
therefore for $A_{\mathcal{H}_{0}}$ we have the equivalence of the statements:
(i) $A_{\mathcal{H}_{0}}$ generates a contraction semigroup
$(T_{\mathcal{H}_{0}}(t))_{t\geq 0}$, (ii) $A_{\mathcal{H}_{0}}$ is
dissipative, and (iii) $W_{B,c}\Sigma_{8}W_{B,c}^{\top}\geq 0$. By Lemma 3.5,
the operators $A_{\mathcal{Q}_{0}}$ and $A_{\mathcal{H}_{0}}$ are similar
through an isometric isomorphism. By this similarity, the items 1. and 2. of
the assertion and (i), resp. (ii) are equivalent. Finally, by Lemma 3.3, item
3. of the claim is equivalent to (iii). This shows the assertion.

Theorem 3.6 characterizes the boundary and interface conditions for which the
port-Hamiltonian model of two systems of conservation laws coupled by a fixed
interface induces a contraction semigroup. The fact that for all $x_{0}\in
D(A_{\mathcal{Q}_{0}})$, the abstract Cauchy problem (28) has a unique
classical solution $x\colon[0,\infty)\to X$, depending continuously on the
initial data, follows immediately from [8, Corollary II.6.9].

Similarly to Theorem 3.6, we can characterize the boundary and interface
conditions for which the internally stored energy of the system is conserved.
Indeed, we see from the proof of Lemma 3.3 that
$W_{B}\Sigma_{4}W_{B}^{\top}=0$ together with $r=0$ is equivalent to
$W_{B,c}\Sigma_{8}W_{B,c}^{\top}=0$. We thus obtain the following result from
an application of [11, Theorem 4.1].

######  Corollary 3.8.

Consider the port-Hamiltonian operator $A_{\mathcal{Q}_{0}}\colon
D(A_{\mathcal{Q}_{0}})\subset X\to X$ defined in (27) with $r\in\mathbb{R}$
and with a full rank matrix $W_{B}\in\mathbb{R}^{2\times 4}$. Then
$A_{\mathcal{Q}_{0}}$ is the infinitesimal generator of an isometric semigroup
on the energy space $(X,\|\cdot\|_{\mathcal{Q}_{0}})$ if and only if $r=0$ and
$W_{B}$ satisfies $W_{B}\Sigma_{4}W_{B}^{\top}=0$.

##  4 Exponential Stability

In this section, we present conditions that guarantee exponential stability of
the abstract Cauchy problem (28) associated with the port-Hamiltonian operator
$A_{\mathcal{Q}_{0}}$ defined in (27). Recall that a $C_{0}$-semigroup
$(T(t))_{t\geq 0}$ on $X$ is called (uniformly) exponentially stable if there
exist constants $M\geq 1$ and $\alpha>0$ such that
$\|T(t)\|_{\mathcal{L}(X)}\leq Me^{-\alpha t},t\geq 0$, see, e.g., [8,
Definition 3.11]. Throughout this section, we impose the following
assumptions:

  1. (A1)

The matrix operators $\mathcal{Q}^{\pm}$ defining the coercive operator
$\mathcal{Q}_{0}\in\mathcal{L}(X)$ satisfy
$\mathcal{Q}^{\pm}\in\mathcal{C}^{1}([a,b],\mathbb{R}^{2\times 2})$.

  2. (A2)

$r\geq 0$, $\operatorname{rank}(W_{B})=2$, and $W_{B}$ satisfies (39). In
other words, the port-Hamiltonian operator $A_{\mathcal{Q}_{0}}$ generates a
contraction semigroup (see Theorem 3.6).

Exponential stability of boundary port-Hamiltonian systems is already
discussed in [15, Chapter 9] and as we have shown in Lemma 3.5 that our system
is similar via an isometric isomorphism to a system of the type discussed in
that reference, of course, the results there are immediately applicable.
However, the respective conditions translate into estimates on the boundary
values of $w$, that is in terms of the original system in conditions on the
boundary variables as well as the interface variables. In the context of the
systems discussed here, this may be undesirable and we will show that
sufficient conditions can be formulated exclusively in terms of evaluations at
the boundary points at $a$ and $b$.

We begin by showing an essential property that helps to verify the exponential
stability of the port-Hamiltonian system (28). The following lemma is an
extension of the energy estimates in [15, Lemma 9.1.2], which in our setting
removes the necessity to evaluate at the interface. In the case $r=0$ the
difference is that, due to the jump discontinuity at the interface position,
we need to introduce two auxiliary functions defined on two complementary
subintervals to exploit the system dynamics. However, the case $r>0$ yields an
improved energy estimate that only requires the evaluation at one of the
boundary points. The result is similar in appearance to [15, Lemma 9.1.2] for
boundary port-Hamiltonian systems without an interface and the proof technique
relies in a substantial manner on the tools developed there.

######  Lemma 4.1.

Let the assumptions (A1)-(A2) hold. Denote by $(T(t))_{t\geq 0}$ the
contraction semigroup generated by $A_{\mathcal{Q}_{0}}$.

  1. (i)

If $r=0$, then there exist constants $\tau>0$ and $C>0$ such that for every
initial value $x_{0}\in D(A_{\mathcal{Q}_{0}})$, the trajectory
$x(\cdot)=T(\cdot)x_{0}$ satisfies

(40) $\|x(\tau)\|_{\mathcal{Q}_{0}}^{2}\leq
C\int_{0}^{\tau}|\mathcal{Q}^{-}(a)x(a,t)|^{2}+|\mathcal{Q}^{+}(b)x(b,t)|^{2}\,dt.$

  2. (ii)

If $r>0$, then there exist constants $\tau>0$ and $C>0$ such that for every
initial value $x_{0}\in D(A_{\mathcal{Q}_{0}})$, the trajectory
$x(\cdot)=T(\cdot)x_{0}$ satisfies both

(41) $\begin{split}\|x(\tau)\|_{\mathcal{Q}_{0}}^{2}&\leq
C\int_{0}^{\tau}|\mathcal{Q}^{+}(b)x(b,t)|^{2}\,dt\quad\text{and}\\\
\|x(\tau)\|_{\mathcal{Q}_{0}}^{2}&\leq
C\int_{0}^{\tau}|\mathcal{Q}^{-}(a)x(a,t)|^{2}\,dt.\end{split}$

######  Proof 4.2.

The proof of $(i)$ can be found in [16, pp. 120-122] and we will concentrate
on the second assertion. We will prove the first inequality in (41); the proof
of the second being similar.

Since (41) is clear for $x_{0}=0$, assume that $0\neq x_{0}\in
D(A_{\mathcal{Q}_{0}})$ and let $x(\cdot,\cdot)$ be the corresponding
solution. Choose $\gamma>0$, $\tau>0$ such that $\tau>2\gamma(b-a)$ and define
the function $F\colon[a,b]\setminus\\{0\\}\to[0,\infty)$ by

(42)
$F(z):=\int_{\gamma(b-z)}^{\tau-\gamma(b-z)}x^{\top}(z,t)\mathcal{Q}_{0}(z)x(z,t)\,dt,\quad
z\in[a,b]\setminus\\{0\\}.$

In general, $F$ does not have a continuous extension to $[a,b]$, as
$F(0^{-})\neq F(0^{+})$.

Recall that on domains not containing the interface position, the operator
$\mathcal{J}_{0}$ defined in (12) acts as the differential operator
$P_{1}\frac{d}{dz}$, where $P_{1}$ is given in (2). Following the proof of
[15, Lemma 9.1.2], it can be shown that if $\gamma$ and $\tau$ are
sufficiently large, then there is a $\kappa>0$ such that

$\frac{d}{dz}F(z)\geq-\kappa F(z),\quad z\in[a,b]\setminus\\{0\\}.$

Applying Grönwall’s Lemma (see, e.g., [25, Section 2.4]), the following holds
on the respective subdomain:

(43) $\displaystyle\begin{split}F(z)&\leq F(b)e^{\kappa(b-z)},\quad
z\in(0,b],\\\ F(z)&\leq F(0^{-})e^{-\kappa z},\quad z\in[a,0).\end{split}$

We want to show that there is some $\tilde{C}>0$ such that for all
$z\in[a,b]\setminus\\{0\\}$ we have

(44) $\displaystyle F(z)\leq\tilde{C}F(b)e^{\kappa(b-a)}.$

Using the notation (cf. (26))
$f_{I}:=f_{I,\mathcal{Q}_{0}x}=\left(\mathcal{Q}^{-}x\right)_{2}(0^{-})$,
continuity of $\mathcal{Q}^{-}$, as well as the fact that $\mathcal{Q}^{-}(0)$
is symmetric, we obtain that

$\displaystyle F(0^{-})$ $\displaystyle=\int_{\gamma b}^{\tau-\gamma
b}x^{\top}(0^{-},t)\mathcal{Q}^{-}(0)x(0^{-},t)\,dt$
$\displaystyle=\int_{\gamma b}^{\tau-\gamma
b}\begin{bmatrix}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}\\\
f_{I}\end{bmatrix}^{\top}\left(\mathcal{Q}^{-}(0)\right)^{-1}\begin{bmatrix}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}\\\
f_{I}\end{bmatrix}\,dt.$

As $mI\leq\mathcal{Q}^{\pm}(z)\leq MI$ for all $z\in[a,b]$, it follows for all
$t\geq 0$ that

$\displaystyle\frac{1}{M}\int_{\gamma b}^{\tau-\gamma
b}\left|\begin{bmatrix}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}\\\
f_{I}\end{bmatrix}\right|^{2}\,dt\leq F(0^{-})\leq\frac{1}{m}\int_{\gamma
b}^{\tau-\gamma
b}\left|\begin{bmatrix}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}\\\
f_{I}\end{bmatrix}\right|^{2}\,dt.$

Analogously, using $f_{I}=\left(\mathcal{Q}^{+}x\right)_{2}(0^{+})$, see (26),

$\displaystyle\frac{1}{M}\int_{\gamma b}^{\tau-\gamma
b}\left|\begin{bmatrix}\left(\mathcal{Q}^{+}(0)x(0^{+},t)\right)_{1}\\\
f_{I}\end{bmatrix}\right|^{2}\,dt\leq F(0^{+})\leq\frac{1}{m}\int_{\gamma
b}^{\tau-\gamma
b}\left|\begin{bmatrix}\left(\mathcal{Q}^{+}(0)x(0^{+},t)\right)_{1}\\\
f_{I}\end{bmatrix}\right|^{2}\,dt.$

For $t\geq 0$, define
$\alpha(t):=\begin{bmatrix}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}&\left(\mathcal{Q}^{+}(0)x(0^{+},t)\right)_{1}\end{bmatrix}^{\top}$.
Then the interface condition (25) can be rewritten as

$f_{I}=re_{I}=r\Big{(}\left(\mathcal{Q}^{-}(0)x(0^{-},t)\right)_{1}-\left(\mathcal{Q}^{+}(0)x(0^{+},t)\right)_{1}\Big{)}=r\left(\alpha_{1}(t)-\alpha_{2}(t)\right).$

Now consider the quadratic norms $|\cdot|_{V_{1}}$ and $|\cdot|_{V_{2}}$
defined via the quadratic forms given by the symmetric, positive definite
matrices

$V_{1}:=\begin{bmatrix}1+r^{2}&-r^{2}\\\ -r^{2}&r^{2}\end{bmatrix}\,,\quad
V_{2}:=\begin{bmatrix}r^{2}&-r^{2}\\\ -r^{2}&1+r^{2}\end{bmatrix},$

and let $k_{1}\geq 1$ be such that always $|\alpha|^{2}_{V_{1}}\leq
k_{1}|\alpha|^{2}_{V_{2}}$. With this we have

$F(0^{-})\leq\frac{1}{m}\int_{\gamma b}^{\tau-\gamma
b}\alpha_{1}(t)^{2}+r^{2}(\alpha_{1}(t)-\alpha_{2}(t))^{2}\,dt=\frac{1}{m}\int_{\gamma
b}^{\tau-\gamma b}\left|\alpha(t)\right|^{2}_{V_{1}}\,dt\\\ \leq
k_{1}\frac{M}{m}\frac{1}{M}\int_{\gamma b}^{\tau-\gamma
b}\left|\alpha(t)\right|^{2}_{V_{2}}\,dt\leq k_{1}\frac{M}{m}F(0^{+}).$

Reversing the roles of $F(0^{-})$ and $F(0^{+})$ there also exists a
$k_{2}\geq 1$ such that $F(0^{+})\leq k_{2}(M/m)F(0^{-})$. In particular,
$F(0^{-})=0$ if and only if $F(0^{+})=0$.

We will now consider different instances concerning the choice of the initial
state.

Case 1: $x_{0}$ is chosen such that $F(b)=0$. Then (43) yields
$F(0^{-})=F(0^{+})=0$, and so $F\equiv 0$. Hence, the estimate (44) is
trivially satisfied.

Case 2: $x_{0}$ is chosen such that $F(b)\neq 0$ and $F(0^{-})=0$. Then (43)
yields that $F(z)=0$ holds for all $z\in[a,0)$, and for the choice
$\tilde{C}=1$ the estimate (44) is satisfied.

Case 3: $x_{0}$ is chosen such that $F(b)\neq 0$ and $F(0^{-})\neq 0$. Then we
also have $F(0^{+})\neq 0$. From our previous considerations we have

$\displaystyle F(0^{-})\leq k_{1}\frac{M}{m}F(0^{+}),$

and (43) yields

$\displaystyle F(0^{-})\leq k_{1}\frac{M}{m}F(0^{+})\leq
k_{1}\frac{M}{m}F(b)e^{\kappa b}.$

In summary, we have shown

$\displaystyle F(z)$ $\displaystyle\leq F(b)e^{\kappa(b-z)},\quad z\in(0,b],$
$\displaystyle F(z)$ $\displaystyle\leq
k_{1}\frac{M}{m}F(b)e^{\kappa(b-z)},\quad z\in[a,0).$

Choosing $\tilde{C}=k_{1}\frac{M}{m}$, we conclude that (44) holds for all
$z\in[a,b]\setminus\\{0\\}$.

Now, we are able to verify the first estimate in (41). Following the proof of
[15, Lemma 9.1.2] and using the estimate (44), it may be shown that

$\displaystyle
2(\tau-2\gamma(b-a))\|x(\tau)\|_{\mathcal{Q}_{0}}^{2}\leq\frac{(b-a)\tilde{C}}{m}e^{\kappa(b-a)}\int_{0}^{\tau}|\mathcal{Q}^{+}(b)x(b,t)|^{2}\,dt.$

Letting $C=\frac{(b-a)\tilde{C}e^{\kappa(b-a)}}{2m(\tau-2\gamma(b-a))}$, we
conclude that the first inequality in (41) holds. The second estimate in (41)
can be shown in a similar fashion by defining

$\displaystyle\tilde{F}(z)=\int_{\gamma(z-a)}^{\tau-\gamma(z-a)}x^{\top}(z,t)\mathcal{Q}_{0}(z)x(z,t)\,dt,\quad
z\in[a,b]\setminus\\{0\\}.$

######  Remark 4.3.

It can be shown that the uniform bound $k_{1}$ in the above proof has to be
chosen such that $k_{1}>\frac{r^{2}+1}{r^{2}}$.

Lemma 4.1 shows that there exists some time $\tau>0$ such that the energy of a
state can be bounded by the sum of the accumulated energies at both boundaries
during the time interval $[0,\tau]$, and in the case $r>0$, it can be bounded
by the accumulated energy at either boundary. This result allows us to prove
the following sufficient condition for exponential stability of the semigroup
generated by $A_{\mathcal{Q}_{0}}$.

######  Theorem 4.4.

Let the assumptions (A1)-(A2) hold. Assume there exists some $k>0$ such that

(45) $\langle
A_{\mathcal{Q}_{0}}x_{0},x_{0}\rangle_{\mathcal{Q}_{0}}\leq-k\left(|(\mathcal{Q}_{0}x_{0})(a)|^{2}+\|(\mathcal{Q}_{0}x_{0})(b)|^{2}\right)$

holds for all $x_{0}\in D(A_{\mathcal{Q}_{0}})$. Then $A_{\mathcal{Q}_{0}}$
generates an exponentially stable $C_{0}$-semigroup. In addition, if $r>0$,
then exponential stability of the $C_{0}$-semigroup generated by
$A_{\mathcal{Q}_{0}}$ follows if one of the following estimates holds for all
$x_{0}\in D(A_{\mathcal{Q}_{0}})$:

$\displaystyle\langle A_{\mathcal{Q}_{0}}x_{0},x_{0}\rangle_{\mathcal{Q}_{0}}$
$\displaystyle\leq-k|(\mathcal{Q}_{0}x_{0})(b)|^{2},$ $\displaystyle\langle
A_{\mathcal{Q}_{0}}x_{0},x_{0}\rangle_{\mathcal{Q}_{0}}$
$\displaystyle\leq-k|(\mathcal{Q}_{0}x_{0})(a)|^{2}.$

######  Proof 4.5.

This follows immediately from the energy estimates in Lemma 4.1 and from
Theorem 9.1.3 in [15].

Lastly, we want to present another sufficient condition for the exponential
stability of the port-Hamiltonian system (9). To prove the following theorem,
one follows the exposition of Lemma 9.1.4 in [15] and exploits the passivity
relation at the interface, as elaborated in [16, Theorem 5.19]. The proof
relies on the fact that the estimate (9.25) in [15] has been shown with
respect to the evaluation at both spatial boundaries.

######  Theorem 4.6.

Let the assumptions (A1)-(A2) hold. If the matrix $W_{B}$ satisfies
$W_{B}\Sigma_{4}W_{B}^{\top}>0$, then $A_{\mathcal{Q}_{0}}$ generates an
exponentially stable $C_{0}$-semigroup.

In this section, we have shown that if we impose certain boundary conditions,
then we can ensure that the contraction semigroup $(T(t))_{t\geq 0}$ generated
by the port-Hamiltonian operator $A_{\mathcal{Q}_{0}}$ is of type
$C_{0}(1,-\alpha)$ for some $\alpha>0$, i.e.,

$\|T(t)\|_{\mathcal{L}(X,\|\cdot\|_{\mathcal{Q}_{0}})}\leq e^{-\alpha t},\quad
t\geq 0.$

As opposed to the boundary conditions, the interface condition plays a minor
role in the proof of Theorem 4.6. It is only required that there is no power
inflow at the interface position.

##  5 Example: Coupled Acoustic Waveguides

In this section, we show how the results developed so far may be applied to
specific situations. We consider the example of two acoustic waveguides
coupled by an interface consisting in some membrane which acts as a local
acoustic impedance as it appears, for instance, in [21].

Let $a<0<b$. Consider two acoustic waveguides on the spatial intervals $[a,0)$
and $(0,b]$, respectively. On each subdomain, the model of the acoustic wave
is given by the coupled system of linear balance equations consisting in the
mass balance equation and in the momentum balance equation. One may choose as
energy variables the condensation $s$, which is the relative change of mass
density of the air, and the momentum $\pi$. Then the system of balance
equation may be expressed on each subdomain by

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}s^{\pm}(z,t)\\\
\pi^{\pm}(z,t)\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0&-\frac{\partial}{\partial z}\\\
-\frac{\partial}{\partial z}&0\end{bmatrix}\begin{bmatrix}p^{\pm}(z,t)\\\
v^{\pm}(z,t)\end{bmatrix},$

where $s^{\pm}(z,t)\in\mathbb{R}$ is the condensation and
$p^{\pm}(z,t)\in\mathbb{R}$ is the relative pressure at time $t\geq 0$, and
$\pi^{\pm}(z,t)\in\mathbb{R}$ denotes the momentum and
$v^{\pm}(z,t)\in\mathbb{R}$ denotes the velocity at the positions $z\in[a,0)$
and $z\in(0,b]$, respectively. The variables $e^{\pm}=(p^{\pm},v^{\pm})$ are
related to the state variables $x^{\pm}=(s^{\pm},\pi^{\pm})$ by the linear map
defined by the coercive multiplication operators $\mathcal{Q}^{\pm}$ as
follows:

$e^{\pm}(\cdot)=(\mathcal{Q}^{\pm}x^{\pm})(\cdot)=\begin{bmatrix}B^{\pm}(\cdot)&0\\\
0&\frac{1}{\rho_{0}^{\pm}(\cdot)}\end{bmatrix}\begin{bmatrix}s^{\pm}(\cdot)\\\
\pi^{\pm}(\cdot)\end{bmatrix},\quad x^{\pm}\in X^{\pm},$

where $\rho_{0}^{\pm}(z)>0$ and $B^{\pm}(z)>0$ are the mean mass density and
the adiabatic bulk modulus at $z$, respectively. We assume that
$\frac{1}{\rho_{0}^{-}},B^{-}\in L^{\infty}((a,l),\mathbb{R})$, and
$\frac{1}{\rho_{0}^{+}},B^{+}\in L^{\infty}((l,b),\mathbb{R})$.

The mechanical energy of the corresponding system is given by the following
Hamiltonian functionals:

$\displaystyle H^{-}(x^{-})$
$\displaystyle=\int_{a}^{0}\mathcal{H}^{-}(x^{-}(z))\,dz=\frac{1}{2}\langle
x^{-},\mathcal{Q}^{-}x^{-}\rangle_{L^{2}(a,0)},\quad x^{-}\in
X^{-}=L^{2}((a,0),\mathbb{R}^{2}),$ $\displaystyle H^{+}(x^{+})$
$\displaystyle=\int_{0}^{b}\mathcal{H}^{+}(x^{+}(z))\,dz=\frac{1}{2}\langle
x^{+},\mathcal{Q}^{+}x^{+}\rangle_{L^{2}(0,b)},\quad x^{+}\in
X^{+}=L^{2}((0,b),\mathbb{R}^{2}).$

The two acoustic waveguides are coupled at $z=0$ by an acoustic impedance
which we assume to be of resistive type with resistance $R_{I}>0$ .
Furthermore, we set the (relative) pressure at $z=a$ to zero, and we put
another resistive impedance with resistance $R_{b}>0$ at $z=b$.

We want to show by means of our results that the coupled system is well-posed
and exponentially stable.

Well-posedness. The model of the coupling is precisely given by the system (9)
(with $l=0$) defined on the state space $X=L^{2}([a,b],\mathbb{R}^{2})$, where
$[a,b]$ is the composed spatial domain. Defining the coercive matrix operator
$\mathcal{Q}_{0}=c_{0}^{-}\mathcal{Q}^{-}+c_{0}^{+}\mathcal{Q}^{+}\in\mathcal{L}(X)$,
we may endow $X$ with the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{Q}_{0}}$ defined in (10).

Due to the resistive acoustic impedance placed at the interface, we expect a
pressure drop, and therefore have a discontinuity of the pressure
$p=c_{0}^{-}p^{-}+c_{0}^{+}p^{+}$, whereas the velocity is continuous at the
interface $v=c_{0}^{-}v^{-}+c_{0}^{+}v^{+}$. Thus, the interface port
variables are given by

$\displaystyle f_{I,\mathcal{Q}_{0}x}$ $\displaystyle=v(0^{+})=v(0^{-}),$
$\displaystyle-e_{I,\mathcal{Q}_{0}x}$ $\displaystyle=p(0^{+})-p(0^{-}),$

and are related by the passivity relation

$f_{I}(t)=\frac{1}{R_{I}}e_{I}(t),\quad t\geq 0.$

Furthermore, we have the boundary conditions

$p(a,t)=0\quad\text{and}\quad p(b,t)=R_{b}v(b,t),\quad t\geq 0.$

Using the definition of the boundary port variables (16) (and neglecting the
time-dependence), we may write equivalently

$0=\begin{bmatrix}0&0&1&0\\\ -1&R_{b}&0&0\end{bmatrix}\begin{bmatrix}p(b)\\\
v(b)\\\ p(a)\\\
v(a)\end{bmatrix}=\tilde{W}_{B}\begin{bmatrix}(\mathcal{Q}_{0}x)(b)\\\
(\mathcal{Q}_{0}x)(a)\end{bmatrix}=W_{B}\begin{bmatrix}f_{\partial,\mathcal{Q}_{0}x}\\\
e_{\partial,\mathcal{Q}_{0}x}\end{bmatrix},$

with
$W_{B}=\tilde{W}_{B}R_{\operatorname{ext}}^{-1}=\frac{1}{\sqrt{2}}\begin{bmatrix}0&1&1&0\\\
-R_{b}&1&-1&R_{b}\end{bmatrix}\in\mathbb{R}^{2\times 4}$. Note that
$\operatorname{rank}(W_{B})=2$. Furthermore, one computes that

$W_{B}\Sigma_{4}W_{B}^{\top}=\begin{bmatrix}0&0\\\ 0&2R_{b}\end{bmatrix}\geq
0.$

Summing up, the port-Hamiltonian operator $A_{\mathcal{Q}_{0}}$ with domain
specified by $W_{B}$ defined above for the coupled acoustic waves, satisfies
the assumptions of Theorem 3.6. Hence, $A_{\mathcal{Q}_{0}}$ generates a
contraction semigroup on the energy space $(X,\|\cdot\|_{\mathcal{Q}_{0}})$.
In particular, this system is well-posed.

Exponential stability. Theorem 4.4 may be used to infer exponential stability.
By virtue of the power balance (22), the associated port-Hamiltonian operator
$A_{\mathcal{Q}_{0}}$ satisfies

$\displaystyle 2\langle
A_{\mathcal{Q}_{0}}x,x\rangle_{\mathcal{Q}_{0}}=\langle
e_{\partial,\mathcal{Q}_{0}x},f_{\partial,\mathcal{Q}_{0}x}\rangle-
e_{I,\mathcal{Q}_{0}x}f_{I,\mathcal{Q}_{0}x}\leq\langle
e_{\partial,\mathcal{Q}_{0}x},f_{\partial,\mathcal{Q}_{0}x}\rangle$

for all $x\in D(A_{\mathcal{Q}_{0}})$. Using the boundary conditions, simple
computations yield that

$\displaystyle\langle
e_{\partial,\mathcal{Q}_{0}x},f_{\partial,\mathcal{Q}_{0}x}\rangle=-R_{b}v(b)^{2}.$

Furthermore,

$\displaystyle|(\mathcal{Q}^{+}x)(b)|^{2}=p(b)^{2}+v(b)^{2}=(1+R_{b}^{2})v(b)^{2}.$

In summary, we have

$\displaystyle\langle
A_{\mathcal{Q}_{0}}x,x\rangle_{\mathcal{Q}_{0}}\leq\frac{1}{2}\langle
e_{\partial,\mathcal{Q}_{0}x},f_{\partial,\mathcal{Q}_{0}x}\rangle=\frac{-R_{b}}{2(1+R_{b}^{2})}|(\mathcal{Q}^{+}x)(b)|^{2}$

for all $x\in D(A_{\mathcal{Q}_{0}})$. Hence, by Theorem 4.4, this system is
exponentially stable.

##  6 Conclusions

Based on the port-Hamiltonian model of systems with moving interfaces
suggested in [6], we have given a mathematically rigorous definition of the
nonlinear Hamiltonian operator, depending on the color functions, appearing in
such models. For given color functions, we have derived the balance equation
of the Hamiltonian functional in terms of interface variables at the boundary
of the spatial domain and at the interface. For the case that the interface
position is fixed, a set of tools is provided that fully characterize
contractivity and give conditions for exponential stability of such systems.
We have worked out necessary and sufficient conditions that guarantee that the
associated port-Hamiltonian operator generates a contraction semigroup on the
respective energy space. These passivity conditions have been defined with
respect to the interface and boundary port variables and are easy to check in
practice. Moreover, we have presented criteria for exponential stability of
the $C_{0}$-semigroup generated by the associated port-Hamiltonian operator.
The study can be seen as a first step towards the development of the theory
for the case of moving interfaces. At the time of writing this, there are
still a number of notable obstacles towards this goal. It is tempting to let
$l(\cdot)$ as a function of $t$ denote the moving interface and to consider
the operators $A_{\mathcal{Q}_{l(t)}}$ as a family of operators that should
generate an evolution family. Standard tools in this direction fail
immediately, however, most notably because there is no subspace $Y$ dense in
$L^{2}([a,b],\mathbb{R}^{2})$ in the intersection of the domains
$D(A_{\mathcal{Q}_{l(t)}})$, compare, e.g., [23, Chapter 5]. Clearly, the
energy flow generated by the moving interface has to be taken into account
here as it has been done in [6] by defining an additional pair of port
variables. A route towards the generation of evolution families could be
provided by using these addtional port variables and using passivity
properties. A very special case of this problem will be discussed in [17].

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