concept of stability for this class of systems, as this is a key property that
has to be satisfied for proving well-posedness. This section relies
substantially on [23, Chapter 5].
As in the autonomous case, let us begin with the definition of an evolution
problem.

######  Definition 13.1 (Evolution Problem).

Let $0\leq t\leq\tau$, and let $A(t)\colon D(A(t))\subset X\to X$ be a linear
operator. The time-variant initial value problem

$\displaystyle\begin{split}\dot{x}(t)&=A(t)x(t),\hskip 14.22636pt0\leq
s<t\leq\tau,\\\ x(s)&=x_{0}\in X\end{split}$ (32)

is called an evolution problem. A function $x\colon[s,\tau]\to X$ is called a
classical solution of the evolution problem (32) if the following holds:

  1. (i)

$x\in\mathcal{C}([s,\tau],X)$.

  2. (ii)

$x_{|(s,\tau)}\in\mathcal{C}^{1}((s,\tau),X)$.

  3. (iii)

$x(t)\in D(A(t))$ for all $s<t\leq\tau$.

  4. (iv)

$x$ satisfies (32).

The proof of the following theorem is based on using Picard’s iteration method
and applying the Banach contraction principle.

######  Theorem 13.1 (Theorem 5.1.1 in [23]).

For $0\leq t\leq\tau$, let $A(t)\in\mathcal{L}(X)$. If the mapping

$\displaystyle[0,\tau]\to\mathcal{L}(X),\hskip 8.5359ptt\mapsto A(t)$

is continuous, then for every $x_{0}\in X$ the initial value problem (32) has
a unique classical solution.

######  Proof 13.2.

We make use of Picard’s iteration theorem. Since the mapping $t\mapsto A(t)$
is continuous, we may define

$\displaystyle\alpha=\max_{t\in[0,\tau]}\|A(t)\|_{\mathcal{L}(X)}.$

Furthermore, for $x_{0}\in X$ define the mapping

$\displaystyle S_{x_{0}}\colon\mathcal{C}([s,\tau],X)$
$\displaystyle\to\mathcal{C}([s,\tau],X),$ $\displaystyle u$
$\displaystyle\mapsto x_{0}+\int_{s}^{\cdot}A(\eta)u(\eta)\,d\eta,$

where $\mathcal{C}([s,\tau],X)$ is endowed with the norm

$\displaystyle\|u\|_{\infty}=\max_{t\in[s,\tau]}\|u(t)\|_{X},\hskip
14.22636ptu\in\mathcal{C}([s,\tau],X),$

making $(\mathcal{C}([s,\tau],X),\|\cdot\|_{\infty})$ a Banach space. For all
$u,v\in\mathcal{C}([s,\tau],X)$ and for all $s\leq t\leq\tau$ the following
estimate holds:

$\displaystyle\|S_{x_{0}}u(t)-S_{x_{0}}v(t)\|_{X}$
$\displaystyle=\left\|x_{0}+\int_{s}^{t}A(\eta)u(\eta)\,d\eta-
x_{0}-\int_{s}^{t}A(\eta)v(\eta)\,d\eta\right\|_{X}$
$\displaystyle=\left\|\int_{s}^{t}A(\eta)\left[u(\eta)-v(\eta)\right]\,d\eta\right\|_{X}$
$\displaystyle\leq\int_{s}^{t}\|A(\eta)\|_{\mathcal{L}(X)}\|u(\eta)-v(\eta)\|_{X}\,d\eta$
$\displaystyle\leq\alpha(t-s)\|u-v\|_{\infty}.$

Furthermore, we have

$\displaystyle\hskip 14.22636pt\|S_{x_{0}}^{2}u(t)-S_{x_{0}}^{2}v(t)\|_{X}$
$\displaystyle=\left\|S_{x_{0}}\left(x_{0}+\int_{s}^{\cdot}A(\eta_{1})u(\eta_{1})\,d\eta_{1}\right)(t)-S_{x_{0}}\left(x_{0}+\int_{s}^{\cdot}A(\eta_{1})v(\eta_{1})\,d\eta_{1}\right)(t)\right\|_{X}$
$\displaystyle=\left\|\int_{s}^{t}A(\eta_{2})\left(x_{0}+\int_{s}^{\eta_{2}}A(\eta_{1})u(\eta_{1})\,d\eta_{1}\right)\,d\eta_{2}-\int_{s}^{t}A(\eta_{2})\left(x_{0}+\int_{s}^{\eta_{2}}A(\eta_{1})v(\eta_{1})\,d\eta_{1}\right)\,d\eta_{2}\right\|_{X}$
$\displaystyle=\left\|\int_{s}^{t}A(\eta_{2})\left(\int_{s}^{\eta_{2}}A(\eta_{1})\left[u(\eta_{1})-v(\eta_{1})\right]d\eta_{1}\right)\,d\eta_{2}\right\|_{X}$
$\displaystyle\leq\alpha\int_{s}^{t}\int_{s}^{\eta_{2}}\|A(\eta_{1})\left[u(\eta_{1})-v(\eta_{1})\right]\|_{X}\,d\eta_{1}\,d\eta_{2}$
$\displaystyle\leq\alpha^{2}\|u-v\|_{\infty}\int_{s}^{t}\int_{s}^{\eta_{2}}1\,d\eta_{1}\,d\eta_{2}$
$\displaystyle=\frac{\alpha^{2}(t-s)^{2}}{2}\|u-v\|_{\infty}.$

By induction it follows that

$\displaystyle\|S_{x_{0}}^{n}u(t)-S_{x_{0}}^{n}v(t)\|_{X}\leq\frac{\alpha^{n}(\tau-s)^{n}}{n!}\|u-v\|_{\infty}$

for all $n\in\mathbb{N}$, $u,v\in\mathcal{C}([s,\tau],X)$ and $s\leq
t\leq\tau$. Consequently, we have

$\displaystyle\|S_{x_{0}}^{n}u-S_{x_{0}}^{n}v\|_{\infty}\leq\frac{\alpha^{n}(\tau-s)^{n}}{n!}\|u-v\|_{\infty}.$

Note that for $N\in\mathbb{N}$ large enough we get
$\frac{\alpha^{N}(\tau-s)^{N}}{N!}<1$, and so $S_{x_{0}}^{N}$ is a contraction
mapping. By the Banach contraction principle, $S_{x_{0}}^{N}$ has a unique
fixed point $x=x(\cdot;x_{0})\in\mathcal{C}([s,\tau],X)$. One can readily see
that

$\displaystyle
S_{x_{0}}^{N}(S_{x_{0}}(x))=S_{x_{0}}(S_{x_{0}}^{N}(x))=S_{x_{0}}(x).$

Thus, $S_{x_{0}}(x)$ is a fixed point of $S_{x_{0}}^{N}$. As $x$ is the unique
fixed point of $S_{x_{0}}^{N}$, we infer that $S_{x_{0}}(x)=x$ and, hence, $x$
is a fixed point of $S_{x_{0}}$ as well. Finally, the uniqueness of the fixed
point of $S_{x_{0}}^{N}$ together with the fact that every fixed point of
$S_{x_{0}}$ is a fixed point of $S_{x_{0}}^{N}$ yield that $x$ is the unique
fixed point of $S_{x_{0}}$ as well, and therefore satisfies

$\displaystyle x(t)=x_{0}+\int_{s}^{t}A(\eta)x(\eta)\,d\eta,\hskip
14.22636ptt\in[s,\tau].$ (33)

Now, since $x$ is continuous, the right-hand side of (33) is differentiable.
Thus, $x$ is differentiable with derivative

$\displaystyle\frac{d}{dt}x(t)=\frac{d}{dt}\left(x_{0}+\int_{s}^{t}A(\eta)x(\eta)\,d\eta\right)=A(t)x(t),\hskip
14.22636ptt\in[s,\tau].$

Furthermore, we have $x(s)=x_{0}$, and so $x$ is a solution of the evolution
problem (32). Note that every solution of (32) is in particular a solution of
(33). Due to the uniqueness of the fixed point $x$ of the mapping $S_{x_{0}}$,
we conclude that the solution of (32) is unique.

Recall that in Theorem 13.1 we required the operators $A(t)$, $0\leq
t\leq\tau$, to be bounded linear operators on $X$. However, this result gives
us some intuition regarding the behavior of the solutions of the evolution
problem (32). Using the representation (33) of such a solution, we may define
a solution operator $U(t,s)\colon X\to X$, $0\leq s\leq t\leq\tau$, of the
initial value problem (32) as follows:

$\displaystyle
U(t,s)x_{0}=x(t;x_{0})=x_{0}+\int_{s}^{t}A(\eta)x(\eta)\,d\eta,\hskip
14.22636pt0\leq s\leq t\leq\tau,\,x_{0}\in X.$ (34)

The family $(U(t,s))_{0\leq s\leq t\leq\tau}$ is a two-parameter family of
operators on $X$. The main properties of the family of solution operators are
given in the next theorem.

######  Theorem 13.2 (Theorem 5.1.2 in [23]).

For $0\leq t\leq\tau$, let $A(t)\in\mathcal{L}(X)$, and let the mapping
$t\mapsto A(t)$ be continuous on $[0,\tau]$. Then for every $0\leq s\leq
t\leq\tau$, the operator $U(t,s)$ defined in (34) is an element of
$\mathcal{L}(X)$ and it satisfies the following:

  1. (i)

$\|U(t,s)\|_{\mathcal{L}(X)}\leq\exp\left(\int_{s}^{t}\|A(\eta)\|_{\mathcal{L}(X)}\,d\eta\right)$.

  2. (ii)

$U(t,t)=I_{X}$ and for $0\leq s\leq r\leq t\leq\tau$ we have

$\displaystyle U(t,s)=U(t,r)U(r,s).$

  3. (iii)

The mapping $(t,s)\mapsto U(t,s)$ is continuous for $0\leq s\leq t\leq\tau$.

  4. (iv)

For $0\leq s\leq t\leq\tau$ we have

$\displaystyle\frac{\partial}{\partial t}U(t,s)=A(t)U(t,s).$

  5. (v)

For $0\leq s\leq t\leq\tau$ we have

$\displaystyle\frac{\partial}{\partial s}U(t,s)=-U(t,s)A(s).$

These properties resemble the ones of (uniformly continuous) one-parameter
semigroups studied in Section 10. Indeed, in the time-variant case the two-
parameter family of operators $(U(t,s))_{0\leq s\leq t\leq\tau}$ replaces the
one-paramter semigroup $(U(t))_{t\geq 0}$ of the autonomous case, which is the
$C_{0}$-semigroup generated by the (autonomous) operator $A$. This motivates
to introduce the notion of evolution systems, which play a substantial role in
the framework of evolution problems.

######  Definition 13.3 (Evolution System).

A two-parameter family of operators $U(t,s)\in\mathcal{L}(X)$, $0\leq s\leq
t\leq\tau$, is called an evolution system if the following holds:

  1. (i)

$U(s,s)=I_{X}$ and $U(t,s)=U(t,r)U(r,s)$ for $0\leq s\leq r\leq t\leq\tau$.

  2. (ii)

The mapping $(t,s)\mapsto U(t,s)$ is strongly continuous for $0\leq s\leq
t\leq\tau$.

Just as for one-paramter semigroups, the strong continuity of $(U(t,s))_{0\leq
s\leq t\leq\tau}$ is equivalent to the mapping $(t,s)\mapsto U(t,s)x$ being
continuous for all $x\in X$.
In the following, we want to present one important criterion on a given family
of (usually unbouded) operators $(A(t))_{t\in[0,\tau]}$ on $X$ that guarantees
the existence of a unique classical solution of the evolution problem (32) for
a dense subset of initial values in $X$. The existence of a unique solution
allows us to specify an evolution system associated with the family
$(A(t))_{t\in[0,\tau]}$. Note that the uniqueness of the solution yields
property (i), whereas the continuity of the solution at the initial data
yields property (ii) of the corresponding evolution system. The aforementioned
criterion is called stability, and is defined as follows.

######  Definition 13.4 (Stable Family of Generators).

A family $(A(t))_{t\in[0,\tau]}$ of infinitesimal generators of
$C_{0}$-semigroups on $X$ is called stable if there exist constants $M\geq 1$
and $\omega\in\mathbb{R}$, called stability constants, such that the following
holds:

  1. (i)

For $0\leq t\leq\tau$ we have $(\omega,\infty)\subset\rho(A(t))$.

  2. (ii)

For every finite sequence $0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{k}\leq\tau$,
$k\in\mathbb{N}$, it holds that

$\displaystyle\left\|\prod_{j=1}^{k}R(\lambda,A(t_{j}))\right\|_{\mathcal{L}(X)}\leq\frac{M}{(\lambda-\omega)^{k}},\hskip
14.22636pt\lambda>\omega.$ (35)

######  Remark 13.5.

We give some short remarks concerning Definition 13.4.

  1. (i)

In general, the operators $R(\lambda,A(t_{j}))$ do not commute. Hence, the
order of the terms given in (35) is important.

  2. (ii)

The stability of $(A(t))_{t\in[0,\tau]}$ is preserved when the norm
$\|\cdot\|_{X}$ is replaced by an equivalent one. However, the stability
constants depend on the particular norm $X$ is endowed with.

  3. (iii)

If for $0\leq t\leq\tau$, $A(t)$ generates a $C_{0}$-semigroup of type
$C_{0}(1,\omega)$ (see Theorem 10.2), then one can readily see that the family
$(A(t))_{t\in[0,\tau]}$ is stable with stability constants $M=1$ and
$\omega\in\mathbb{R}$. In particular, every family $(A(t))_{t\in[0,\tau]}$ of
generators of contraction semigroups is stable.

In general, it is not easy to determine whether a given family
$(A(t))_{t\in[0,\tau]}$ of infinitesimal generators is stable. In the next
theorem we give a characterization regarding the stability of such a family.
For the proof, one exploits Theorem 10.11.

######  Theorem 13.5 (Theorem 5.2.2 in [23]).

For $0\leq t\leq\tau$, let $A(t)$ be the generator of the $C_{0}$-semigroup
$(S_{t}(s))_{s\geq 0}$ on $X$. Then the family $(A(t))_{t\in[0,\tau]}$ is
stable if and only if there exist constants $M\geq 1$ and
$\omega\in\mathbb{R}$ such that $(\omega,\infty)\subset\rho(A(t))$ for all
$t\in[0,\tau]$, and either one of the following conditions is satisfied:

  1. (i)

For every finite sequence $0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{k}\leq\tau$,
$k\in\mathbb{N}$, and for $s_{j}\geq 0$, $j=1,\ldots,k$,

$\displaystyle\left\|\prod_{j=1}^{k}S_{t_{j}}(s_{j})\right\|_{\mathcal{L}(X)}\leq
M\exp\left(\omega\sum_{j=1}^{k}s_{j}\right).$

  2. (ii)

For every finite sequence $0\leq t_{1}\leq t_{2}\leq\ldots\leq t_{k}\leq\tau$,
$k\in\mathbb{N}$, and for $\lambda_{j}>\omega$, $j=1,\ldots,k$,

$\displaystyle\left\|\prod_{j=1}^{k}R(\lambda_{j},A(t_{j}))\right\|_{\mathcal{L}(X)}\leq
M\prod_{j=1}^{k}\frac{1}{\lambda_{j}-\omega}.$

We want to emphasize once again that the stability of a family
$(A(t))_{t\in[0,\tau]}$ is not the only condition that has to be satisfied in
order to guarantee the existence of a unique solution of the evolution problem
(32) associated with $(A(t))_{t\in[0,\tau]}$. For instance, there has to exist
a densely and continuously embedded subspace $Y$ of $X$ such that for all
$t\in[0,\tau]$, the space $Y$ is an invariant subspace of the semigroup
$(S_{t}(s))_{s\geq 0}$ generated by $A(t)$, and the family
$(\tilde{A}(t))_{t\in[0,\tau]}$ given by $\tilde{A}(t)=A(t)_{|D(A(t))\cap Y}$,
$t\in[0,\tau]$, has to be stable on $Y$. Together with another condition
related to $Y$, one may show that there exists a unique $Y$-valued solution of
(32). This holds equally for inhomogeneous evolution problems of the form

$\displaystyle\dot{x}(t)$ $\displaystyle=A(t)x(t)+f(t),\hskip 14.22636pt0\leq
s<t\leq\tau,$ $\displaystyle x(s)$ $\displaystyle=x_{0}\in X,$

provided that $f\in\mathcal{C}([s,\tau],X)$. However, we will not provide any
further details concerning this topic. Instead, we refer to [23, Chapter 5]
for more information.
In this chapter, we have defined formally adjoint operators of differential
operators defined on Sobolev spaces, and collected some facts about operator
semigroups and their infinitesimal generators. Furthermore, we have examined
the abstract Cauchy problem associated with operators generating a
$C_{0}$-semigroup. After a quick discussion on abstract control systems and
the concept of admissible control operators, we have studied the extension of
abstract Cauchy problems to evolution problems, where the associated operator
is replaced by a time-dependent family of infinitesimal generators, and
presented one important criterion for well-posedness of such systems. We have
now set the foundation for a comprehensive analysis of the systems considered
in the following chapters, namely infinite-dimensional port-Hamiltonian
systems in Chapter 4, and said systems of order 1 which are coupled by a
moving interface in Chapter 5.

##  Chapter 4 Infinite-Dimensional Port-Hamiltonian Systems

In this chapter, we are going to investigate infinite-dimensional port-
Hamiltonian systems. Throughout the chapter, we will restrict ourselves to
systems with a 1-dimensional spatial domain $[a,b]\subset\mathbb{R}$. We start
again with some motivational examples in Section 14, specifically the model of
a lossless transmission line in Subsection 14.1 and the model of a vibrating
string in Subsection 14.2. We will see that the evaluation of the co-energy
variables at the boundary of the spatial domain play a fundamental role.
Hence, in contrast to the finite-dimensional case, we need to introduce
further port variables in order to derive a port-Hamiltonian formulation of
infinite-dimensional systems, which will be called boundary port-Hamiltonian
systems. The examples can be found in [11, Chapter 4], [15], [27, Chapter 14],
or in the introductory chapter of [31]. Subsequently, in Section 15 we will
generalize the obtained systems considered in Section 14 and study boundary
port-Hamiltonian systems which are associated with formally skew-symmetric
differential operators. To that end, we will define suitable matrices and
operators in order to give a proper representation of the boundary port
variables constituting the infinite-dimensional Dirac structure. Following
this, we are able to define the bond space and a suitable power pairing to
finally define the underlying Dirac structure associated with such systems,
and to specify the dynamics with respect to these so-called Stokes-Dirac
structures. This section heavily relies on [15] and the Sections 2.1 and 2.2
in [31]. At the end of this chapter, we will shortly discuss the analysis of
boundary port-Hamiltonian systems.
We want to stress that we will not introduce an abstract mathematical object
or a particular solution concept for the systems considered in this chapter.
As a consequence, the notion of a system should be understood as a formal
concept rather than a general definition.
Unless stated otherwise, we consider the Hilbert space
$X=L^{2}([a,b],\mathbb{R}^{n})$ equipped with the inner product

$\displaystyle\langle
f,g\rangle_{L^{2}}=\int_{a}^{b}g^{\top}(z)f(z)\,dz,\hskip 14.22636ptf,g\in X.$
(36)

###  14 Motivating Examples

We want to look for a common structure in the models describing the dynamics
of various systems. We seek to rewrite the system models as an abstract Cauchy
problem of the form

$\displaystyle\dot{x}(t)$ $\displaystyle=Ax(t),\hskip 14.22636ptt>0,$
$\displaystyle x(0)$ $\displaystyle=x_{0}\in X,$

where $x\colon[0,\infty)\to X$ is the state variable, and $A\colon D(A)\subset
X\to X$ is a linear operator, as we have discussed in Section 11 in more
detail. For the analysis of such systems we use the framework of operator
semigroups. In this section, we want to present two examples from which we
will infer a general structure for a class of partial differential equations.
Beforehand, we want to clarify the notion of the variational derivative for a
special class of functionals based on [21, Section 4.1], and give a quick
example.

######  Definition 14.1 (Variational Derivative).

Let $H\colon X\to\mathbb{R}$ be a functional defined by

$\displaystyle H(x)=\int_{a}^{b}\mathcal{H}(x(z))\,dz,\hskip 14.22636ptx\in
X,$ (37)

with $\mathcal{H}\in\mathcal{C}^{\infty}({\mathbb{R}^{n},\mathbb{R}})$. The
variational derivative $\frac{\delta H}{\delta x}\colon X\to X$ of $H$ given
by

$\displaystyle\frac{\delta H}{\delta x}(x)=\left(\frac{\delta H}{\delta
x_{1}}(x),\ldots,\frac{\delta H}{\delta x_{n}}(x)\right),\hskip 14.22636ptx\in
X,$

is an operator uniquely determined by the requirement that it satisfies

$\displaystyle H(x+\epsilon\eta)=H(x)+\epsilon\int_{a}^{b}\left(\frac{\delta
H}{\delta x}(x)\right)^{\top}(z)\eta(z)\,dz+\mathcal{O}(\epsilon^{2})$

for any $x\in X,$ and for any variation $\epsilon\eta$, where
$\epsilon\in\mathbb{R}$, $\eta\in X$, and $\mathcal{O}(\epsilon^{2})$ is the
big O notation with respect to $\epsilon^{2}$.

######  Example 14.2 (cf. Section 1.6 in [31]).

Consider the functional

$\displaystyle
H(x)=\frac{1}{2}\int_{a}^{b}x^{\top}(z)\left(\mathcal{Q}x\right)(z)\,dz,\hskip
14.22636ptx\in X,$ (38)

where $\mathcal{Q}\in\mathcal{L}(X)$ is a coercive operator on $X$. Then we
have for all $x,\eta\in X$, $\epsilon\in\mathbb{R}$,

$\displaystyle H(x+\epsilon\eta)$
$\displaystyle=\frac{1}{2}\int_{a}^{b}(x(z)+\epsilon\eta(z))^{\top}\left(\mathcal{Q}(x+\epsilon\eta\right)(z)\,dz$
$\displaystyle=\frac{1}{2}\int_{a}^{b}x^{\top}(z)\left(\mathcal{Q}x\right)(z)+\epsilon\left(x^{\top}(z)\left(\mathcal{Q}\eta\right)(z)+\eta^{\top}(z)\left(\mathcal{Q}x\right)(z)\right)+\epsilon^{2}\eta^{\top}(z)\left(\mathcal{Q}\eta\right)(z)\,dz$
$\displaystyle=H(x)+\epsilon\int_{a}^{b}\left(\mathcal{Q}x\right)^{\top}(z)\eta(z)\,dz+\mathcal{O}(\epsilon^{2}).$

From this we deduce that $\frac{\delta H}{\delta x}(x)=\mathcal{Q}x$.
$\Diamond$

For functionals of the form (37) one can show that the variational derivative
is induced by the gradient of the integrand $\mathcal{H}$ in the following
way:

$\displaystyle\frac{\delta}{\delta x}H(x)(z)=\nabla\mathcal{H}(x(z)),\hskip
14.22636ptx\in X,\,z\in[a,b].$

####  14.1 The Lossless Transmission Line

Figure 4: Transmission Line [17, Figure 7.2]

Consider an ideal transmission line on the spatial interval
$[a,b]\subset\mathbb{R}$, as depicted in Figure 4. Telegrapher’s equations
constitute the following system:

$\displaystyle\begin{split}\frac{\partial Q}{\partial t}(z,t)&=-\frac{\partial
I}{\partial z}(z,t)=-\frac{\partial}{\partial z}\frac{\varphi(z,t)}{L(z)},\\\
\frac{\partial\varphi}{\partial t}(z,t)&=-\frac{\partial V}{\partial
z}(z,t)=-\frac{\partial}{\partial z}\frac{Q(z,t)}{C(z)}.\end{split}$ (39)

Here, $Q(z,t)\in\mathbb{R}$ and $\varphi(z,t)\in\mathbb{R}$ denote the charge
and the magnetic flux-linkage at position $z\in[a,b]$ and at time $t\geq 0$,
respectively, and $C(z)>0$ and $L(z)>0$ are the distributed capacity and the
distributed inductance of the transmission line at $z\in[a,b]$, repsectively.
Furthermore, the current and the voltage are given by $I=\frac{\varphi}{L}$
and $V=\frac{Q}{C}$, respectively. The electro-magnetic energy density
$\mathcal{H}\in\mathcal{C}^{\infty}(\mathbb{R}^{2},\mathbb{R})$ is given by

$\displaystyle\mathcal{H}(Q,\varphi)=\frac{1}{2}\left(\frac{Q^{2}}{C}+\frac{\varphi^{2}}{L}\right),\hskip
14.22636pt(Q,\varphi)\in\mathbb{R}^{2}.$

The system’s Hamiltonian $H\colon X\to\mathbb{R}$, describing the internally
stored energy, is the integral of the electro-magnetic energy density
$\mathcal{H}$: letting $x=(Q,\varphi)\in X$ we have

$\displaystyle
H(Q,\varphi)=\frac{1}{2}\int_{a}^{b}\frac{Q(z)^{2}}{C(z)}+\frac{\varphi(z)^{2}}{L(z)}\,dz=\frac{1}{2}\int_{a}^{b}\begin{bmatrix}Q(z)&\varphi(z)\end{bmatrix}\begin{bmatrix}\frac{1}{C(z)}&0\\\
0&\frac{1}{L(z)}\end{bmatrix}\begin{bmatrix}Q(z)\\\
\varphi(z)\end{bmatrix}\,dz.$ (40)

Assuming that both the capacity $C$ and the inductance $L$ are bounded on
$[a,b]$, we may define the coercive multiplication operator
$\mathcal{Q}\in\mathcal{L}(X)$ as follows:

$\displaystyle(\mathcal{Q}x)(z)=\mathcal{Q}(z)x(z)=\begin{bmatrix}\frac{1}{C(z)}&0\\\
0&\frac{1}{L(z)}\end{bmatrix}\begin{bmatrix}Q(z)\\\
\varphi(z)\end{bmatrix},\hskip 14.22636ptx\in X,\,z\in[a,b].$

Thus, the Hamiltonian $H$ from (40) can be written as $H(x)=\frac{1}{2}\langle
x,\mathcal{Q}x\rangle_{L^{2}}$ for $x\in X$.
As we have seeen in the finite-dimensional case (cf. Section 5), the co-energy
variables correspond to the gradient of the system’s Hamiltonian function. In
the infinite-dimensional case, this generalizes to the variational derivative
of the Hamiltonian functional. As shown in Example 14.2, if a functional is of
the form (38), then the variational derivative is given by

$\displaystyle\frac{\delta H}{\delta
x}(x)=\mathcal{Q}x=\begin{bmatrix}\frac{Q}{C}\\\
\frac{\varphi}{L}\end{bmatrix},\hskip 14.22636ptx=(Q,\varphi)\in X.$

Altogether, we may equivalently rewrite the system (39) as

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}Q\\\
\varphi\end{bmatrix}=\begin{bmatrix}0&-\frac{\partial}{\partial z}\\\
-\frac{\partial}{\partial z}&0\end{bmatrix}\begin{bmatrix}\frac{\delta
H}{\delta Q}(Q,\varphi)\\\ \frac{\delta
H}{\delta\varphi}(Q,\varphi)\end{bmatrix}.$ (41)

This system is called an _infinite-dimensional Hamiltonian system_ defined
with respect to the matrix differential operator $\mathcal{J}\colon
D(\mathcal{J})\subset X\to X$ given by

$\displaystyle D(\mathcal{J})$ $\displaystyle=H^{1}([a,b],\mathbb{R}^{2}),$
$\displaystyle\mathcal{J}x$ $\displaystyle=\begin{bmatrix}0&-\frac{d}{dz}\\\
-\frac{d}{dz}&0\end{bmatrix}x,\hskip 14.22636ptx\in D(\mathcal{J}),$

and with respect to the Hamiltonian $H$ defined in (40). Necessarily, the
matrix differential operator $\mathcal{J}$ has to satisfy two properties,
namely formal skew-symmetry and the Jacobi identity, as explained in [21,
Chapter 7]. We will only verify the formal skew-symmetry in the sense of
Definition 9.4. To this end, let $e=(e_{1},e_{2})$,
$\tilde{e}=(\tilde{e}_{1},\tilde{e}_{2})\in H_{0}^{1}([a,b],\mathbb{R}^{2})$
be some vectors vanishing at the boundary. Then integration by parts yields

$\displaystyle\begin{split}&\hskip
14.22636pt\langle\mathcal{J}e,\tilde{e}\rangle_{L^{2}}+\langle
e,\mathcal{J}\tilde{e}\rangle_{L^{2}}\\\
&=-\int_{a}^{b}\tilde{e}_{1}(z)\frac{de_{2}}{dz}(z)+\tilde{e}_{2}(z)\frac{de_{1}}{dz}(z)+e_{1}(z)\frac{d\tilde{e}_{2}}{dz}(z)+e_{2}(z)\frac{d\tilde{e}_{1}}{dz}(z)\,dz\\\
&=-\big{[}e_{1}(z)\tilde{e}_{2}(z)+e_{2}(z)\tilde{e}_{1}(z)\big{]}_{a}^{b}\\\
&=0.\end{split}$ (42)

Since $\mathcal{J}$ is not modulated by the state variable $x$, Corollary 7.5
in [21] yields that $\mathcal{J}$ is a Hamiltonian operator (in the sense of
[21, Definition 7.1]).
Analogously to the finite-dimensional case, the Hamiltonian structure provides
energy conservation: assume that $x\colon[0,\infty)\to X$ is a classical
solution of (41) which is vanishing at the boundary. Then by symmetry of the
operator $\mathcal{Q}$ and by formal skew-symmetry of $\mathcal{J}$ we obtain

$\displaystyle\begin{split}\frac{d}{dt}H(x(\cdot,t))&=\frac{1}{2}\frac{d}{dt}\langle
x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}\\\
&=\frac{1}{2}\left\langle\frac{\partial x}{\partial
t}(\cdot,t),\mathcal{Q}x(\cdot,t)\right\rangle_{L^{2}}+\frac{1}{2}\left\langle\mathcal{Q}x(\cdot,t),\frac{\partial
x}{\partial t}(\cdot,t)\right\rangle_{L^{2}}\\\
&=\frac{1}{2}\langle\mathcal{J}\mathcal{Q}x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}+\frac{1}{2}\langle\mathcal{Q}x(\cdot,t),\mathcal{J}\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}\\\
&=0.\end{split}$ (43)

We see that the Hamiltonian formulation imposes that there must not occur any
energy exchange at the boundaries of the transmission line with the rest of
the circuit, and therefore the total energy is conserved. However, this
assumption is quite restrictive, and for the purpose of control and for the
interconnection of two or more Hamiltonian systems it is essential to take the
interaction with its environment into account. This motivates to introduce a
new type of port variables, called _boundary port variables_ , and thus to
extend the (port-)Hamiltonian framework to boundary port-Hamiltonian systems
in the infinite-dimensional case.
Now, if the state variables do not vanish at the boundary of the spatial
domain, then from (42) we find that some boundary terms appear. Indeed, the
energy is not conserved in general, but satisfies the following energy-balance
equation, which again can be derived from (42) and (43):

$\displaystyle\begin{split}\frac{d}{dt}H(x(\cdot,t))&=\frac{1}{2}\langle\mathcal{J}\mathcal{Q}x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}+\frac{1}{2}\langle\mathcal{Q}x(\cdot,t),\mathcal{J}\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}\\\
&=-\frac{1}{2}\left[2\frac{Q(z,t)}{C(z)}\frac{\varphi(z,t)}{L(z)}\right]_{a}^{b}\\\
&=-\big{[}V(z,t)I(z,t)\big{]}_{a}^{b}.\end{split}$ (44)

Note that the product of voltage and current equals power. Hence, equation
(44) states that the change of internally stored energy solely occurs via the
spatial boundary. This suggests to introduce the evaluation not of the state
variables, but of the co-energy variables $V=\frac{Q}{C}$ and
$I=\frac{\varphi}{L}$ at the spatial boundary as new port variables in order
to involve the power flow through the boundary of the system (41). We will
postpone the definition of the boundary port variables to Section 15.

######  Remark 14.3 (Energy Space).

As $H(x)=\frac{1}{2}\langle x,\mathcal{Q}x\rangle_{L^{2}}$, $x\in X$, and as
$\mathcal{Q}\in\mathcal{L}(X)$ is a bounded and coercive operator, one usually
endows the state space $X$ with the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{Q}}$ defined by

$\displaystyle\langle x,y\rangle_{\mathcal{Q}}:=\frac{1}{2}\langle
x,\mathcal{Q}y\rangle_{L^{2}},\hskip 14.22636ptx,y\in X.$ (45)

Note that for all $z\in[a,b]$ we have $mI\leq\mathcal{Q}(z)\leq MI$ for some
$0<m\leq M$. Thus, the standard $L^{2}$-norm is equivalent to the norm defined
in (45), as we have for all $f\in L^{2}([a,b],\mathbb{R})$,

$\displaystyle\frac{m}{2}\|f\|_{L^{2}}^{2}\leq\|f\|_{\mathcal{Q}}^{2}\leq\frac{M}{2}\|f\|_{L^{2}}^{2}.$

The main reason for doing so is that in this way, the squared norm of the
state $x\in X$ equals the Hamiltonian, i.e., the internally stored energy, of
the system:

$\displaystyle\|x(\cdot,t)\|_{\mathcal{Q}}^{2}=\frac{1}{2}\langle
x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}=H(x(\cdot,t)),\hskip
14.22636ptt\geq 0.$ (46)

For this choice of the inner product, $X$ is known as the energy space. We
have already mentioned the usefulness of the relation (46) in Example 11.4.

####  14.2 The Vibrating String

Figure 5: Vibrating String

Consider a vibrating string of the length $L=b-a>0$, $a,b\in\mathbb{R}$, held
stationary at both ends and free to vibrate transversely. The scenario is
illustrated in Figure 5. The vibration of the string can be modeled by

$\displaystyle\frac{\partial w^{2}}{\partial t^{2}}(z,t)=\frac{1}{(\rho
A)(z)}\frac{\partial}{\partial z}\left(T(z)\frac{\partial w}{\partial
z}(z,t)\right),$ (47)

where $w(z,t)\in\mathbb{R}$ denotes the vertical displacement of the string at
position $z\in[a,b]$, and at time $t\geq 0$, and $T(z)>0$ and $(\rho A)(z)>0$,
$z\in[a,b]$ denote the Young’s modulus of the string and the linear mass
density, respectively, which are assumed to be bounded. We take the momentum
$p=(\rho A)\frac{\partial w}{\partial t}$ and the strain
$\epsilon=\frac{\partial w}{\partial z}$ to constitute the state variable
$x=(p,\epsilon)\in X$. The Hamiltonian $H\colon X\to\mathbb{R}$ of this system
is then given by the sum of the kinetic and the elastic energies in the
following way:

$\displaystyle H(p,\epsilon)=\frac{1}{2}\int_{a}^{b}\frac{1}{(\rho
A)(z)}p(z)^{2}+T(z)\epsilon(z)^{2}\,dz=\frac{1}{2}\int_{a}^{b}\begin{bmatrix}p(z)&\epsilon(z)\end{bmatrix}\begin{bmatrix}\frac{1}{(\rho
A)(z)}&0\\\ 0&T(z)\end{bmatrix}\begin{bmatrix}p(z)\\\
\epsilon(z)\end{bmatrix}\,dz.$ (48)

Again, we may define the coercive multiplication operator
$\mathcal{Q}\in\mathcal{L}(X)$ given by

$\displaystyle(\mathcal{Q}x)(z)=\mathcal{Q}(z)x(z)=\begin{bmatrix}\frac{1}{(\rho
A)(z)}&0\\\ 0&T(z)\end{bmatrix}\begin{bmatrix}p(z)\\\
\epsilon(z)\end{bmatrix},\hskip 14.22636ptx\in X,\,z\in[a,b].$

Additionally, as the Hamiltonian is once again of the form (38), the
variational derivative of the Hamiltonian $H$ equals the coercive matrix
operator $\mathcal{Q}$:

$\displaystyle\frac{\delta H}{\delta x}(x)=\begin{bmatrix}\frac{\delta
H}{\delta p}(p,\epsilon)\\\ \frac{\delta
H}{\delta\epsilon}(p,\epsilon)\end{bmatrix}=\begin{bmatrix}\frac{p}{\rho A}\\\
T\epsilon\end{bmatrix}=\begin{bmatrix}\frac{\partial w}{\partial t}\\\
T\frac{\partial w}{\partial z}\end{bmatrix},\hskip 14.22636ptx\in X.$

Altogether, we may equivalently rewrite equation (47) as an infinite-
dimensional, first-order Hamiltonian system of the form

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}p\\\
\epsilon\end{bmatrix}=\begin{bmatrix}0&\frac{\partial}{\partial z}\\\
\frac{\partial}{\partial z}&0\end{bmatrix}\begin{bmatrix}\frac{\delta
H}{\delta p}(p,\epsilon)\\\ \frac{\delta
H}{\delta\epsilon}(p,\epsilon)\end{bmatrix}$ (49)

with respect to the (Hamiltonian) matrix differential operator
$\mathcal{J}\colon D(\mathcal{J})\subset X\to X$ given by

$\displaystyle D(\mathcal{J})$ $\displaystyle=H^{1}([a,b],\mathbb{R}^{2}),$
$\displaystyle\mathcal{J}x$ $\displaystyle=\begin{bmatrix}0&\frac{d}{dz}\\\
\frac{d}{dz}&0\end{bmatrix}x,\hskip 14.22636ptx\in D(\mathcal{J}),$

and with respect to the Hamiltonian $H$ given in (48). The proof that
$\mathcal{J}$ is indeed a Hamiltonian operator (see again [21, Definition
7.1]) works exactly as in the previous example. For a classical solution
$x\colon[0,\infty)\to X$ of (49), the energy-balance equation of the system
(49) is given by

$\displaystyle\frac{d}{dt}H(x(\cdot,t))=\left[\frac{p(z,t)}{(\rho
A)(z)}T(z)\epsilon(z,t)\right]_{a}^{b},$ (50)

where the product of the co-energy variables, namely the velocity
$\frac{\delta H}{\delta p}(p,\epsilon)=\frac{p}{\rho A}$ and the stress
$\frac{\delta H}{\delta\epsilon}(p,\epsilon)=T\epsilon$, again equals power.
In our particular scenario, the ends of the string are both fixed, that is,
the velocity at the boundary is zero. In particular, $p(a,t)=p(b,t)=0$, $t\geq
0$, which is why in this case, the energy is conserved. However, if at least
one end of the string may move freely or some force may be applied to it, the
power flow through the boundary has to be encompassed, as depicted in (50).

######  Remark 14.4.

We note that the model (49) of this system has the same structure as the model
(41) of the lossless transmission line: the multiplication operator
$\mathcal{Q}\in\mathcal{L}(X)$ is coercive and represents the mechanical and
physical properties of the system. Furthermore, the Hamiltonian operators can
be written as $\mathcal{J}=P_{1}\frac{d}{dz}$, where the matrix
$P_{1}\in\mathbb{R}^{2\times 2}$ is symmetric (and invertible). Given the
vibrating string example, the formal skew-symmetry of the differential
operator $\mathcal{J}$ implies that the kinetic energy is transformed into
elastic energy and vice versa. Hence, the internally stored energy is
conserved in the absence of any power flow through the boundary. The formally
skew-symmetric differential operators can be regarded as the infinite-
dimensional counterpart of the skew-symmetric matrices in the finite-
dimensional case.

Just as in the finite-dimensional case there may also occur (internal) energy
dissipation in infinite-dimensional systems. We refer to [11, Subsection
4.1.3] and [31, Section 1.4] and the references mentioned there. In the
sequel, we will generalize the systems considered in Section 14 and only study
idealized systems where any dissipation phenomena are neglected.

###  15 Boundary Port-Hamiltonian Systems Associated with Skew-symmetric
Operators

We infer from the preceding examples that it is possible to describe a variety
of PDEs with the following structure:

$\displaystyle\begin{split}\frac{\partial x}{\partial
t}(z,t)&=\mathcal{J}(\mathcal{Q}(z)x(z,t)),\hskip 14.22636ptt>0,\\\
x(z,0)&=x_{0}(z)\end{split}$ (51)

on the state space $X=L^{2}([a,b],\mathbb{R}^{n})$, where
$\frac{\delta}{\delta x}H=\mathcal{Q}\in\mathcal{L}(X)$ is the variational
derivative of the Hamiltonian $H\colon X\to\mathbb{R}$ given by

$\displaystyle
H(x)=\int_{a}^{b}\mathcal{H}(x(z))\,dz=\frac{1}{2}\int_{a}^{b}x^{\top}(z)(\mathcal{Q}x)(z)\,dz,$
(52)

where $\mathcal{H}\in\mathcal{C}^{\infty}(\mathbb{R}^{n},\mathbb{R})$ is the
Hamiltonian density. Furthermore, the (multiplication) operator $\mathcal{Q}$
is assumed to be coercive. Let the differential operator $\mathcal{J}\colon
D(\mathcal{J})\subset X\to X$ be given by

$\displaystyle\begin{split}D(\mathcal{J})&=H^{N}\left([a,b],\mathbb{R}^{n}\right),\\\
\mathcal{J}x&=\sum_{i=0}^{N}P_{i}\frac{d^{i}x}{dz^{i}},\hskip 14.22636ptx\in
D(\mathcal{J}),\end{split}$ (53)

with $P_{i}\in\mathbb{R}^{n\times n}$, $i=0,\ldots,N$, where $P_{N}$ is
supposed to be invertible. Furthermore, we assume that

$\displaystyle P_{i}=(-1)^{i+1}P_{i}^{\top},\hskip 14.22636pti=0,\ldots,N,$
(54)

that is, the constituting matrices of $\mathcal{J}$ are skew-symmetric and
symmetric in an alternating fashion. We will see that condition (54) implies
that the differential operator $\mathcal{J}$ is formally skew-symmetric.
Recall from Remark 14.4 that the formal skew-symmetry corresponds to an
energy-conserving exchange of energy between different physical domains.
Finally, following Remark 14.3, we endow the state space $X$ with the inner
product $\langle\cdot,\cdot\rangle_{\mathcal{Q}}$ defined in (45), so that the
Hamiltonian $H$ can be rewritten as in (46).
In this section, we want to define the Dirac structure which is associated
with systems of the form (51). As the examples in Section 14 suggested, we
need to introduce boundary port variables to encompass the power flow through
the boundary as well. In Subsection 15.1 we will define this new type of port
variables by means of suitably chosen matrices and operators, and give a
representation of the evaluation of the effort variables at the spatial
boundary with respect to the boundary port variables. Subsection 15.2 deals
with the right choice of the bond space
$\mathcal{B}=\mathcal{F}\times\mathcal{E}$ and a suitable power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
constituting the bilinear form (the plus pairing)
$\ll\cdot,\cdot\gg\colon\mathcal{B}\times\mathcal{B}\to\mathbb{R}$ in order to
define the underlying Stokes-Dirac structure associated with infinite-
dimensional linear systems of the form (51), leading to the notion of boundary
port-Hamiltonian systems.

####  15.1 Boundary Port Variables

Let us, to begin with, show that the differential operator $\mathcal{J}$
defined by (53)-(54) is indeed formally skew-symmetric.

######  Theorem 15.0 (Theorem 2.1 in [31]).

Let $\mathcal{J}$ be the differential operator defined in (53)-(54). Then for
any two functions $e_{1},e_{2}\in D(\mathcal{J})$, we have

$\displaystyle\langle\mathcal{J}e_{1},e_{2}\rangle_{L^{2}}+\langle
e_{1},\mathcal{J}e_{2}\rangle_{L^{2}}=\left[\begin{bmatrix}e_{1}^{\top}(z),\ldots,\frac{d^{N-1}e_{1}^{\top}}{dz^{N-1}}(z)\end{bmatrix}\mathcal{P}\begin{bmatrix}e_{2}(z)\\\
\vdots\\\ \frac{d^{N-1}e_{2}}{dz^{N-1}}(z)\end{bmatrix}\right]_{a}^{b},$ (55)

with the symmetric matrix $\mathcal{P}\in\mathbb{R}^{nN\times nN}$ given by

$\displaystyle\mathcal{P}=\begin{bmatrix}P_{1}&P_{2}&P_{3}&\cdots&P_{N-1}&P_{N}\\\
-P_{2}&-P_{3}&-P_{4}&\cdots&-P_{N}&0\\\ P_{3}&P_{4}&\ddots&\ddots&0&0\\\
\vdots&\ddots&\ddots&\ddots&&\vdots\\\
(-1)^{N-1}P_{N}&0&\cdots&\cdots&&0\end{bmatrix}.$ (56)

In particular, $\mathcal{J}$ is formally skew-symmetric.

######  Proof 15.1.

The proof is rather tedious. The claim basically follows from iteratively
applying integration by parts. We refer to [14, Theorem 3.1].

In order to define a power-conserving interconnection structure in the
infinite-dimensional case, we need to incorporate the evaluation of the effort
variables at the spatial boundary into the definition of the respective power
pairing, which we will specify later on. To this end, we will write the right-
hand side of (55) subject to the boundary port variables. Note that these port
variables will depend on the constituting matrices of $\mathcal{J}$, which are
captured in the matrix $\mathcal{P}$ defined in (56). Since we suppose that
the matrix $P_{N}$ is invertible, it is easy to check that $\mathcal{P}$ is
invertible as well.
In order to define the boundary port variables, we need to introduce further
matrices. First, let the matrix
$\mathcal{P}_{\operatorname{ext}}\in\mathbb{R}^{2nN\times 2nN}$ associated
with the differential operator $\mathcal{J}$ be defined as

$\displaystyle\mathcal{P}_{\operatorname{ext}}=\begin{bmatrix}\mathcal{P}&0\\\
0&-\mathcal{P}\end{bmatrix},$

where $\mathcal{P}$ is the symmetric and invertible matrix from (56). Next, we
factorize $\mathcal{P}_{\operatorname{ext}}$ in such a way that it allows us
to define the port variables, and simultaneously guarantees that the bilinear
form - which has yet to be defined - becomes independent of the (matrix)
coefficients of the operator $\mathcal{J}$, as we will see shortly.

######  Lemma 15.2 (Lemma 2.4 in [31]).

The matrix $R_{\operatorname{ext}}\in\mathbb{R}^{2nN\times 2nN}$ defined as

$\displaystyle
R_{\operatorname{ext}}=\frac{1}{\sqrt{2}}\begin{bmatrix}\mathcal{P}&-\mathcal{P}\\\
I&I\end{bmatrix}$ (57)

is invertible and satisfies

$\displaystyle\mathcal{P}_{\operatorname{ext}}=R_{\operatorname{ext}}^{\top}\Sigma
R_{\operatorname{ext}},$

where

$\displaystyle\Sigma=\begin{bmatrix}0&I\\\
I&0\end{bmatrix}\in\mathbb{R}^{2nN\times 2nN}.$ (58)

Furthermore, all possible matrices $R\in\mathbb{R}^{2nN\times 2nN}$ satisfying
$\mathcal{P}_{\operatorname{ext}}=R^{\top}\Sigma R$ are of the form

$\displaystyle R=UR_{\operatorname{ext}},$

with $U\in\mathbb{R}^{2nN\times 2nN}$ satisfying $U^{\top}\Sigma U=\Sigma$.

The proof of Lemma 15.2 is fairly easy. For details we refer to the proof of
[15, Lemma 3.4].
The essential part of defining the Dirac structure associated with the
operator $\mathcal{J}$ is to introduce the boundary port variables, which we
are finally able to do.

######  Definition 15.3.

The boundary port variables associated with the differential operator
$\mathcal{J}$ given by (53)-(54) are the vectors
$f_{\partial},e_{\partial}\in\mathbb{R}^{nN}$ defined by

$\displaystyle\begin{bmatrix}f_{\partial,e}\\\
e_{\partial,e}\end{bmatrix}:=R_{\operatorname{ext}}\operatorname{tr}(e),\hskip
14.22636pte\in D(\mathcal{J}),$ (59)

where $\operatorname{tr}\colon H^{N}([a,b],\mathbb{R}^{n})\to\mathbb{R}^{2nN}$
is the boundary trace operator defined in (20).

We usually write $\begin{bmatrix}f_{\partial}\\\ e_{\partial}\end{bmatrix}$
instead of $\begin{bmatrix}f_{\partial,e}\\\ e_{\partial,e}\end{bmatrix}$
whenever the dependence of the boundary port variables on the variable $e\in
D(\mathcal{J})$ is obvious. We call $f_{\partial}\in\mathbb{R}^{nN}$ and
$e_{\partial}\in\mathbb{R}^{nN}$ the _boundary flow_ and the _boundary effort_
, respectively.
Using Definition 15.3 and Lemma 15.2, we can rewrite the right-hand side of
(55) in terms of the boundary port variables in the following way: for
$e_{1},e_{2}\in H^{N}\left([a,b],\mathbb{R}^{n}\right)$ we have

$\displaystyle\begin{bmatrix}f_{\partial,e_{1}}^{\top}&e_{\partial,e_{1}}^{\top}\end{bmatrix}\Sigma\begin{bmatrix}f_{\partial,e_{2}}\\\
e_{\partial,e_{2}}\end{bmatrix}$
$\displaystyle=\operatorname{tr}(e_{1})^{\top}R_{\operatorname{ext}}^{\top}\Sigma
R_{\operatorname{ext}}\operatorname{tr}(e_{2})$
$\displaystyle=\operatorname{tr}(e_{1})^{\top}\mathcal{P}_{\operatorname{ext}}\operatorname{tr}(e_{2})$
$\displaystyle=\operatorname{tr}(e_{1})^{\top}\begin{bmatrix}\mathcal{P}&0\\\
0&-\mathcal{P}\end{bmatrix}\operatorname{tr}(e_{2})$
$\displaystyle=\left[\begin{bmatrix}e_{1}^{\top}(z)&\cdots&\frac{d^{N-1}e_{1}^{\top}}{dz^{N-1}}(z)\end{bmatrix}\mathcal{P}\begin{bmatrix}e_{2}(z)\\\
\vdots\\\ \frac{d^{N-1}e_{2}}{dz^{N-1}}(z)\end{bmatrix}\right]_{a}^{b}.$

Thus, for all $e_{1},e_{2}\in D(\mathcal{J})$, equation (55) becomes

$\displaystyle\begin{split}\langle\mathcal{J}e_{1},e_{2}\rangle_{L^{2}}+\langle
e_{1},\mathcal{J}e_{2}\rangle_{L^{2}}&=\begin{bmatrix}f_{\partial,e_{1}}^{\top}&e_{\partial,e_{1}}^{\top}\end{bmatrix}\Sigma\begin{bmatrix}f_{\partial,e_{2}}\\\
e_{\partial,e_{2}}\end{bmatrix}\\\
&=f_{\partial,e_{1}}^{\top}e_{\partial,e_{2}}+e_{\partial,e_{1}}^{\top}f_{\partial,e_{2}}.\end{split}$
(60)

Now, we are able to define the Dirac structure on a suitable bond space
corresponding to the formally skew-symmetric operator $\mathcal{J}$.

####  15.2 Dirac Structure and Boundary Port-Hamiltonian Systems

We have already discussed that in the finite-dimensional case the space of
flows $\mathcal{F}$ and the space of efforts $\mathcal{E}$ correspond to a
finite-dimensional vector space and its dual space, with the power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
of an element of the flow space and of an element of the effort space being
equal to physical power (see Section 4). In the present case the flow space
$\mathcal{F}$ is now an infinite-dimensional Hilbert space, and the effort
space $\mathcal{E}$ is again to be defined as the dual space of the space of
flows, $\mathcal{E}=\mathcal{F}^{\prime}$. Throughout, we will identify
$\mathcal{F}$ with its dual $\mathcal{E}$.
In order to define a Dirac structure, we need to endow the bond space
$\mathcal{B}=\mathcal{F}\times\mathcal{E}$ not with the canonical inner
product $\langle\cdot,\cdot\rangle_{\mathcal{B}}$, but with the plus pairing
$\ll\cdot,\cdot\gg$ defined in (5), i.e.,

$\displaystyle\ll(f^{1},e^{1}),(f^{2},e^{2})\gg\hskip 2.84544pt=\langle
e^{1}\mid f^{2}\rangle+\langle e^{2}\mid f^{1}\rangle,\hskip
14.22636pt((f^{1},e^{1}),(f^{2},e^{2}))\in\mathcal{B}\times\mathcal{B}.$ (61)

This symmetric bilinear form depends on the choice of the power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$,
which we will, together with the flow and effort spaces, specify shortly.
We may now define the notion of a Dirac structure on an infinite-dimensional
Hilbert space.

######  Definition 15.4.

Let the flow space $\mathcal{F}$ be an infinite-dimensional Hilbert space, and
let the effort space $\mathcal{E}=\mathcal{F}^{\prime}$ be its dual space. A
linear subspace $\mathcal{D}\subset\mathcal{B}$ of the bond space
$\mathcal{B}=\mathcal{F}\times\mathcal{E}$ is called a Dirac structure if

$\displaystyle\mathcal{D}=\mathcal{D}^{\perp\\!\\!\\!\perp},$

where $\mathcal{D}^{\perp\\!\\!\\!\perp}$ denotes the orthogonal complement of
$\mathcal{D}$ with respect to the plus pairing $\ll\cdot,\cdot\gg$ defind in
(61).

Note that this is precisely the characterization of a Dirac structure in the
finite-dimensional case (cf. Proposition 2.4), and that this implies that the
Dirac structure $\mathcal{D}$ captures the power-conserving interconnection
structure of a system. For a general definition of an infinite-dimensional
port-Hamiltonian system with total energy (52), see [31, Definition 1.13].
In Subsection 15.1 we have defined boundary port variables which we need to
incorporate into the definition of the power pairing and thus the plus
pairing. By Definition 15.4, this is necessary for defining the Dirac
structure associated with $\mathcal{J}$ defined by (53)-(54). First, we need
to select the flow space and the effort space appropriately. Because the
bilinear form contains both state space elements and their evaluation at the
boundary, we set

$\displaystyle\mathcal{F}=\mathcal{E}=L^{2}\left([a,b],\mathbb{R}^{n}\right)\times\mathbb{R}^{nN},$
(62)

endowed with the canonical inner product:

$\displaystyle\langle(f^{1},f_{\partial}^{1}),(f^{2},f_{\partial}^{2})\rangle_{\mathcal{F}}=\langle
f^{1},f^{2}\rangle_{L^{2}}+\langle
f_{\partial}^{1},f_{\partial}^{2}\rangle_{2},\hskip
14.22636pt(f^{1},f_{\partial}^{1}),(f^{2},f_{\partial}^{2})\in\mathcal{F}.$

In contrast to the finite-dimensional case, the power pairing does not
necessarily coincide with the canonical duality pairing/inner product, but is
slightly modified. Here, we choose the power pairing

$\displaystyle\left\langle\begin{pmatrix}e\\\
e_{\partial}\end{pmatrix}\Bigg{|}\begin{pmatrix}f\\\
f_{\partial}\end{pmatrix}\right\rangle=\langle e,f\rangle_{L^{2}}-\langle
e_{\partial},f_{\partial}\rangle_{2},\hskip
14.22636pt\left(\begin{pmatrix}e\\\
e_{\partial}\end{pmatrix},\begin{pmatrix}f\\\
f_{\partial}\end{pmatrix}\right)\in\mathcal{E}\times\mathcal{F}.$ (63)

The reason behind that modification will be clear soon. Altogether, we endow
the bond space $\mathcal{B}=\mathcal{F}\times\mathcal{E}$ with the following
plus pairing:

$\displaystyle\ll(f^{1},f_{\partial}^{1},e^{1},e_{\partial}^{1}),(f^{2},f_{\partial}^{2},e^{2},e_{\partial}^{2})\gg\hskip
2.84544pt:=\langle e^{2},f^{1}\rangle_{L^{2}}-\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}+\langle
e^{1},f^{2}\rangle_{L^{2}}-\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2},$ (64)

where $(f^{i},f_{\partial}^{i},e^{i},e_{\partial}^{i})\in\mathcal{B}$,
$i=1,2$.
Now we have everything required to define the Dirac structure associated with
the formally skew-symmetric operator $\mathcal{J}$ given by (53)-(54).

######  Theorem 15.4 (Theorem 2.7 in [31]).

Consider the formally skew-symmetric operator $\mathcal{J}$ given by
(53)-(54). Let $\mathcal{B}=\mathcal{F}\times\mathcal{E}$ be the bond space
with respect to the flow space $\mathcal{F}$ and the effort space
$\mathcal{E}$ defined in (62). Then the subspace
$\mathcal{D}_{\mathcal{J}}\subset\mathcal{B}$ given by

$\displaystyle\mathcal{D}_{\mathcal{J}}:=\left\\{\left(\begin{pmatrix}f\\\
f_{\partial}\end{pmatrix},\begin{pmatrix}e\\\
e_{\partial}\end{pmatrix}\right)\in\mathcal{B}\hskip 2.84544pt\Bigg{|}\hskip
2.84544pte\in H^{N}\left([a,b],\mathbb{R}^{n}\right),\hskip
2.84544ptf=\mathcal{J}e,\hskip 2.84544pt\begin{bmatrix}f_{\partial,e}\\\
e_{\partial,e}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}(e)\right\\}$
(65)

is a Dirac structure with respect to the plus pairing
$\ll\cdot,\cdot\gg\colon\mathcal{B}\times\mathcal{B}\to\mathbb{R}$ defined in
(64), and is called the Stokes-Dirac structure.

For the proof one has to verify that
$\mathcal{D}_{\mathcal{J}}=\mathcal{D}_{\mathcal{J}}^{\perp\\!\\!\\!\perp}$
holds with respect to the plus pairing $\ll\cdot,\cdot\gg$, according to
Definition 15.4. We refer to the proof of Theorem 3.6 in [15].
The Stokes-Dirac structure $\mathcal{D}_{\mathcal{J}}$ is the underlying Dirac
structure of the class of systems of the form (51). Consequently, and
analogously to the finite-dimensional case, the dynamics of these systems can
be geometrically specified with respect to $\mathcal{D}_{\mathcal{J}}$, which
we are going to do next.
Beforehand, we explain the reason behind the specific choice of the power
pairing (63): assume that
$(f^{i},f_{\partial}^{i},e^{i},e_{\partial}^{i})\in\mathcal{D}_{\mathcal{J}}$
for $i=1,2$. As $\mathcal{D}=\mathcal{D}^{\perp\\!\\!\\!\perp}$, we get

$\displaystyle 0$ $\displaystyle=\hskip
2.84544pt\ll(f^{1},f_{\partial}^{1},e^{1},e_{\partial}^{1}),(f^{2},f_{\partial}^{2},e^{2},e_{\partial}^{2})\gg$
$\displaystyle=\hskip
2.84544pt\ll(\mathcal{J}e^{1},f_{\partial}^{1},e^{1},e_{\partial}^{1}),(\mathcal{J}e^{2},f_{\partial}^{2},e^{2},e_{\partial}^{2})\gg$
$\displaystyle=\langle\mathcal{J}e^{1},e^{2}\rangle_{L^{2}}+\langle
e^{1},\mathcal{J}e^{2}\rangle_{L^{2}}-\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}-\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2}$
$\displaystyle=\langle\mathcal{J}e^{1},e^{2}\rangle_{L^{2}}+\langle
e^{1},\mathcal{J}e^{2}\rangle_{L^{2}}-\begin{bmatrix}f_{\partial,e^{1}}^{\top}&e_{\partial,e^{1}}^{\top}\end{bmatrix}\Sigma\begin{bmatrix}f_{\partial,e^{2}}\\\
e_{\partial,e^{2}}\end{bmatrix},$

with $\Sigma$ defined in (58). This is exactly equation (60), which, in
particular, has been exploited in the proof of Theorem 15.4. Now, comparing
the system description (51) and the constitutive relations (65) of
$\mathcal{D}_{\mathcal{J}}$, we infer:

  1. (i)

Define the temporal change of the state variables as the flow variables, i.e.,
$f=\frac{\partial x}{\partial t}$.

  2. (ii)

Define the variational derivative of $H$ as the effort variables,
$e=\frac{\delta H}{\delta x}(x)=\mathcal{Q}x$.

  3. (iii)

Encompass the two external boundary port variables
$f_{\partial,e},e_{\partial,e}$ defined with respect to the effort variables
$e$, see Definition 15.3.

Thus, we are able to specify the dynamics of the system (51) with respect to
the Stokes-Dirac structure $\mathcal{D}_{\mathcal{J}}$ in (65). By doing so,
the power flow through the boundary of the spatial domain is encompassed,
which is why we call such systems _boundary port-Hamiltonian systems_.

######  Corollary 15.5.

Consider the Hamiltonian system (51) with Hamiltonian (52), which is augmented
with the boundary port variables given in Definition 15.3. The dynamics of the
resulting boundary port-Hamiltonian system are geometrically specified by the
requirement that for all $t>0$ we have

$\displaystyle\left(\begin{pmatrix}\frac{\partial}{\partial t}x(\cdot,t)\\\
f_{\partial}(t)\end{pmatrix},\begin{pmatrix}\mathcal{Q}x(\cdot,t)\\\
e_{\partial}(t)\end{pmatrix}\right)\in\mathcal{D}_{\mathcal{J}},$

with $f_{\partial}(t)=f_{\partial,\mathcal{Q}x(\cdot,t)}$,
$e_{\partial}(t)=e_{\partial,\mathcal{Q}x(\cdot,t)}$, and
$\mathcal{D}_{\mathcal{J}}$ the Stokes-Dirac structure (65) defined with
respect to the plus pairing (64).

Recall that the energy space $X$ is endowed with the inner product
$\langle\cdot,\cdot\rangle_{\mathcal{Q}}$ defined in (45). Thus, for a
classical solution $x\colon[0,\infty)\to X$ of the system (51) with the
Hamiltonian $H\colon X\to\mathbb{R}$ given in (52) we obtain the following
balance equation with respect to the boundary port variables: applying (60),
we have for all $t\geq 0$,

$\displaystyle\begin{split}\frac{d}{dt}\|x(\cdot,t)\|_{\mathcal{Q}}^{2}=\frac{d}{dt}H(x(\cdot,t))&=\frac{1}{2}\frac{d}{dt}\langle
x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}\\\
&=\frac{1}{2}\langle\mathcal{J}\mathcal{Q}x(\cdot,t),\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}+\frac{1}{2}\langle\mathcal{Q}x(\cdot,t),\mathcal{J}\mathcal{Q}x(\cdot,t)\rangle_{L^{2}}\\\
&=\frac{1}{2}\begin{bmatrix}f_{\partial}^{\top}(t)&e_{\partial}^{\top}(t)\end{bmatrix}\Sigma\begin{bmatrix}f_{\partial}(t)\\\
e_{\partial}(t)\end{bmatrix}\\\ &=\langle
e_{\partial}(t),f_{\partial}(t)\rangle_{2}.\end{split}$ (66)

This equation expresses the fact that the change of internal power solely
happens at the boundary of the spatial domain, and therefore generalizes the
balance equations (44) and (50) we have derived in the introductory section of
this chapter.

######  Remark 15.6.

In Section 5 we pointed out that, in order to distinguish between the power
flowing into the energy-storing elements and the power flowing into the Dirac
structure, we need to put a minus sign in front of the rate of change of the
state $x$, see (9). In the infinite-dimensional case, the power flowing into
the energy-storing elements would correspond to the inner product
$\langle\frac{\delta H}{\delta x}(x),\frac{\partial x}{\partial
t}\rangle_{L^{2}}$, and the power flow into the Dirac structure would
correspond to the inner product $\langle e_{S},f_{S}\rangle_{L^{2}}$, leading
to the relations $f=f_{S}=-\frac{\partial x}{\partial t}$ and
$e=e_{S}=\frac{\delta H}{\delta x}(x)$. Necessarily, some minor changes have
to be made concerning the Dirac structure and the plus pairing. To be more
accurate, the plus pairing in (64) would need to be changed to

$\displaystyle\ll(f^{1},f_{\partial}^{1},e^{1},e_{\partial}^{1}),(f^{2},f_{\partial}^{2},e^{2},e_{\partial}^{2})\gg\hskip
2.84544pt:=\langle f^{1},e^{2}\rangle_{L^{2}}+\langle
f_{\partial}^{1},e_{\partial}^{2}\rangle_{2}+\langle
e^{1},f^{2}\rangle_{L^{2}}+\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2},$

that is, the plus pairing is induced by the canonical power pairing
$\langle\cdot\mid\cdot\rangle$ on the Hilbert space $\mathcal{F}=\mathcal{E}$,
and not by the one defined in (63). Moreover, the Stokes-Dirac structure given
in (65) would be of the form

$\displaystyle\mathcal{D}_{-\mathcal{J}}=\left\\{\left(\begin{pmatrix}f\\\
f_{\partial}\end{pmatrix},\begin{pmatrix}e\\\
e_{\partial}\end{pmatrix}\right)\in\mathcal{B}\hskip 2.84544pt\Bigg{|}\hskip
2.84544pte\in H^{N}\left([a,b],\mathbb{R}^{n}\right),\hskip
2.84544pt-f=\mathcal{J}e,\hskip 2.84544pt\begin{bmatrix}f_{\partial,e}\\\
e_{\partial,e}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}(e)\right\\}.$

With these simple adjustments one could continue the aforementioned power flow
convention from Chapter 2. The proof that $\mathcal{D}_{-\mathcal{J}}$ is
indeed a Dirac structure with respect to the modified plus pairing works
analogously. However, as both sign conventions are common in the literature
(cf. [15, 20] and [29, 30]), we will stick with the sign convention applied
throughout this chapter for the rest of the thesis.

This is a good starting point for the analysis of the system (51) with
Hamiltonian (52). To this end, one defines the (maximal) _port-Hamiltonian
operator_ $A_{\mathcal{Q}}\colon D(A_{\mathcal{Q}})\subset X\to X$ given by

$\displaystyle D(A_{\mathcal{Q}})$ $\displaystyle=\left\\{x\in
X\mid\mathcal{Q}x\in D(\mathcal{J})\right\\},$ $\displaystyle
A_{\mathcal{Q}}x$ $\displaystyle=\mathcal{J}(\mathcal{Q}x),\hskip
14.22636ptx\in D(A_{\mathcal{Q}}),$

where $\mathcal{J}$ is the formally skew-symmetric operator defined in
(53)-(54), and studies the abstract Cauchy problem

$\displaystyle\begin{split}\dot{x}(t)&=A_{\mathcal{Q}}x(t),\hskip
14.22636ptt>0,\\\ x(0)&=x_{0}\in X,\end{split}$ (67)

as discussed in the beginning of Chapter 4. As for every partial differential
equation, it is clear that one has to impose boundary conditions in order to
guarantee well-posedness of the system (67). Indeed, in [15, Section 4] it has
been shown that formulating the boundary conditions with respect to the
boundary port variables, one can characterize those boundary conditions for
which the associated port-Hamiltonian operator generates a contraction
semigroup.
What we may deduce from equation (66) is that the change of energy, and
therefore the change of the state, is equal to the power flow at the boundary.
Hence, it seems natural to let some input (or at least a part of it) act on
the boundary of the spatial domain, and to define boundary control systems
associated with boundary port-Hamiltonian systems, see [17, Chapter 11]. Some
control related problems of such systems have been successfully studied in the
PhD theses [4] and [31].
In this chapter, we have worked out the main differences of port-based
modeling of finite-dimensional systems to infinite-dimensional systems on a
1-dimensional spatial domain. We give a short overview of the major changes:

  1. (i)

The gradient of the Hamiltonian function is replaced by the variational
derivative $\frac{\delta H}{\delta x}$ of the Hamiltonian functional (the
gradient $\nabla\mathcal{H}$ of the Hamiltonian density).

  2. (ii)

The skew-symmetric matrix $J$ is replaced by the formally skew-symmetric
differential operator $\mathcal{J}$.

  3. (iii)

One has to incorporate the power flow at the boundary of the spatial domain
(by means of the boundary port variables $f_{\partial}$ and $e_{\partial}$)
into the definition of the (Stokes-) Dirac structure.

  4. (iv)

The power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
does, in general, not coincide with the canonical inner product on
$\mathcal{F}=\mathcal{E}$, but is slightly modified.

Moreover, in Theorem 15.4 we have defined the underlying power-conserving
interconnection structure, namely the Stokes-Dirac structure
$\mathcal{D}_{\mathcal{J}}$, of the system (51).
Now that we have gained a basic understanding concerning port-based modeling
of infinite-dimensional systems, we can finally move on to the main part of
this thesis. We want to define two first order Hamiltonian systems that are
coupled through a moving interface as a single boundary port-Hamiltonian
system.

At this stage it is significant to mention two points:

##  Chapter 5 Boundary Port-Hamiltonian Systems with a Moving Interface

The main chapter of this thesis is devoted to the formulation of two systems
of two conservation laws defined on two complementary intervals that are
coupled by some moving interface as a single boundary port-Hamiltonian system
on the composed domain. Given the great success of the study of boundary port-
Hamiltonian systems from Chapter 4, in [10] one wanted to investigate whether
the port-Hamiltonian formulation can be extended to two systems that are
coupled by a moving interface. As we have already mentioned in the
introductory chapter of this thesis, this approach has been motivated by the
fact that there are several cases where the system is heterogeneous in the
considered spatial domain, and thus may be subdivided in several phases. Due
to the distinct physical properties on the respective subdomains, there might
arise some discontinuities of some variables at the interface, e.g., some
pressure discontinuities of two gases separated by a moving piston, or
different heat fluxes at the interface of a liquid-solid transition. These
discontinuities are modeled by a set of interface relations, and result in a
change of the position of the interface. According to [10], the port-
Hamiltonian formulation of this scenario yields an abstract control system,
where the input is defined as the velocity of the moving interface position.
_Our utmost objective is to rewrite the systems defined in[10] as evolution
problems and to analyze them by means of the framework of strongly continuous
semigroups_, see Chapter 3. It should be mentioned beforehand that we will not
be able to cover all problems regarding this fairly new system description, as
this is beyond the scope of this thesis. Furthermore, the concept of a system
used in this chapter remains purely formal. However, we seek to provide some
useful results that may help to prove well-posedness of this class of systems,
which one may call boundary port-Hamiltonian interface control systems. This
chapter is organized as follows. In Section 16 we recall the port-Hamiltonian
formulation of a system of two conservation laws with an arbitrary Hamiltonian
functional. Section 17 deals with the port-Hamiltonian formulation of two
systems that are coupled by some interface at a fixed position. We will
introduce interface port variables associated with the predefined interface
relations, and so-called color functions in order to express the state
variables on the total spatial domain. Furthermore, we will define the
interconnected system as a port-Hamiltonian system associated with a formally
skew-symmetric operator, and present the underyling Dirac structure of this
class of systems. Section 18 aims to repeat the preceding procedure in case of
a moving interface. We will see that this approach yields a system that has
the form of an abstract control system, with the input being the velocity of
the displacement of the interface position. However, there are several
difficulties concerning the proper formulation and analysis of this class of
systems. This is why we tackle some of these problems in the upfollowing
sections. In Section 19 we will greatly simplify the systems obtained in
Section 17, and show that, under certain boundary and interface conditions,
the associated port-Hamiltonian operator generates a contraction semigroup,
see Theorem 19.9. Afterwards, we present criteria for when this operator even
generates an exponentially stable (contraction) semigroup. Furthermore, we
specify its Hilbert space adjoint and the associated resolvent operator.
Lastly, in Subsection 19.6 we will come back to the case where the interface
is moving over time. Using the simplified system discussed throughout Section
19, we define a family of port-Hamiltonian operators that encompasses the
position of the moving interface, and prove the stability of this family of
operators.
We assume that some basic concepts about weak derivatives and distributions
are known to the reader. For a brief overview, we refer to the first two
sections in [25, Chapter 13]. For a comprehensive exposition we suggest [33].
We will denote the space of test functions by
$\mathcal{D}(a,b):=\mathcal{C}_{c}^{\infty}((a,b),\mathbb{R})$, and the vector
space of distributions by $\mathcal{D}^{\prime}(a,b)$. Throughout, we will
identify a function $f\in L^{2}([a,b],\mathbb{R})$ with its regular
distribution $\Lambda_{f}\colon\mathcal{D}(a,b)\to\mathbb{R}$. In particular,
we will sometimes regard the space $L^{2}([a,b],\mathbb{R})$ as a subspace of
$\mathcal{D}^{\prime}(a,b)$. The action of a regular distribution
$\Lambda_{f}$ on a test function $\varphi\in\mathcal{D}(a,b)$ is denoted by

$\displaystyle\langle\Lambda_{f},\varphi\rangle_{\mathcal{D}^{\prime}(a,b)}:=\langle
f,\varphi\rangle_{L^{2}}=\int_{a}^{b}f(z)\varphi(z)\,dz.$

###  16 Port-Hamiltonian System of two Conservation Laws

In this section, we briefly recall the port-Hamiltonian formulation of a
system of two conservation laws describing an energy-conserving exchange of
energy between different physical domains. We have already come across this
type of systems in Section 14, namely the first-order systems of the lossless
transmission line and the vibrating string. However, in this section, we will
slightly generalize these systems by considering arbitrary Hamiltonian
functionals.
Let $X=L^{2}([a,b],\mathbb{R}^{2})$ be endowed with the standard $L^{2}$-inner
product $\langle\cdot,\cdot\rangle_{L^{2}}$ defined in (36). As mentioned
before, let $H\colon X\to\mathbb{R}$ be some arbitrary Hamiltonian functional
of the form

$\displaystyle H(x)=\int_{a}^{b}\mathcal{H}(x(z))\,dz,\hskip 14.22636ptx\in
X,$ (68)

with Hamiltonian density
$\mathcal{H}\in\mathcal{C}^{\infty}(\mathbb{R}^{2},\mathbb{R})$ and
variational derivative $\frac{\delta}{\delta x}H\colon X\to X$, as described
in the beginning of Section 14. Now, consider a system of two conservation
laws

$\displaystyle\frac{\partial}{\partial t}x(z,t)+\frac{\partial}{\partial
z}\mathcal{N}(x(\cdot,t))(z)=0,\hskip 14.22636ptz\in[a,b],\,t>0,$ (69)

on the state space $X$, with the effort variables defined as

$\displaystyle\mathcal{N}(x)=\begin{bmatrix}\mathcal{N}_{1}(x)\\\
\mathcal{N}_{2}(x)\end{bmatrix}=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\begin{bmatrix}\delta_{x_{1}}H(x)\\\
\delta_{x_{2}}H(x)\end{bmatrix},\hskip 14.22636ptx\in X.$

Then the system of conservation laws (69) may be equivalently rewritten as the
Hamiltonian system

$\displaystyle\frac{\partial}{\partial
t}x(z,t)=\mathcal{J}\frac{\delta}{\delta x}H(x(\cdot,t))(z)$ (70)

generated by the Hamiltonian (68) and defined with respect to the differential
operator $\mathcal{J}\colon D(\mathcal{J})\subset X\to X$ given by

$\displaystyle\begin{split}D(\mathcal{J})&=H^{1}([a,b],\mathbb{R}^{2}),\\\
\mathcal{J}x&=\begin{bmatrix}0&-\frac{d}{dz}\\\
-\frac{d}{dz}&0\end{bmatrix}x,\hskip 14.22636ptx\in
D(\mathcal{J}).\end{split}$

Since the matrix

$\displaystyle P_{1}=\begin{bmatrix}0&-1\\\
-1&0\end{bmatrix}\in\mathbb{R}^{2\times 2}$ (71)

is symmetric, recall that, by Theorem 15.0, the matrix differential operator
$\mathcal{J}=P_{1}\frac{d}{dz}$ is formally skew-symmetric.
Next, we need to augment the Hamiltonian system (70) with boundary port
variables. In contrast to [10], we will stick with the definition from Section
15. This will have an impact on the definition of the Dirac structure and the
corresponding plus pairing $\langle\cdot\mid\cdot\rangle$, and hence slightly
deviates from the execution in [10]. Now, since $N=1$, the matrix
$\mathcal{P}$ defined in (56) reduces to $\mathcal{P}=P_{1}$, and the
invertible matrix $R_{\operatorname{ext}}$ defined in (57) is therefore given
by

$\displaystyle
R_{\operatorname{ext}}=\frac{1}{\sqrt{2}}\begin{bmatrix}P_{1}&-P_{1}\\\
I&I\end{bmatrix}\in\mathbb{R}^{4\times 4}.$ (72)

According to (59), the boundary flow variable
$f_{\partial}=f_{\partial,e}\in\mathbb{R}^{2}$ and boundary effort variable
$e_{\partial}=e_{\partial,e}\in\mathbb{R}^{2}$ are for all effort variables
$e\in D(\mathcal{J})$ given by

$\displaystyle\begin{bmatrix}f_{\partial}\\\
e_{\partial}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}(e)=\begin{bmatrix}\frac{1}{\sqrt{2}}(P_{1}e(b)-P_{1}e(a))\\\
\frac{1}{\sqrt{2}}(e(b)+e(a))\end{bmatrix}.$

The only difference is that the geometric specification of the dynamics of the
system (70) with respect to the Stokes-Dirac structure
$\mathcal{D}_{\mathcal{J}}$ defined in (65) becomes

$\displaystyle\left(\begin{pmatrix}\partial_{t}x_{1}(\cdot,t)\\\
\partial_{t}x_{2}(\cdot,t)\\\
f_{\partial}(t)\end{pmatrix},\begin{pmatrix}\delta_{x_{1}}H(x(\cdot,t))\\\
\delta_{x_{2}}H(x(\cdot,t))\\\
e_{\partial}(t)\end{pmatrix}\right)\in\mathcal{D}_{\mathcal{J}},\hskip
14.22636ptt>0,$

where $f_{\partial}(t)=f_{\partial,\delta_{x}H(x(\cdot,t))}$ and
$e_{\partial}(t)=e_{\partial,\delta_{x}H(x(\cdot,t))}$.
Due to the constitutive relation of the Dirac structure
$\mathcal{D}_{\mathcal{J}}$, the rate of energy change along a solution
$x\colon[0,\infty)\to X$ of (70) is still given for all $t\geq 0$ by the
following balance equation:

$\displaystyle\hskip 14.22636pt\frac{d}{dt}H(x(\cdot,t))$
$\displaystyle=\frac{d}{dt}\int_{a}^{b}\mathcal{H}(x(z,t))\,dz$
$\displaystyle=\int_{a}^{b}\frac{\partial}{\partial t}\mathcal{H}(x(z,t))\,dz$
$\displaystyle=\int_{a}^{b}\delta_{x_{1}}H(x(z,t))\frac{\partial}{\partial
t}x_{1}(z,t)+\delta_{x_{2}}H(x(z,t))\frac{\partial}{\partial t}x_{2}(z,t)\,dz$
$\displaystyle=\int_{a}^{b}-\delta_{x_{1}}H(x(z,t))\frac{\partial}{\partial
z}\big{(}\delta{x_{2}}H(x(z,t))\big{)}-\delta_{x_{2}}H(x(z,t))\frac{\partial}{\partial
z}\big{(}\delta_{x_{1}}H(x(z,t))\big{)}\,dz$
$\displaystyle=\int_{a}^{b}-\frac{\partial}{\partial
z}\left[\delta_{x_{1}}H(x(z,t))\delta_{x_{2}}H(x(z,t))\right]\,dz$
$\displaystyle=-\big{[}\delta_{x_{1}}H(x(z,t))\delta_{x_{2}}H(x(z,t))\big{]}_{a}^{b}$
$\displaystyle=\frac{1}{2}\big{[}\delta_{x}H(x(z,t))^{\top}P_{1}\delta_{x}H(x(z,t))\big{]}_{a}^{b}$
$\displaystyle=\langle e_{\partial}(t),f_{\partial}(t)\rangle_{2},$

where $\delta_{x}H(x(z,t))=\delta_{x}H(x(\cdot,t))(z)$. So, just as in
Subsection 15.2, this equality states that the system’s change of energy
equals the power supplied to the system through the boundary.
Now that we have clarified the port-Hamiltonian formulation of systems of two
conservation laws defined by an arbitrary Hamiltonian functional $H$ of the
form (68), we may deal with our main concern next.

###  17 Two Port-Hamiltonian Systems Coupled by an Interface

In the next two sections we examine two boundary port-Hamiltonian systems
defined on two complementary intervals which are coupled by a moving
interface. We seek to derive a formulation of the interconnected system as a
boundary port-Hamiltonian system. It is well-known that the (power-conserving)
interconnection of port-Hamiltonian systems again defines a port-Hamiltonian
system. This interconnection is realized by the composition of the underyling
Dirac structures 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 [27, Chapter 6] for the finite-
dimensional case as well as to [31, Chapter 7] and the references specified
therein for the infinite-dimensional case. In [10, Section 2.2] it was
depicted how port-Hamiltonian systems are coupled through their (fixed)
boundaries.
However, as we consider moving interfaces, we will have to deal with time-
varying spatial domains, and thus we have to take a different approach. In
order to keep track of the interface position, we are going to take the
characteristic functions of the time-varying spatial domains of each subsystem
to be part of the state variables. In preparation for the time-varying case
studied in Section 18, this section is devoted to the aforementioned
formulation as a boundary port-Hamiltonian system in case of a fixed interface
position, and is organized as follows: In Subsection 17.1 we introduce the
characteristic functions of the complementary subdomains of the interconnected
subsystems, called color-functions, as part of the state variables.
Furthermore, we define the Hamiltonian and its variational derivative on the
extended state space, and we specify interface port variables with respect to
some predefined interface relations. In Subsection 17.2 we state the balance
equations of the constituting state variables. The main difficulty relies on
the integration of the interface relations into the definition of these
balance equations. Finally, in Subsection 17.3 we are going to write the
system of balance equations as a (generalized) Hamiltonian system with respect
to some formally skew-symmetric operator on the extended state space.
Moreover, by taking the interface and boundary port variables into account, we
may define the underlying Dirac structure and state a balance equation of the
total system with respect to the boundary and the interface port variables.

####  17.1 Color Functions and Interface Port Variables

Our starting situation is as follows: let $H^{-}\colon
L^{2}((a,0),\mathbb{R}^{2})\to\mathbb{R}$ and $H^{+}\colon
L^{2}((0,b),\mathbb{R}^{2})\to\mathbb{R}$ be two Hamiltonian functionals given
by

$\displaystyle H^{-}(x^{-})$
$\displaystyle=\int_{a}^{0}\mathcal{H}^{-}(x^{-}(z))\,dz,\hskip
14.22636ptx^{-}\in L^{2}((a,0),\mathbb{R}^{2}),\hskip 8.5359pt\text{and}$ (73)
$\displaystyle H^{+}(x^{+})$
$\displaystyle=\int_{0}^{b}\mathcal{H}^{+}(x^{+}(z))\,dz,\hskip
14.22636ptx^{+}\in L^{2}((0,b),\mathbb{R}^{2}),$ (74)

with the distinct energy densities
$\mathcal{H}^{\pm}\in\mathcal{C}^{\infty}(\mathbb{R}^{2},\mathbb{R})$,
respectively, indicating that the physical properties of the system differ on
the respective domain. We assume that we may, in a way, extend these
Hamiltonians to Hamiltonian functionals $H^{\pm}\colon X\to\mathbb{R}$. Now,
consider the two systems of conservation laws

$\displaystyle\begin{split}\frac{\partial}{\partial
t}x^{-}+\frac{\partial}{\partial z}\mathcal{N}^{-}(x^{-})&=0,\hskip
14.22636ptz\in[a,0),\hskip 2.84544ptt>0,\\\ \frac{\partial}{\partial
t}x^{+}+\frac{\partial}{\partial z}\mathcal{N}^{+}(x^{+})&=0,\hskip
14.22636ptz\in(0,b],\hskip 2.84544ptt>0,\end{split}$ (75)

defined on the respective interval $[a,0)$ or $(0,b]$, where

$\displaystyle\mathcal{N}^{-}(x^{-})=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\frac{\delta}{\delta x^{-}}H^{-}(x^{-})\hskip
8.5359pt\text{and}\hskip 8.5359pt\mathcal{N}^{+}(x^{+})=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\frac{\delta}{\delta x^{+}}H^{+}(x^{+})$ (76)

are the _flux variables_ corresponding to the Hamiltonians $H^{-}$ and $H^{+}$
of the respective system defined in (73) and (74), respectively. 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 $z=0$. In order to do so, we need to
introduce suitable state variables defined on the composed domain, and we need
to define some _interface relations_ at $z=0$. These interface relations
simultaneously constitute the _interface port variables_
$(e_{I},f_{I})\in\mathbb{R}^{2}$ which we will need do define the underyling
Dirac structure of the coupled system. These are given by

$\displaystyle f_{I}$
$\displaystyle=\mathcal{N}_{1}^{+}(x^{+})(0^{+})=\mathcal{N}_{1}^{-}(x^{-})(0^{-}),$
(77) $\displaystyle-e_{I}$
$\displaystyle=\mathcal{N}_{2}^{+}(x^{+})(0^{+})-\mathcal{N}_{2}^{-}(x^{-})(0^{-}).$
(78)

Equation (77) is a continuity equation of the flux variable $\mathcal{N}_{1}$,
called the privileged variable, and (78) is a balance equation of the flux
variable $\mathcal{N}_{2}$ with external (source) term $e_{I}$.
In order to keep track of the interface position, we use the characteristic
functions of the domains of the two systems given in (75), called _color
functions_ :

$\displaystyle c_{0}(z,t)=\begin{cases}1,&z\in[a,0),\\\
0,&z\in[0,b],\end{cases}\hskip 8.5359pt\text{and}\hskip
8.5359pt\overline{c}_{0}(z,t)=\begin{cases}0,&z\in[a,0],\\\
1,&z\in(0,b].\end{cases}$ (79)

This allows us to write the state variables of the coupled system as the sum
of prolongations of the variables of each subsystem to the composed domain
$[a,b]$:

$\displaystyle\begin{split}x(z,t)&=x^{-}(z,t)+x^{+}(z,t)\\\
&=c_{0}(z,t)x(z,t)+\overline{c}_{0}(z,t)x(z,t).\end{split}$ (80)

Next, we want to define the Hamiltonian and, hence, the flux variables of the
total system. To this end, we introduce the extended state space
$\tilde{X}=L^{2}([a,b],\mathbb{R}^{4})$. The extended state variable is
denoted by

$\displaystyle\tilde{x}=(x,c,\overline{c})\in\tilde{X},$ (81)

where we want to stress that $\overline{c}$ is fully independent of $c$. While
in the fixed interface scenario, it might be unnecessary to consider the
extended state space $\tilde{X}$ (the color functions $c_{0}$ and
$\overline{c}_{0}$ are time-independent), this is essential in case of a
moving interface discussed in Section 18. So, in preparation for the moving
interface scenario, we will (for now) formulate the coupled system on the
extended state space $\tilde{X}$.
We start with the energy density
$\mathcal{H}\in\mathcal{C}^{\infty}(\mathbb{R}^{4},\mathbb{R})$, which is
given by

$\displaystyle\mathcal{H}(x,c,\overline{c})=c\mathcal{H}^{-}(x)+\overline{c}\mathcal{H}^{+}(x),\hskip
14.22636pt\tilde{x}\in\mathbb{R}^{4}.$ (82)

Next, define $H\colon\tilde{X}\to\mathbb{R}$ as

$\displaystyle
H(x,c,\overline{c})=\int_{a}^{b}\mathcal{H}(x(z),c(z),\overline{c}(z))\,dz,\hskip
14.22636pt\tilde{x}\in\tilde{X}.$ (83)

Since $H^{\pm}\colon X\to\mathbb{R}$ are functionals, one can readily see that
$H$ is a functional as well. The variational derivative
$\delta_{\tilde{x}}H\colon\tilde{X}\to\tilde{X}$ is again induced by the
gradient of the energy density $\mathcal{H}$ defined in (82), i.e.,

$\displaystyle\delta_{\tilde{x}}H(\tilde{x})=\begin{bmatrix}\delta_{x}H(x,c,\overline{c})\\\
\delta_{c}H(x,c,\overline{c})\\\
\delta_{\overline{c}}H(x,c,\overline{c})\end{bmatrix}=\begin{bmatrix}c\delta_{x}H^{-}(x)+\overline{c}\delta_{x}H^{+}(x)\\\
\mathcal{H}^{-}(x)\\\ \mathcal{H}^{+}(x)\end{bmatrix},\hskip
14.22636pt\tilde{x}\in\tilde{X}.$ (84)

If we now choose $c$ and $\overline{c}$ as the color functions defined in
(79), then we obtain for all $(x,c_{0},\overline{c}_{0})\in\tilde{X}$,

$\displaystyle H(x,c_{0},\overline{c}_{0})$
$\displaystyle=\int_{a}^{b}c_{0}(z)\mathcal{H}^{-}(x(z))+\overline{c}_{0}(z)\mathcal{H}^{+}(x(z))\,dz$
$\displaystyle=\int_{a}^{0}\mathcal{H}^{-}(x(z))\,dz+\int_{0}^{b}\mathcal{H}^{+}(x(z))\,dz$
$\displaystyle=H^{-}(x^{-})+H^{+}(x^{+}).$

So, $H(\cdot,c_{0},\overline{c}_{0})$ describes the internally stored energy,
which is given by the sum of the Hamiltonians of the subsystems defined in
(75), and therefore is the Hamiltonian of the coupled system.
The vector of flux variables of the coupled system becomes

$\displaystyle\mathcal{N}(x,c_{0},\overline{c}_{0})=c_{0}\mathcal{N}^{-}(x)+\overline{c}_{0}\mathcal{N}^{+}(x),\hskip
14.22636ptx\in X,$ (85)

which obviously satisfies

$\displaystyle
c_{0}\mathcal{N}(x,c_{0},\overline{c}_{0})=c_{0}\mathcal{N}^{-}(x)\hskip
8.5359pt\text{and}\hskip
8.5359pt\overline{c}_{0}\mathcal{N}(x,c_{0},\overline{c}_{0})=\overline{c}_{0}\mathcal{N}^{+}(x).$

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

$\displaystyle\mathcal{N}_{i}(x,c_{0},\overline{c}_{0})(z,t)=\mathcal{N}_{i}(x(\cdot,t),c_{0}(\cdot,t),\overline{c}_{0}(\cdot,t))(z).$

Note that the first row of (84) corresponds to the definition of the flux
variable (85) in the following way:

$\displaystyle\begin{split}\begin{bmatrix}0&1\\\
1&0\end{bmatrix}\delta_{x}H(x,c_{0},\overline{c}_{0})&=\begin{bmatrix}0&1\\\
1&0\end{bmatrix}c_{0}\delta_{x}H^{-}(x)+\overline{c}_{0}\delta_{x}H^{+}(x)\\\
&=c_{0}\mathcal{N}^{-}(x)+\overline{c}_{0}\mathcal{N}^{+}(x)\\\
&=\mathcal{N}(x,c_{0},\overline{c}_{0}).\end{split}$ (86)

In order to define the coupled system as a Hamiltonian-like system on the
extended state space $\tilde{X}$, we need to define the balance equation of
each variable constituting the extended state variable
$\tilde{x}=(x,c_{0},\overline{c}_{0})$, and to include the interface relations
(77)-(78). This is what we are going to do next.

####  17.2 Balance Equations of the State Variables

The coupling of the two systems of conservation laws (75) at the (interface)
position $z=0$ with the interface relations (77)-(78) led us to the
introduction of the prolongated flux variables (85) defined on the composed
domain $[a,b]$. Consequently, the flux variable
$\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})$ satisfies a continuity equation at
the interface, whereas the flux variable
$\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})$ satisfies a balance equation at
the interface. In the following, we want to state a conservation law for each
of the variables $x_{1}$, $x_{2}$, $c_{0}$, and $\overline{c}_{0}$ on the
whole domain $[a,b]$ having regard to the interface relations.

##### Conservation laws of $\mathbf{c_{0}}$ and $\mathbf{\overline{c}_{0}}$.

First of all, let us state the balance equations of the color functions
$c_{0}$ and $\overline{c}_{0}$ defined in (79) which are part of the extended
state variable $\tilde{x}\in\tilde{X}=L^{2}([a,b],\mathbb{R}^{4})$ defined in
(81) as well. It is clear that, for all $t\geq 0$, they satisfy the trivial
conservation laws

$\displaystyle\partial_{t}c_{0}=0\hskip 8.5359pt\text{and}\hskip
8.5359pt\partial_{t}\overline{c}_{0}=0,$ (87)

respectively.

##### Conservation law of $\mathbf{x_{1}}$.

Next, consider the conservation law of the state variable $x_{1}$ on $[a,b]$.
Assuming that the individual conservation laws (75) are satisfied, with the
aid of the flux variable (85), it may be formally written as

$\displaystyle\begin{split}\partial_{t}x_{1}&=-\partial_{z}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})\\\
&=-\partial_{z}(c_{0}\mathcal{N}_{1}^{-}(x)+\overline{c}_{0}\mathcal{N}_{1}^{+}(x))\\\
&=-\partial_{z}\left(c_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})+\overline{c}_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})\right)\\\
&=\underbrace{-\left[\partial_{z}(c_{0}\cdot)+\partial_{z}(\overline{c}_{0}\cdot)\right]}_{=:\mathbf{d}_{0}}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0}).\end{split}$
(88)

To begin with, for all $t>0$, equation (88) holds in
$\mathcal{D}^{\prime}(a,b)$: let $\varphi\in\mathcal{D}(a,b)$. Then, due to
(75) and the continuity condition (77), the following holds for all $t>0$:

$\displaystyle\hskip
14.22636pt\int_{a}^{b}\left[\mathbf{d}_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(z,t)\right]\varphi(z)\,dz$
$\displaystyle=\int_{a}^{b}-\left[\partial_{z}(c_{0}(z,t)\mathcal{N}_{1}(z,t))+\partial_{z}(\overline{c}_{0}(z,t)\mathcal{N}_{1}(z,t))\right]\varphi(z)\,dz$
$\displaystyle=\int_{a}^{b}-\left[\partial_{z}(c_{0}(z,t)\mathcal{N}_{1}(z,t))\right]\varphi(z)\,dz+\int_{a}^{b}-\left[\partial_{z}(\overline{c}_{0}(z,t)\mathcal{N}_{1}(z,t))\right]\varphi(z)\,dz$
(part. int.)
$\displaystyle=\int_{a}^{b}c_{0}(z,t)\mathcal{N}_{1}(z,t)\frac{d}{dz}\varphi(z)\,dz+\int_{a}^{b}\overline{c}_{0}(z,t)\mathcal{N}_{1}(z,t)\frac{d}{dz}\varphi(z)\,dz$
$\displaystyle=\int_{a}^{0}\mathcal{N}_{1}^{-}(z,t)\frac{d}{dz}\varphi(z)\,dz+\int_{0}^{b}\mathcal{N}_{1}^{+}(z,t)\frac{d}{dz}\varphi(z)\,dz$
(part. int.)
$\displaystyle=\mathcal{N}_{1}^{-}(0^{-},t)\varphi(0)+\int_{a}^{0}-\partial_{z}\mathcal{N}_{1}^{-}(z,t)\varphi(z)\,dz$
$\displaystyle\hskip
11.38092pt-\mathcal{N}_{1}^{+}(0^{+},t)\varphi(0)+\int_{0}^{b}-\partial_{z}\mathcal{N}_{1}^{+}(z,t)\varphi(z)\,dz$
(75)
$\displaystyle=\mathcal{N}_{1}^{-}(0^{-},t)\varphi(0)+\int_{a}^{0}\partial_{t}x_{1}^{-}(z,t)\varphi(z)\,dz\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle\hskip
11.38092pt-\mathcal{N}_{1}^{+}(0^{+},t)\varphi(0)+\int_{0}^{b}\partial_{t}x_{1}^{+}(z,t)\varphi(z)\,dz$
(77) $\displaystyle=\int_{a}^{b}\partial_{t}x_{1}(z,t)\varphi(z)\,dz.$

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

$\displaystyle-\int_{a}^{b}\frac{\partial}{\partial
z}c_{0}(z,t)\mathcal{N}_{1}(z,t)\varphi(z)\,dz=\int_{a}^{0}\frac{\partial}{\partial
z}(\mathcal{N}_{1}^{-}(z,t)\varphi(z))\,dz=\mathcal{N}_{1}^{-}(0^{-},t)\varphi(0),$

and

$\displaystyle-\int_{a}^{b}\frac{\partial}{\partial
z}\overline{c}_{0}(z,t)\mathcal{N}_{1}(z,t)\varphi(z)\,dz=\int_{0}^{b}\frac{\partial}{\partial
z}(\mathcal{N}_{1}^{+}(z,t)\varphi(z))\,dz=-\mathcal{N}_{1}^{+}(0^{+},t)\varphi(0).$

This implies that, due to the continuity condition (77), the evaluation of the
flux variables of the respective subdomain at the interface cancel each other
out, and equation (88) becomes

$\displaystyle\begin{split}\partial_{t}x_{1}(\cdot,t)&=\mathbf{d}_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(\cdot,t)\\\
&=-c_{0}(\cdot,t)\partial_{z}\mathcal{N}_{1}^{-}(x)(\cdot,t)-\partial_{z}c_{0}(\cdot,t)\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(\cdot,t)\\\
&\hskip
13.51523pt-\overline{c}_{0}(\cdot,t)\partial_{z}\mathcal{N}_{1}^{+}(x)(\cdot,t)-\partial_{z}\overline{c}_{0}(\cdot,t)\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(\cdot,t)\\\
&=-c_{0}(\cdot,t)\partial_{z}\mathcal{N}_{1}^{-}(x)(\cdot,t)-\overline{c}_{0}(\cdot,t)\partial_{z}\mathcal{N}_{1}^{+}(x)(\cdot,t).\end{split}$
(89)

As a consequence, equation (89) holds, in fact, in $L^{2}([a,b],\mathbb{R})$.
This motivates to define the operator $\mathbf{d}_{0}\colon
D(\mathbf{d}_{0})\subset L^{2}([a,b],\mathbb{R})\to L^{2}([a,b],\mathbb{R})$
in the following way:

$\displaystyle D(\mathbf{d}_{0})$ $\displaystyle=\left\\{x\in
L^{2}([a,b],\mathbb{R})\mid x_{|(a,0)}\in H^{1}((a,0),\mathbb{R}),\hskip
2.84544ptx_{|(0,b)}\in H^{1}((0,b),\mathbb{R}),\hskip
2.84544ptx\in\mathcal{C}([a,b],\mathbb{R})\right\\}$ $\displaystyle\hskip
3.41418pt=H^{1}([a,b],\mathbb{R}),$ (90) $\displaystyle\mathbf{d}_{0}x$
$\displaystyle=-\left[\frac{d}{dz}(c_{0}x)+\frac{d}{dz}(\overline{c}_{0}x)\right],\hskip
14.22636ptx\in D(\mathbf{d}_{0}).$

For a function $x=c_{0}x^{-}+\overline{c}_{0}x^{+}\in D(\mathbf{d}_{0})$ we
have in particular that

$\displaystyle\mathbf{d}_{0}x=-c_{0}\frac{d}{dz}x^{-}-\overline{c}_{0}\frac{d}{dz}x^{+}.$

This way, the continuity equation (77) of the flux variable
$\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})$ is incorporated into the domain of
the operator $\mathbf{d}_{0}$. This allows us to define classical solutions of
the conservation law of the state variable $x_{1}$ in the following way.

######  Definition 17.1.

Let $x=(x_{1},x_{2})\colon[0,\infty)\to X$. Let $Y=L^{2}([a,b],\mathbb{R})$.
We call $x_{1}\colon[0,\infty)\to Y$ a classical solution of the conservation
law

$\displaystyle\begin{split}\partial_{t}x_{1}(t)&=\mathbf{d}_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(\cdot,t),\hskip
14.22636ptt>0,\\\ x_{1}(0)&=x_{0,1}\in Y,\end{split}$ (91)

where $\mathbf{d}_{0}$ is the operator given by (90), and with the flux
variable

$\displaystyle\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})=c_{0}\mathcal{N}_{1}^{-}(x)+\overline{c}_{0}\mathcal{N}_{1}^{+}(x)$

of the systems of conservation laws (75) coupled at $z=0$, if:

  1. (i)

$x_{1}\in\mathcal{C}([0,\infty),Y)$.

  2. (ii)

$x_{1_{|(0,\infty)}}\in\mathcal{C}^{1}((0,\infty),Y)$.

  3. (iii)

$\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(\cdot,t)\in D(\mathbf{d}_{0})$ for
all $t>0$.

  4. (iv)

$x_{1}$ satisfies (91).

Note that the operator $\mathbf{d}_{0}$ acts as the differential operator
$-\frac{d}{dz}$ on each subdomain $[a^{\prime},b^{\prime}]\subset[a,b]$ not
containing the interface, i.e., with either $a\leq a^{\prime}<b^{\prime}<0$,
or with $0<a^{\prime}<b^{\prime}\leq b$. Indeed, according to (89), for all
$t>0$ it holds that

$\displaystyle\frac{d}{dt}\int_{a^{\prime}}^{b^{\prime}}x_{1}(z,t)\,dz=\int_{a^{\prime}}^{b^{\prime}}\partial_{t}x_{1}^{-}(z,t)\,dz=\int_{a^{\prime}}^{b^{\prime}}-\partial_{z}\mathcal{N}_{1}^{-}(x)(z,t)\,dz=\mathcal{N}_{1}^{-}(a^{\prime},t)-\mathcal{N}_{1}^{-}(b^{\prime},t)$

or

$\displaystyle\frac{d}{dt}\int_{a^{\prime}}^{b^{\prime}}x_{1}(z,t)\,dz=\int_{a^{\prime}}^{b^{\prime}}\partial_{t}x_{1}^{+}(z,t)\,dz=\int_{a^{\prime}}^{b^{\prime}}-\partial_{z}\mathcal{N}_{1}^{+}(x)(z,t)\,dz=\mathcal{N}_{1}^{+}(a^{\prime},t)-\mathcal{N}_{1}^{+}(b^{\prime},t),$

respectively. Now, if we consider an interval $[a^{\prime},b^{\prime}]$ with
$a\leq a^{\prime}<0<b^{\prime}\leq b$, then the continuity of the flux
variable $\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})$ at the interface implies
that

$\displaystyle\frac{d}{dt}\int_{a^{\prime}}^{b^{\prime}}x_{1}(z,t)\,dz$
$\displaystyle=\int_{a^{\prime}}^{b^{\prime}}\mathbf{d}_{0}\mathcal{N}_{1}(x,c_{0},\overline{c}_{0})(z,t)\,dz\,$
$\displaystyle=\int_{a^{\prime}}^{0}-\partial_{z}\mathcal{N}_{1}^{-}(x)(z,t)\,dz+\int_{0}^{b^{\prime}}-\partial_{z}\mathcal{N}_{1}^{+}(x)(z,t)\,dz$
$\displaystyle=\mathcal{N}_{1}^{-}(a^{\prime},t)\underbrace{-\mathcal{N}_{1}^{-}(0^{-},t)+\mathcal{N}_{1}^{+}(0^{+},t)}_{\stackrel{{\scriptstyle(\ref{Interface
Paper - Interface Variable fI})}}{{=0}}}-\mathcal{N}_{1}^{+}(b^{\prime},t)$
$\displaystyle=\mathcal{N}_{1}^{-}(a^{\prime},t)-\mathcal{N}_{1}^{+}(b^{\prime},t)$
$\displaystyle=\mathcal{N}_{1}(a^{\prime},t)-\mathcal{N}_{1}(b^{\prime},t).$

##### Conservation law of $\mathbf{x_{2}}$.

Lastly, let us consider the conservation law of the state variable $x_{2}$.
Recall that at the interface, the associated flux variable
$\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})=c_{0}\mathcal{N}_{2}^{-}(x)+\overline{c}_{0}\mathcal{N}_{2}^{+}(x)$
is supposed to satisfy the balance equation (78), where $e_{I}\in\mathbb{R}$
is defined as the difference of the flux variables $\mathcal{N}_{2}^{+}(x)$
and $\mathcal{N}_{2}^{-}(x)$ localized at the interface. In order to formulate
this conservation law, we have to calculate the formal adjoint
$\mathbf{d}_{0}^{\ast}\colon D(\mathbf{d}_{0}^{\ast})\subset
L^{2}([a,b],\mathbb{R})\to L^{2}([a,b],\mathbb{R})$ of the operator
$\mathbf{d}_{0}$ defined in (90). This will allow us to define a formally
skew-symmetric differential operator, as we will see over the course of this
chapter.
Recall from Section 9 that the domain of the adjoint operator is given by

$\displaystyle\left\\{y\in
X\mid\langle\mathbf{d}_{0}\cdot,y\rangle_{L^{2}}\colon
D(\mathbf{d}_{0})\to\mathbb{R},\hskip
2.84544ptx\mapsto\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}\text{ is
continuous}\right\\}.$

However, we need to find the formally adjoint operator, as the system is
considered to be open. Since, in contrast to the differential operators
considered in Section 15, this encompasses both the power flow at the boundary
and the interface, we seek an operator $\mathbf{d}_{0}^{\ast}$ such that

$\displaystyle\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}=\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}\hskip 5.69046pt+\text{boundary and
interface terms},\hskip 8.5359ptx\in D(\mathbf{d}_{0}),\,y\in
D(\mathbf{d}_{0}^{\ast}).$

Therefore, the general domain of the formal adjoint $\mathbf{d}_{0}^{\ast}$ is
given by

$\displaystyle D(\mathbf{d}_{0}^{\ast})=\left\\{y\in
X\mid\langle\mathbf{d}_{0}\cdot,y\rangle_{L^{2}}\colon\mathbf{H}_{0}^{1}(a,b)\to\mathbb{R},\hskip
2.84544ptx\mapsto\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}\text{ is
continuous}\right\\},$ (92)

where

$\displaystyle\mathbf{H}_{0}^{1}(a,b):=\left\\{f\in
H_{0}^{1}([a,b],\mathbb{R})\mid f(0)=0\right\\}\subset D(\mathbf{d}_{0}).$
(93)

Note that the set $\mathbf{H}_{0}^{1}(a,b)$ is dense in
$L^{2}([a,b],\mathbb{R})$. Hence, the formal adjoint of $\mathbf{d}_{0}$, as
we wish to define it, actually exists.
We first want to specify the domain $D(\mathbf{d}_{0}^{\ast})$ given in (92).
To that end, let $x\in\mathbf{H}_{0}^{1}(a,b)$ and let $y\in
L^{2}([a,b],\mathbb{R})$. Then we get

$\displaystyle\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}$
$\displaystyle=\int_{a}^{b}(\mathbf{d}_{0}x(z))y(z)\,dz$
$\displaystyle=\int_{a}^{b}-\left[\frac{d}{dz}(c_{0}x)(z)+\frac{d}{dz}(\overline{c}_{0}x)(z)\right]y(z)\,dz$
$\displaystyle=-\int_{a}^{0}\frac{d}{dz}(c_{0}x)(z)y(z)\,dz-\int_{0}^{b}\frac{d}{dz}(\overline{c}_{0}x)(z)y(z)\,dz.$

If we additionally assume that $y_{|(a,0)}\in H^{1}((a,0),\mathbb{R})$ and
$y_{|(0,b)}\in H^{1}((0,b),\mathbb{R})$, then integration by parts yields

$\displaystyle\begin{split}\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}&=-x(0^{-})y(0^{-})+x(a)y(a)+x(0^{+})y(0^{+})-x(b)y(b)\\\
&\hskip
14.22636pt+\int_{a}^{0}(xc_{0})(z)\frac{d}{dz}y(z)\,dz+\int_{0}^{b}(x\overline{c}_{0})(z)\frac{d}{dz}y(z)\,dz\\\
&=-\underbrace{\big{[}(c_{0}+\overline{c}_{0})(z)x(z)y(z)\big{]}_{a}^{b}}_{=0}+\underbrace{x(0)}_{=0}\left[y(0^{+})-y(0^{-})\right]\\\
&\hskip
14.22636pt+\int_{a}^{0}x(z)\frac{d}{dz}y(z)\,dz+\int_{0}^{b}x(z)\frac{d}{dz}y(z)\,dz.\end{split}$
(94)

By virtue of Hölder’s inequality, we have for all
$x\in\mathbf{H}_{0}^{1}(a,b)$ and for all such $y\in L^{2}([a,b],\mathbb{R})$,

$\displaystyle|\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}|$
$\displaystyle\leq\int_{0}^{a}\left|x(z)\frac{d}{dz}y(z)\right|\,dz+\int_{0}^{b}\left|x(z)\frac{d}{dz}y(z)\right|\,dz$
$\displaystyle\leq\|x\|_{L^{2}(a,0)}\left\|\frac{d}{dz}y\right\|_{L^{2}(a,0)}+\|x\|_{L^{2}(0,b)}\left\|\frac{d}{dz}y\right\|_{L^{2}(0,b)}$
$\displaystyle\leq\|x\|_{L^{2}(a,b)}\left(\left\|\frac{d}{dz}y\right\|_{L^{2}(a,0)}+\left\|\frac{d}{dz}y\right\|_{L^{2}(0,b)}\right)$
$\displaystyle\leq C_{y}\|x\|_{L^{2}(a,b)},$

whence

$\displaystyle\|\langle\mathbf{d}_{0}\cdot,y\rangle_{L^{2}}\|_{\text{op}\colon\mathbf{H}_{0}^{1}(a,b)\to\mathbb{R}}=\sup_{\begin{subarray}{c}x\in\mathbf{H}_{0}^{1}(a,b)\\\
\|x\|_{L^{2}}=1\end{subarray}}|\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}|\leq
C_{y}.$

Consequently, each element $y\in L^{2}([a,b],\mathbb{R})$ that is an
$H^{1}$-function on the respective subdomains is an element of the domain
$D(\mathbf{d}_{0}^{\ast})$. From the preceding calculations one can readily
see that we cannot relax this requirement. As a result, the (maximal) domain
of the operator $\mathbf{d}_{0}^{\ast}$ can be specified as follows:

$\displaystyle D(\mathbf{d}_{0}^{\ast})=\left\\{y\in
L^{2}([a,b],\mathbb{R})\mid y_{|(a,0)}\in H^{1}((a,0),\mathbb{R}),\hskip
2.84544pty_{|(0,b)}\in H^{1}((0,b),\mathbb{R})\hskip 2.84544pt\right\\}.$

It is left to determine the elements $\mathbf{d}_{0}^{\ast}y\in
L^{2}([a,b],\mathbb{R})$ for $y\in D(\mathbf{d}_{0}^{\ast})$. We have already
calculated in (94) that for all $x\in D(\mathbf{d}_{0})$ and for all $y\in
D(\mathbf{d}_{0}^{\ast})$ we get

$\displaystyle\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}$
$\displaystyle=-\big{[}(c_{0}+\overline{c}_{0})(z)x(z)y(z)\big{]}_{a}^{b}+x(0)\left[y(0^{+})-y(0^{-})\right]$
$\displaystyle\hskip
14.22636pt+\int_{a}^{0}x(z)\frac{d}{dz}y(z)\,dz+\int_{0}^{b}x(z)\frac{d}{dz}y(z)\,dz$
$\displaystyle\stackrel{{\scriptstyle!}}{{=}}-\big{[}(c_{0}+\overline{c}_{0})(z)x(z)y(z)\big{]}_{a}^{b}+x(0)\left[y(0^{+})-y(0^{-})\right]+\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}.$

Thus, our objective is to rewrite the second line as a single $L^{2}$-inner
product on the composed domain $[a,b]$ with respect to $x$ and
$\mathbf{d}_{0}^{\ast}y$. We claim that $\mathbf{d}_{0}^{\ast}y\in
L^{2}([a,b],\mathbb{R})$ is given as

$\displaystyle\mathbf{d}_{0}^{\ast}y=\left[\frac{d}{dz}(c_{0}y)+\frac{d}{dz}(\overline{c}_{0}y)\right]-\left[\frac{d}{dz}c_{0}+\frac{d}{dz}\overline{c}_{0}\right]y,\hskip
14.22636pty\in D(\mathbf{d}_{0}^{\ast}).$

Although at first glance, the co-domain of this operator seems to be
$\mathcal{D}^{\prime}(a,b)$, we will see that the distributional terms of this
expression, that is, the Dirac masses arising from taking the derivatives of
the characteristic functions $c_{0}$ and $\overline{c}_{0}$, cancel each other
out, so that the remaining terms lie again in
$L^{2}([a,b],\mathbb{R})\subset\mathcal{D}^{\prime}(a,b)$.
Initially, let $\varphi\in\mathcal{D}(a,b)$, which is a dense subset of
$D(\mathbf{d}_{0})=H^{1}([a,b],\mathbb{R})$. Then for all $y\in
D(\mathbf{d}_{0}^{\ast})$ it holds that (omitting the dependence on the
variable $z$)

$\displaystyle\hskip
14.22636pt\int_{a}^{b}\left(\left[\frac{d}{dz}(c_{0}y)+\frac{d}{dz}(\overline{c}_{0}y)\right]-\left[\frac{d}{dz}c_{0}+\frac{d}{dz}\overline{c}_{0}\right]y\right)\varphi\,dz$
$\displaystyle=\int_{a}^{b}\frac{d}{dz}(c_{0}y)\varphi\,dz+\int_{a}^{b}\frac{d}{dz}(\overline{c}_{0}y)\varphi\,dz-\int_{a}^{b}\frac{d}{dz}c_{0}\varphi
y\,dz-\int_{a}^{b}\frac{d}{dz}\overline{c}_{0}\varphi y\,dz$
$\displaystyle(\text{part. int.})\hskip 8.5359pt$
$\displaystyle=\big{[}(c_{0}+\overline{c}_{0})\varphi
y\big{]}_{a}^{b}-\int_{a}^{0}c_{0}y\frac{d}{dz}\varphi\,dz-\int_{0}^{b}\overline{c}_{0}y\frac{d}{dz}\varphi\,dz$
$\displaystyle\hskip 12.80365pt-\big{[}(c_{0}+\overline{c}_{0})\varphi
y\big{]}_{a}^{b}+\int_{a}^{0}\frac{d}{dz}(\varphi
y)\,dz+\int_{0}^{b}\frac{d}{dz}(\varphi y)\,dz$
$\displaystyle=-\int_{a}^{0}y\frac{d}{dz}\varphi\,dz-\int_{0}^{b}y\frac{d}{dz}\varphi\,dz$
$\displaystyle\hskip
14.22636pt+\int_{a}^{0}\varphi\frac{d}{dz}y+y\frac{d}{dz}\varphi\,dz+\int_{0}^{b}\varphi\frac{d}{dz}y+y\frac{d}{dz}\varphi\,dz$
$\displaystyle=\int_{a}^{0}\varphi\frac{d}{dz}y\,dz+\int_{0}^{b}\varphi\frac{d}{dz}y\,dz.$

Next, we want to show that for all $x\in D(\mathbf{d}_{0})$, $y\in
D(\mathbf{d}_{0}^{\ast})$ the inner product $\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}$ exists in a classical sense by
proving that the operation $x\mapsto\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}$ is closed. Let $y\in
D(\mathbf{d}_{0}^{\ast})$, pick any $x\in D(\mathbf{d}_{0})$, and let
$(\varphi_{n})_{n\in\mathbb{N}}\in\mathcal{D}(a,b)^{\mathbb{N}}$ be a sequence
with $\lim\limits_{n\to\infty}\varphi_{n}=x$ in $L^{2}$. Since the embedding
$L^{2}([a,b],\mathbb{R})\hookrightarrow L^{1}([a,b],\mathbb{R})$ is
continuous, this convergence holds in $L^{1}$ as well. Following [18, Chapter
6], we may apply the dominated convergence theorem. This yields

$\displaystyle\hskip
14.22636pt\lim_{n\to\infty}\int_{a}^{b}\left(\left[\frac{d}{dz}(c_{0}y)+\frac{d}{dz}(\overline{c}_{0}y)\right]-\left[\frac{d}{dz}c_{0}+\frac{d}{dz}\overline{c}_{0}\right]y\right)\varphi_{n}\,dz$
$\displaystyle=\lim_{n\to\infty}\int_{a}^{0}\varphi_{n}\frac{d}{dz}y\,dz+\lim_{n\to\infty}\int_{0}^{b}\varphi_{n}\frac{d}{dz}y\,dz$
$\displaystyle=\int_{a}^{0}x\frac{d}{dz}y\,dz+\int_{0}^{b}x\frac{d}{dz}y\,dz.$

Thus, for all $x\in D(\mathbf{d}_{0})$, $y\in D(\mathbf{d}_{0}^{\ast})$ the
inner product $\langle x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}$ exists with

$\displaystyle\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}}=\int_{a}^{0}x(z)\frac{d}{dz}y(z)\,dz+\int_{0}^{b}x(z)\frac{d}{dz}y(z)\,dz.$

Altogether, the formal adjoint $\mathbf{d}_{0}^{\ast}\colon
D(\mathbf{d}_{0}^{\ast})\subset L^{2}([a,b],\mathbb{R})\to
L^{2}([a,b],\mathbb{R})$ is given by

$\displaystyle\begin{split}D(\mathbf{d}_{0}^{\ast})&=\left\\{y\in
L^{2}([a,b],\mathbb{R})\mid y_{|(a,0)}\in H^{1}((a,0),\mathbb{R}),\hskip
2.84544pty_{|(0,b)}\in H^{1}((0,b),\mathbb{R})\hskip 2.84544pt\right\\},\\\
\mathbf{d}_{0}^{\ast}y&=\left[\frac{d}{dz}(c_{0}y)+\frac{d}{dz}(\overline{c}_{0}y)\right]-\left[\frac{d}{dz}c_{0}+\frac{d}{dz}\overline{c}_{0}\right]y\\\
&\hskip
3.41418pt=\left(-\mathbf{d}_{0}-\left[\frac{d}{dz}c_{0}+\frac{d}{dz}\overline{c}_{0}\right]\right)y,\hskip
14.22636pty\in D(\mathbf{d}_{0}^{\ast}).\end{split}$ (95)

In particular, for a function $y=c_{0}y^{-}+\overline{c}_{0}y^{+}\in
D(\mathbf{d}_{0}^{\ast})$ we have

$\displaystyle\mathbf{d}_{0}^{\ast}y=c_{0}\frac{d}{dz}y^{-}+\overline{c}_{0}\frac{d}{dz}y^{+}.$

Moreover, we obtain the following relation between the operators
$\mathbf{d}_{0}$ and $\mathbf{d}_{0}^{\ast}$:

$\displaystyle\langle\mathbf{d}_{0}x,y\rangle_{L^{2}}=-\big{[}x(z)y(z)\big{]}_{a}^{b}+x(0)\left[y(0^{+})-y(0^{-})\right]+\langle
x,\mathbf{d}_{0}^{\ast}y\rangle_{L^{2}},\hskip 14.22636ptx\in
D(\mathbf{d}_{0}),\,y\in D(\mathbf{d}_{0}^{\ast}).$ (96)

We are finally able to state the conservation law of the state variable
$x_{2}$ with respect to the operator $\mathbf{d}_{0}^{\ast}$. Assume that for
all $t>0$, the flux variable
$\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})(\cdot,t)$ on the composed domain is
an element of $D(\mathbf{d}_{0}^{\ast})$. From the preceding discussion and
due to the individual conservation laws (75), we deduce that for all
$\varphi\in\mathcal{D}(a,b)$ we have

$\displaystyle\hskip
14.22636pt\int_{a}^{b}\left[-\mathbf{d}_{0}^{\ast}\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})(z,t)\right]\varphi(z)\,dz$
$\displaystyle=\int_{a}^{0}-\partial_{z}\mathcal{N}_{2}^{-}(z,t)\varphi(z)\,dz+\int_{0}^{b}-\partial_{z}\mathcal{N}_{2}^{+}(z,t)\varphi(z)\,dz$
$\displaystyle=\int_{a}^{0}\partial_{t}x_{2}^{-}(z,t)\varphi(z)\,dz+\int_{0}^{b}\partial_{t}x_{2}^{+}(z,t)\varphi(z)\,dz$
$\displaystyle=\int_{a}^{b}\partial_{t}x_{2}(z,t)\varphi(z)\,dz.$

In conclusion, the conservation law of the state variable $x_{2}$ is given by

$\displaystyle\partial_{t}x_{2}=-\mathbf{d}_{0}^{\ast}\mathcal{N}_{2}(x,c_{0},\overline{c}_{0}),$
(97)

and it holds in a classical sense.

######  Remark 17.2.

We need to point out that the balance equation (78) the flux variable
$\mathcal{N}_{2}$ is supposed to satisfy is incorporated neither into the
domain of the operator $\mathbf{d}_{0}^{\ast}$ nor into the conservation law
(97) itself. So far, we have only allowed a jump discontinuity of
$\mathcal{N}_{2}$ at the interface. One possible way is to rewrite the
conservation law as

$\displaystyle\partial_{t}x_{2}=\mathbf{d}_{0}\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})-e_{I}\delta_{0},$

with $\delta_{0}\in\mathcal{D}^{\prime}(a,b)$ the Dirac mass at $z=0$, and
$e_{I}\colon[0,\infty)\to\mathbb{R}$. By doing so, $e_{I}$ may be regarded as
an external source term, and the equation holds (in a distributional sense) if
and only if $\mathcal{N}_{2}$ satisfies the balance equation (78). However,
since this will only complicate the modeling in the further course and since
we want to define a formally skew-symmetric matrix differential operator later
on, we will stick with the conservation law (97) with respect to the formally
adjoint operator $\mathbf{d}_{0}^{\ast}$ of $\mathbf{d}_{0}$, and take the
interface port variable $e_{I}$ to be implicitly defined by means of the jump
discontinuity of $\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})$ at the interface.
As we will see, this does not affect the dynamics of the resulting system over
the one proposed in [10].

Let us now define classical solutions of the conservation law of $x_{2}$ with
respect to $\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})$.

######  Definition 17.3.

Let $x=(x_{1},x_{2})\colon[0,\infty)\to X$. Let $Y=L^{2}([a,b],\mathbb{R})$.
We call $x_{2}\colon[0,\infty)\to Y$ a classical solution of the conservation
law

$\displaystyle\begin{split}\partial_{t}x_{2}(t)&=-\mathbf{d}_{0}^{\ast}\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})(\cdot,t),\hskip
14.22636ptt>0,\\\ x_{2}(0)&=x_{0,2}\in Y,\end{split}$ (98)

where $\mathbf{d}_{0}^{\ast}$ is the operator given in (95), and with the flux
variable

$\displaystyle\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})=c_{0}\mathcal{N}_{2}^{-}(x)+\overline{c}_{0}\mathcal{N}_{2}^{+}(x)$

of the systems of conservation laws (75) coupled at $z=0$, if:

  1. (i)

$x_{2}\in\mathcal{C}([0,\infty),Y)$.

  2. (ii)

$x_{2_{|(0,\infty)}}\in\mathcal{C}^{1}((0,\infty),Y)$.

  3. (iii)

$\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})(\cdot,t)\in
D(\mathbf{d}_{0}^{\ast})$ for all $t>0$.

  4. (iv)

$x_{2}$ satisfies (98).

Analogously to the operator $\mathbf{d}_{0}$, its formal adjoint
$\mathbf{d}_{0}^{\ast}$ acts as the differential operator $\frac{d}{dz}$ on
the respective subdomains. Assuming that the flux variable $\mathcal{N}_{2}$
satisfies the balance equation (78), on an arbitrary interval
$[a^{\prime},b^{\prime}]\subset[a,b]$ containing the interface we get the
following for all $t>0$:

$\displaystyle\frac{d}{dt}\int_{a^{\prime}}^{b^{\prime}}x_{2}(z,t)\,dz$
$\displaystyle=\int_{a^{\prime}}^{b^{\prime}}\partial_{t}x_{2}(z,t)\,dz$
$\displaystyle=\int_{a^{\prime}}^{b^{\prime}}-\mathbf{d}_{0}^{\ast}\mathcal{N}_{2}(x,c_{0},\overline{c}_{0})(z,t)\,dz$
$\displaystyle=\int_{a^{\prime}}^{0}-\partial_{z}\mathcal{N}_{2}^{-}(x)(z,t)\,dz+\int_{0}^{b^{\prime}}-\partial_{z}\mathcal{N}_{2}^{+}(x)(z,t)\,dz$
$\displaystyle=\mathcal{N}_{2}^{-}(a^{\prime},t)-\mathcal{N}_{2}^{+}(b^{\prime},t)-\mathcal{N}_{2}^{-}(0^{-},t)+\mathcal{N}_{2}^{+}(0^{+},t)$
$\displaystyle=\mathcal{N}_{2}(a^{\prime},t)-\mathcal{N}_{2}(b^{\prime},t)-e_{I}(t).$

Finally, we can formulate the interconnection of the conservation laws (75) as
a port-Hamiltonian system defined on the extended state space $\tilde{X}$,
which we are going to do next.

####  17.3 Port-Hamiltonian Formulation and Dirac Structure

In the previous section, we have formulated the dynamics of the state
variables $x_{1}$ and $x_{2}$ on the composed domain $[a,b]$ through the
conservation laws (91) and (98) using the operator $\mathbf{d}_{0}$ and its
formal adjoint $\mathbf{d}_{0}^{\ast}$ given by (90) and (95), respectively.
Furthermore, in (87) we stated the conservation laws for the characteristic
functions $c_{0}$ and $\overline{c}_{0}$ defined in (79) of the two subdomains
separated by the interface, which are constituting the extended state variable
as well. Now, we want to formulate a Hamiltonian system off of these equation
on the extended state space $\tilde{X}$, and we want to specify the underlying
Dirac structure of the fixed interface scenario.

######  Remark 17.4.

Recall that both the operator $\mathbf{d}_{0}$ and its formal adjoint
$\mathbf{d}_{0}^{\ast}$ depend on the color functions $c_{0}$ and
$\overline{c}_{0}$. Strictly speaking, the operators $\mathbf{d}_{0}$ and
$\mathbf{d}_{0}^{\ast}$ are therefore modulated by the state
$\tilde{x}=(x,c_{0},\overline{c}_{0})\in\tilde{X}$. However, as the color
functions remain constant over time, we will neglect this dependence of the
aforementioned operators.

With the aid of (86), we may define the agumented system

$\displaystyle\frac{\partial}{\partial t}\begin{bmatrix}x_{1}\\\ x_{2}\\\
c_{0}\\\ \overline{c}_{0}\end{bmatrix}=\begin{bmatrix}0&\mathbf{d}_{0}&0&0\\\
-\mathbf{d}_{0}^{\ast}&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}\begin{bmatrix}c_{0}\delta_{x_{1}}H^{-}(x)+\overline{c}_{0}\delta_{x_{1}}H^{+}(x)\\\
c_{0}\delta_{x_{2}}H^{-}(x)+\overline{c}_{0}\delta_{x_{2}}H^{+}(x)\\\
\mathcal{H}^{-}(x)\\\ \mathcal{H}^{+}(x)\end{bmatrix},$

or short,

$\displaystyle\frac{\partial}{\partial
t}\tilde{x}=\tilde{\mathcal{J}}_{0}\delta_{\tilde{x}}H(\tilde{x}),$ (99)

with the Hamiltonian’s variational derivative $\delta_{\tilde{x}}H$ given by
(84), and with the operator $\tilde{\mathcal{J}}_{0}\colon
D(\tilde{\mathcal{J}}_{0})\subset\tilde{X}\to\tilde{X}$ given by

$\displaystyle\begin{split}D(\tilde{\mathcal{J}}_{0})&=D(\mathbf{d}_{0}^{\ast})\times
D(\mathbf{d}_{0})\times L^{2}([a,b],\mathbb{R}^{2}),\\\
\tilde{\mathcal{J}}_{0}\tilde{x}&=\begin{bmatrix}0&\mathbf{d}_{0}&0&0\\\
-\mathbf{d}_{0}^{\ast}&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\end{bmatrix}\tilde{x},\hskip
14.22636pt\tilde{x}\in\tilde{X}.\end{split}$ (100)

By virtue of the relation (96) between the operators $\mathbf{d}_{0}$ and
$\mathbf{d}_{0}^{\ast}$, we find for all $\tilde{e}^{1},\tilde{e}^{2}\in
D(\tilde{\mathcal{J}}_{0})$ that

$\displaystyle\begin{split}&\hskip
14.22636pt\langle\tilde{\mathcal{J}}_{0}\tilde{e}^{1},\tilde{e}^{2}\rangle_{L^{2}}+\langle\tilde{e}^{1},\tilde{\mathcal{J}}_{0}\tilde{e^{2}}\rangle_{L^{2}}\\\
&=\int_{a}^{b}(e^{2})^{\top}(z)\begin{bmatrix}0&\mathbf{d}_{0}\\\
-\mathbf{d}_{0}^{\ast}&0\end{bmatrix}e^{1}(z)+\left(\begin{bmatrix}0&\mathbf{d}_{0}\\\
-\mathbf{d}_{0}^{\ast}&0\end{bmatrix}e^{2}(z)\right)^{\top}e^{1}(z)\,dz\\\
&=\int_{a}^{b}e_{1}^{2}(z)\mathbf{d}_{0}e_{2}^{1}(z)-e_{2}^{2}(z)\mathbf{d}_{0}^{\ast}e_{1}^{1}(z)+e_{1}^{1}(z)\mathbf{d}_{0}e_{2}^{2}(z)-e_{2}^{1}(z)\mathbf{d}_{0}^{\ast}e_{1}^{2}(z)\,dz\\\
&=-\big{[}e_{1}^{1}(z)e_{2}^{2}(z)\big{]}_{a}^{b}-\big{[}e_{2}^{1}(z)e_{1}^{2}(z)\big{]}_{a}^{b}\\\
&\hskip
14.22636pt+e_{2}^{1}(0)\left[e_{1}^{2}(0^{+})-e_{1}^{2}(0^{-})\right]+e_{2}^{2}(0)\left[e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})\right]\\\
&=\big{[}(e^{1})^{\top}(z)P_{1}e^{2}(z)\big{]}_{a}^{b}+e_{2}^{1}(0)\left[e_{1}^{2}(0^{+})-e_{1}^{2}(0^{-})\right]+e_{2}^{2}(0)\left[e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})\right],\end{split}$
(101)

with $P_{1}$ defined in (71). Hence, the operator $\tilde{\mathcal{J}}_{0}$ is
formally skew-symmetric. We will not check whether $\tilde{\mathcal{J}}_{0}$
satifies the Jacobi identity (see again [21, Chapter 7]), so we refer to the
system (99) as a _generalized Hamiltonian system_.
In order to take the energy exchange at the boundary and the power inflow (or
outflow) at the interface into account, the generalized Hamiltonian system
(99) will now be extended to a boundary port-Hamiltonian system. For this
purpose, we need to define a Stokes-Dirac structure similar to the one given
in (65).
To find the underlying Dirac structure of the system (99), we need to define a
suitable power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
constituting the plus pairing
$\ll\cdot,\cdot\gg\colon\mathcal{B}\times\mathcal{B}\to\mathbb{R}$ on the bond
space $\mathcal{B}=\mathcal{F}\times\mathcal{E}$. We will basically proceed as
in Section 15.
To begin with, consider the operator $\mathcal{J}_{0}\colon
D(\mathcal{J}_{0})\subset X\to X$ given by

$\displaystyle D(\mathcal{J}_{0})$ $\displaystyle=\left\\{x\in X\mid x_{1}\in
D(\mathbf{d}_{0}^{\ast}),\hskip 2.84544ptx_{2}\in D(\mathbf{d}_{0})\right\\},$
$\displaystyle\mathcal{J}_{0}x$
$\displaystyle=\begin{bmatrix}0&\mathbf{d}_{0}\\\
-\mathbf{d}_{0}^{\ast}&0\end{bmatrix}\begin{bmatrix}x_{1}\\\
x_{2}\end{bmatrix},\hskip 14.22636ptx\in D(\mathcal{J}_{0}).$

Note that $\mathcal{J}_{0}$ is the operator we obtain if we reduce the
operator $\tilde{\mathcal{J}}_{0}$ from (100) to $X$, i.e., if we ignore the
color functions we included in the definition of the extended state space
$\tilde{X}$. This operator will be useful for finding the power pairing. Note
that from (101) it immediately follows that $\mathcal{J}_{0}$ is formally
skew-symmetric as well, and that on any interval $(c,d)\subset[a,b]$ not
containing the interface position $z=0$, the operator $\mathcal{J}_{0}$ simply
acts as the matrix differential operator $P_{1}\frac{d}{dz}$. We will come
across this operator in Section 19 again.
For systems associated with a formally skew-symmetric differential operator,
we have already defined the boundary port variables in Definition 15.3. Since
the interface position resides in the interior of the spatial domain $[a,b]$,
the trace operator defined in (20) can be extended to

$\displaystyle\operatorname{tr}_{0}\colon
D(\mathcal{J}_{0})\to\mathbb{R}^{4},\hskip
8.5359pte\mapsto\begin{bmatrix}e(b)\\\ e(a)\end{bmatrix}.$

Following (59), the boundary flow variable
$f_{\partial}=f_{\partial,e}\in\mathbb{R}^{2}$ and the boundary effort
variable $e_{\partial}=e_{\partial,e}\in\mathbb{R}^{2}$ can, for all co-energy
variables $e\in D(\mathcal{J}_{0})$, be defined as

$\displaystyle\begin{bmatrix}f_{\partial}\\\
e_{\partial}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}_{0}(e)=\begin{bmatrix}\frac{1}{\sqrt{2}}(P_{1}e(b)-P_{1}e(a))\\\
\frac{1}{\sqrt{2}}(e(b)+e(a))\end{bmatrix},$

with $R_{\operatorname{ext}}$ defined in (72). Furthermore, we have introduced
the interface relations in (77) and (78) with respect to the flux variables.
In general, the interface flow $f_{I}=f_{I,e}\in\mathbb{R}$ and the interface
effort $e_{I}=e_{I,e}\in\mathbb{R}$ are for all effort variables $e\in
D(\mathcal{J}_{0})$ given as follows:

$\displaystyle f_{I}$ $\displaystyle=e_{2}(0^{+})=e_{2}(0^{-}),$
$\displaystyle-e_{I}$ $\displaystyle=e_{1}(0^{+})-e_{1}(0^{-}).$

Now, let us choose the power pairing in order to define the Dirac structure
associated with the system (99) that is augmented with the boundary and
interface port variables. Define the space of flows $\mathcal{F}$ and the
space of efforts $\mathcal{E}$ as

$\displaystyle\mathcal{F}=\mathcal{E}=L^{2}([a,b],\mathbb{R}^{4})\times\mathbb{R}\times\mathbb{R}^{2},$
(102)

endowed with the canonical inner product
$\langle\cdot,\cdot\rangle_{\mathcal{F}}$. For the right choice of the power
pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
we make use of equation (101). Note that for all $e^{1},e^{2}\in
D(\mathcal{J}_{0})$ we may write the last line of (101) with respect to the
boundary and interface port variables as follows:

$\displaystyle\hskip
14.22636pt\big{[}(e^{1})^{\top}(z)P_{1}e^{2}(z)\big{]}_{a}^{b}+e_{2}^{1}(0)\left[e_{1}^{2}(0^{+})-e_{1}^{2}(0^{-})\right]+e_{2}^{2}(0)\left[e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})\right]$
$\displaystyle=\langle
e_{\partial,e^{2}},f_{\partial,e^{1}}\rangle_{2}+\langle
e_{\partial,e^{1}},f_{\partial,e^{2}}\rangle_{2}-f_{I,e^{1}}e_{I,e^{2}}-f_{I,e^{2}}e_{I,e^{1}}.$

The representaion of the evaluation at the boundary with respect to the
boundary flows and efforts can be looked up in Subsection 15.1 (see page 60).
Consequently, we have for all $\tilde{e}^{1},\tilde{e}^{2}\in
D(\tilde{\mathcal{J}}_{0})$,

$\displaystyle\langle\tilde{\mathcal{J}}_{0}\tilde{e}^{1},\tilde{e}^{2}\rangle_{L^{2}}+\langle\tilde{e}^{1},\tilde{\mathcal{J}}_{0}\tilde{e}^{2}\rangle_{L^{2}}=\langle
e_{\partial,e^{2}},f_{\partial,e^{1}}\rangle_{2}+\langle
e_{\partial,e^{1}},f_{\partial,e^{2}}\rangle_{2}-f_{I,e^{1}}e_{I,e^{2}}-f_{I,e^{2}}e_{I,e^{1}}.$
(103)

Thus, we shall define

$\displaystyle\left\langle\begin{pmatrix}\tilde{e}\\\ e_{I}\\\
e_{\partial}\end{pmatrix}\Bigg{|}\begin{pmatrix}\tilde{f}\\\ f_{I}\\\
f_{\partial}\end{pmatrix}\right\rangle:=\langle\tilde{e},\tilde{f}\rangle_{L^{2}}+e_{I}f_{I}-\langle
e_{\partial},f_{\partial}\rangle_{2},\hskip
14.22636pt\left(\begin{pmatrix}\tilde{e}\\\ e_{I}\\\
e_{\partial}\end{pmatrix},\begin{pmatrix}\tilde{f}\\\ f_{I}\\\
f_{\partial}\end{pmatrix}\right)\in\mathcal{E}\times\mathcal{F},$ (104)

and the plus pairing
$\ll\cdot,\cdot\gg\colon\mathcal{B}\times\mathcal{B}\to\mathbb{R}$ on the bond
space $\mathcal{B}=\mathcal{F}\times\mathcal{E}$ induced by the power pairing
(104) is accordingly defined as in (61). Finally, we can define the Dirac
structure associated with the formally skew-symmetric operator
$\tilde{\mathcal{J}}_{0}$ given by (100). The idea is based on the proofs of
Proposition 4 in [10] and of Theorem 3.6 in [15].

######  Proposition 17.5.

Let $\mathcal{B}=\mathcal{F}\times\mathcal{E}$ be the bond space with respect
to the flow space $\mathcal{F}$ and the effort space $\mathcal{E}$ defined in
(102). The subspace $\mathcal{D}_{\tilde{\mathcal{J}}_{0}}\subset\mathcal{B}$
defined by

$\displaystyle\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$
$\displaystyle:=\left\\{\left(\begin{pmatrix}\tilde{f}\\\ f_{I}\\\
f_{\partial}\end{pmatrix},\begin{pmatrix}\tilde{e}\\\ e_{I}\\\
e_{\partial}\end{pmatrix}\right)\in\mathcal{F}\times\mathcal{E}\hskip
2.84544pt\Bigg{|}\hskip 2.84544pt\tilde{e}\in
D(\tilde{\mathcal{J}}_{0}),\hskip
2.84544pt\tilde{f}=\tilde{\mathcal{J}}_{0}\tilde{e},\hskip
2.84544ptf_{I}=e_{2}(0),\right.$ (105) $\displaystyle\hskip
162.18062pt\left.e_{I}=e_{1}(0^{-})-e_{1}(0^{+}),\hskip
2.84544pt\begin{bmatrix}f_{\partial}\\\
e_{\partial}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}_{0}(e_{1},e_{2})\hskip
2.84544pt\right\\},$

with the flow and effort variables

$\displaystyle\tilde{f}=\begin{bmatrix}f_{1}\\\ f_{2}\\\ f_{c}\\\
f_{\overline{c}}\end{bmatrix}\hskip 8.5359pt\text{and}\hskip
8.5359pt\tilde{e}=\begin{bmatrix}e_{1}\\\ e_{2}\\\ e_{c}\\\
e_{\overline{c}}\end{bmatrix},$

respectively, associated with a system of two conservation laws defined on the
spatial domain $[a,b]\subset\mathbb{R}$ with an interface at the point $z=0$
which imposes the continuity of the effort variable $e_{2}$ and allows for the
discontinuity of the effort variable $e_{1}$, and associated with the extended
state (81), the differential operator $\tilde{\mathcal{J}}_{0}$ defined in
(100), the operator $\mathbf{d}_{0}$ and its formal adjoint
$\mathbf{d}_{0}^{\ast}$ defined in (90) and (95), respectively, is a Dirac
structure with respect to the plus pairing
$\ll\cdot,\cdot\gg\colon\mathcal{B}\times\mathcal{B}\to\mathbb{R}$ induced by
the power pairing
$\langle\cdot\mid\cdot\rangle\colon\mathcal{E}\times\mathcal{F}\to\mathbb{R}$
given by (104).

######  Proof 17.6.

In the following, we denote by

$\displaystyle F=\begin{pmatrix}\tilde{f}\\\ f_{I}\\\
f_{\partial}\end{pmatrix}\in\mathcal{F}\hskip 8.5359pt\text{and}\hskip
8.5359ptE=\begin{pmatrix}\tilde{e}\\\ e_{I}\\\
e_{\partial}\end{pmatrix}\in\mathcal{E}$

the vector of flows and the vector of efforts, respectively, for all
$(F,E)\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$. To prove the claim, we have
to verify that the space $\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ coincides
with its orthogonal complement
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}$ with respect to
the plus pairing $\ll\cdot,\cdot\gg$, see Definition 15.4.
We first show that
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}\subset\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}$.
In order to do so, we have to show that for every $(F^{1},E^{1})$,
$(F^{2},E^{2})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ we have

$\displaystyle\ll(F^{1},E^{1}),(F^{2},E^{2})\gg=\langle E^{1}\mid
F^{2}\rangle+\langle E^{2}\mid F^{1}\rangle\stackrel{{\scriptstyle!}}{{=}}0$

with respect to the pairing $\langle\cdot\mid\cdot\rangle$ defined in (104).
Using the constitutive relations of $\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$
and (103), we compute

$\displaystyle\hskip 14.22636pt\ll(F^{1},E^{1}),(F^{2},E^{2})\gg$
$\displaystyle=\langle\tilde{e}^{2},\tilde{f}^{1}\rangle_{L^{2}}+\langle\tilde{e}^{1},\tilde{f}^{2}\rangle_{L^{2}}+e_{I}^{1}f_{I}^{2}+e_{I}^{2}f_{I}^{1}-\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2}-\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\langle\tilde{e}^{2},\tilde{\mathcal{J}}_{0}\tilde{e}^{1}\rangle_{L^{2}}+\langle\tilde{e}^{1},\tilde{\mathcal{J}}_{0}\tilde{e}^{2}\rangle_{L^{2}}+e_{I}^{1}f_{I}^{2}+e_{I}^{2}f_{I}^{1}-\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2}-\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}$ $\displaystyle=\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}+\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2}-f_{I}^{1}e_{I}^{2}-f_{I}^{2}e_{I}^{1}+e_{I}^{1}f_{I}^{2}+e_{I}^{2}f_{I}^{1}-\langle
e_{\partial}^{1},f_{\partial}^{2}\rangle_{2}-\langle
e_{\partial}^{2},f_{\partial}^{1}\rangle_{2}$ $\displaystyle=0.$

As $(F^{1},E^{1}),(F^{2},E^{2})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ can
be chosen arbitrarily, we conlcude that
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}\subset\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}$.
Next, we want to show the inclusion
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}\subset\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$.
To this end, let

$\displaystyle(F^{2},E^{2})=\left(\begin{pmatrix}\tilde{\phi}\\\ \phi_{I}\\\
\phi_{\partial}\end{pmatrix},\begin{pmatrix}\tilde{\epsilon}\\\
\epsilon_{I}\\\ \epsilon_{\partial}\end{pmatrix}\right)\in\mathcal{B}$

such that for all $(F^{1},E^{1})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ it
holds that

$\displaystyle\ll(F^{1},E^{1}),(F^{2},E^{2})\gg=0.$

We need to verify that
$(F^{2},E^{2})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$. For simplicity, we
will omit the dependence on the variable $z$. For any
$(F^{1},E^{1})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ we have

$\displaystyle\begin{split}0&=\int_{a}^{b}(\tilde{e}^{1})^{\top}\tilde{\phi}+\epsilon_{1}(\mathbf{d}_{0}e_{2}^{1})+\epsilon_{2}(-\mathbf{d}_{0}^{\ast}e_{1}^{1})\,dz+e_{I}^{1}\phi_{I}+\epsilon_{I}f_{I}^{1}\\\
&\hskip 14.22636pt-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}.\end{split}$
(106)

Note that the choice of the variable
$\tilde{e}^{1}=(e_{1}^{1},e_{2}^{1},e_{c}^{1},e_{\overline{c}}^{1})$
determines all remaining variables in
$(F^{1},E^{1})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$. Since (106) has to
hold for all $(F^{1},E^{1})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$, we will
consider different instances for the previously mentioned variable in order to
verify that $(F^{2},E^{2})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$.

  1. (i)

Let $e_{1}^{1}=0$, $e_{2}^{1}=0$, and $e_{\overline{c}}^{1}=0$.
From (105) and (106) we derive

$\displaystyle
0=\int_{a}^{b}(\tilde{e}^{1})^{\top}\tilde{\phi}\,dz=\int_{a}^{b}e_{c}^{1}\phi_{c}\,dz.$

As $e_{c}^{1}$ may be chosen arbitrarily, we conclude that $\phi_{c}=0$. If we
change the roles of $e_{c}^{1}$ and $e_{\overline{c}}^{1}$, then we
analogously obtain $\phi_{\overline{c}}=0$.

  2. (ii)

Let $e_{1}^{1}(a)=e_{1}^{1}(b)=0$, $e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})=0$, and
$e_{2}^{1}=0$.
Then we deduce from (106) that

$\displaystyle
0=\int_{a}^{b}e_{1}^{1}\phi_{1}+\epsilon_{2}(-\mathbf{d}_{0}^{\ast}e_{1}^{1})\,dz.$

Assume that $\epsilon_{2}\in D(\mathbf{d}_{0})$. Then, by virtue of (96), it
follows that

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}e_{1}^{1}\phi_{1}+e_{1}^{1}(-\mathbf{d}_{0}\epsilon_{2})\,dz-\big{[}e_{1}^{1}\epsilon_{2}\big{]}_{a}^{b}+\epsilon_{2}(0)\left[e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})\right]$
$\displaystyle=\int_{a}^{b}e_{1}^{1}(-\mathbf{d}_{0}\epsilon_{2}+\phi_{1})\,dz,$

whence $\phi_{1}=\mathbf{d}_{0}\epsilon_{2}$.

  3. (iii)

Let $e_{1}^{1}=0$ and $e_{2}^{1}(a)=e_{2}^{1}(0)=e_{2}^{1}(b)=0$.
If we assume that $\epsilon_{1}\in D(\mathbf{d}_{0}^{\ast})$ and if we apply
(96) once again, we get

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}e_{2}^{1}\phi_{2}+\epsilon_{1}(\mathbf{d}_{0}e_{2}^{1})\,dz$
$\displaystyle=\int_{a}^{b}e_{2}^{1}\phi_{2}+e_{2}^{1}(\mathbf{d}_{0}^{\ast}\epsilon_{1})\,dz-\big{[}e_{2}^{1}\epsilon_{1}\big{]}_{a}^{b}+e_{2}^{1}(0)\left[\epsilon_{1}(0^{+})-\epsilon_{1}(0^{-})\right]$
$\displaystyle=\int_{a}^{b}e_{2}^{1}\phi_{2}+e_{2}^{1}(\mathbf{d}_{0}^{\ast}\epsilon_{1})\,dz,$

whence $\phi_{2}=-\mathbf{d}_{0}^{\ast}\epsilon_{1}$.

From the preceding instances, we conclude that $\tilde{\epsilon}\in
D(\tilde{\mathcal{J}}_{0})$ and

$\displaystyle\tilde{\phi}=\tilde{\mathcal{J}}_{0}\tilde{\epsilon}.$

It is left to show the last three constitutive relations in (105) concerning
the boundary and the interface port variables.

  1. (iv)

Let $e_{1}^{1}(a)=e_{1}^{1}(b)=0$ and $e_{2}^{1}=0$.
From (96) and (106) we infer

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}e_{1}^{1}(\mathbf{d}_{0}\epsilon_{2})+\epsilon_{2}(-\mathbf{d}_{0}^{\ast}e_{1}^{1})\,dz+e_{I}^{1}\phi_{I}$
$\displaystyle=\epsilon_{2}(0)\left[e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})\right]+e_{I}^{1}\phi_{I}$
$\displaystyle=-e_{I}^{1}\epsilon_{2}(0)+e_{I}^{1}\phi_{I}.$

Hence, $\phi_{I}=\epsilon_{2}(0)$.

  2. (v)

Let $e_{1}^{1}=0$ and $e_{2}^{1}(a)=e_{2}^{1}(b)=0$.
With the same argument we find

$\displaystyle 0$
$\displaystyle=\int_{a}^{b}e_{2}^{1}(-\mathbf{d}_{0}^{\ast}\epsilon_{1})+\epsilon_{1}(\mathbf{d}_{0}e_{2}^{1})\,dz+\epsilon_{I}f_{I}^{1}$
$\displaystyle=e_{2}^{1}(0)\left[\epsilon_{1}(0^{+})-\epsilon_{1}(0^{-})\right]+\epsilon_{I}f_{I}^{1}$
$\displaystyle=f_{I}^{1}\left[\epsilon_{1}(0^{+})-\epsilon_{1}(0^{-})\right]+\epsilon_{I}f_{I}^{1}.$

Consequently, $\epsilon_{I}=\epsilon_{1}(0^{-})-\epsilon_{1}(0^{+})$.

  3. (vi)

Let $e_{1}^{1}(0^{+})-e_{1}^{1}(0^{-})=0$.
Following the discussion in Subsection 15.1 and using the relation (101),
equation (106) can be written as

$\displaystyle 0$
$\displaystyle=\langle\tilde{\epsilon},\tilde{\mathcal{J}}_{0}\tilde{e}^{1}\rangle_{L^{2}}+\langle\tilde{e}^{1},\tilde{\mathcal{J}}_{0}\tilde{\epsilon}\rangle_{L^{2}}+\epsilon_{I}f_{I}^{1}-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\big{[}(e^{1})^{\top}P_{1}\epsilon\big{]}_{a}^{b}-f_{I}^{1}\epsilon_{I}+\epsilon_{I}f_{I}^{1}-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\operatorname{tr}_{0}(e^{1})^{\top}\mathcal{\begin{bmatrix}P_{1}&0\\\
0&-P_{1}\end{bmatrix}}\operatorname{tr}_{0}(\epsilon)-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\operatorname{tr}_{0}(e^{1})^{\top}\mathcal{P}_{\operatorname{ext}}\operatorname{tr}_{0}(\epsilon)-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\operatorname{tr}_{0}(e^{1})^{\top}R_{\operatorname{ext}}^{\top}\Sigma
R_{\operatorname{ext}}\operatorname{tr}_{0}(\epsilon)-\langle
e_{\partial}^{1},\phi_{\partial}\rangle_{2}-\langle\epsilon_{\partial},f_{\partial}^{1}\rangle_{2}$
$\displaystyle=\begin{bmatrix}(f_{\partial}^{1})^{\top}&(e_{\partial}^{1})^{\top}\end{bmatrix}\Sigma
R_{\operatorname{ext}}\operatorname{tr}_{0}(\epsilon)-(e_{\partial}^{1})^{\top}\phi_{\partial}-\epsilon_{\partial}^{\top}f_{\partial}^{1}$
$\displaystyle=\begin{bmatrix}(e_{\partial}^{1})^{\top}&(f_{\partial}^{1})^{\top}\end{bmatrix}\left[R_{\operatorname{ext}}\operatorname{tr}_{0}(\epsilon)-\begin{bmatrix}\phi_{\partial}\\\
\epsilon_{\partial}\end{bmatrix}\right].$

Since a proper choice for the variable $\tilde{e}$ yields arbitary values for
the vectors $e_{\partial}^{1}$ and $f_{\partial}^{1}$, we conclude that

$\displaystyle\begin{bmatrix}\phi_{\partial}\\\
\epsilon_{\partial}\end{bmatrix}=R_{\operatorname{ext}}\operatorname{tr}_{0}(\epsilon).$

Thus, we have $(F^{2},E^{2})\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$, and so
the inclusion
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}\subset\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$
holds as well. Summing up, we have
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}=\mathcal{D}_{\tilde{\mathcal{J}}_{0}}^{\perp\\!\\!\\!\perp}$.
This completes the proof.

Comparing the constitutive relations of the Dirac structure
$\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ defined in (105) with the generalized
Hamiltonian system (99) together with the interface relations (77)-(78)
imposed on the flux variable $\mathcal{N}(x,c_{0},\overline{c}_{0})$, one may
easily see that this system possesses a boundary port-Hamiltonian structure.
Note that it is crucial to pick states of the form

$\displaystyle\tilde{x}=(x,c_{0},\overline{c}_{0})\in\tilde{X},$

with $c_{0}$ and $\overline{c}_{0}$ the color functions defined in (79), as
the conservation laws derived in Subsection 17.2, which are constituting the
generalized Hamiltonian system (99), have been determined with this particular
choice. Denoting $f_{I}(t)=f_{I,\delta_{\tilde{x}}H(\tilde{x}(\cdot,t))}$ and
$e_{I}(t)=e_{I,\delta_{\tilde{x}}H(\tilde{x}(\cdot,t))}$ for $t\geq 0$, the
following holds.

######  Corollary 17.7.

The generalized Hamiltonian system (99) of four conservation laws may be
defined as a generalized boundary port-Hamiltonian system, and is
geometrically specified by the requirement that for all $t>0$ we have

$\displaystyle\left(\begin{pmatrix}\partial_{t}\tilde{x}(\cdot,t)\\\
f_{I}(t)\\\
f_{\partial}(t)\end{pmatrix},\begin{pmatrix}\delta_{\tilde{x}}H(\tilde{x}(\cdot,t))\\\
e_{I}(t)\\\
e_{\partial}(t)\end{pmatrix}\right)\in\mathcal{D}_{\tilde{\mathcal{J}}_{0}},$

with $\mathcal{D}_{\tilde{\mathcal{J}}_{0}}$ the Dirac structure defined in
(105), the extended state vector $\tilde{x}=(x,c_{0},\overline{c}_{0})$, where
$c_{0}$ and $\overline{c}_{0}$ are precisely the color functions defined in
(79), the variational derivate (84) of the Hamiltonian
$H\colon\tilde{X}\to\mathbb{R}$ defined in (83), the pair of port variables
$(f_{I},e_{I})\in\mathbb{R}^{2}$ associated with the interface, and the pair
of port variables $(f_{\partial},e_{\partial})\in\mathbb{R}^{4}$ associated
with the boundary of the spatial domain $[a,b]$.

As a consequence of the boundary port-Hamiltonian structure, we can state a
power balance equation along solutions $\tilde{x}\colon[0,\infty)\to\tilde{X}$
of (99) of the form

$\displaystyle\tilde{x}(t)=\begin{bmatrix}x_{1}(\cdot,t)\\\ x_{2}(\cdot,t)\\\
c_{0}(\cdot,t)\\\ \overline{c}_{0}(\cdot,t)\end{bmatrix},\hskip
14.22636ptt\geq 0,$

by means of the boundary and interface port variables: using (103), for all
$t>0$ it holds that

$\displaystyle\hskip 14.22636pt\frac{d}{dt}H(\tilde{x}(\cdot,t))$
$\displaystyle=\frac{d}{dt}\int_{a}^{b}c_{0}(z,t)\mathcal{H}^{-}(x(z,t))+\overline{c}_{0}(z,t)\mathcal{H}^{+}(x(z,t))\,dz$
$\displaystyle=\int_{a}^{b}(\partial_{t}c_{0}(z,t))\mathcal{H}^{-}(x(z,t))+c_{0}(z,t)\partial_{t}\mathcal{H}^{-}(x(z,t))\,dz$
$\displaystyle\hskip
14.22636pt+\int_{a}^{b}(\partial_{t}\overline{c}_{0}(z,t))\mathcal{H}^{+}(x(z,t))+\overline{c}_{0}(z,t)\partial_{t}\mathcal{H}^{+}(x(z,t))\,dz$
$\displaystyle=\int_{a}^{b}c_{0}(z,t)\left[\delta_{x_{1}}H^{-}(x(z,t))\partial_{t}x_{1}(z,t)+\delta_{x_{2}}H^{-}(x(z,t))\partial_{t}x_{2}(z,t)\right]\,dz$
$\displaystyle\hskip
14.22636pt+\int_{a}^{b}\overline{c}_{0}(z,t)\left[\delta_{x_{1}}H^{+}(x(z,t))\partial_{t}x_{1}(z,t)+\delta_{x_{2}}H^{+}(x(z,t))\partial_{t}x_{2}(z,t)\right]\,dz$
$\displaystyle=\int_{a}^{b}c_{0}(z,t)\left[\delta_{x_{1}}H^{-}(x(z,t))(\mathbf{d}_{0}\delta_{x_{2}}H^{-}(x(z,t)))+\delta_{x_{2}}H^{-}(x(z,t))\left(-\mathbf{d}_{0}^{\ast}\delta_{x_{1}}H^{-}(x(z,t))\right)\right]\,dz$
$\displaystyle\hskip
14.22636pt+\int_{a}^{b}\overline{c}_{0}(z,t)\left[\delta_{x_{1}}H^{+}(x(z,t))(\mathbf{d}_{0}\delta_{x_{2}}H^{+}(x(z,t)))+\delta_{x_{2}}H^{+}(x(z,t))\left(-\mathbf{d}_{0}^{\ast}\delta_{x_{1}}H^{+}(x(z,t))\right)\right]\,dz$
$\displaystyle=\frac{1}{2}\left(\langle\tilde{\mathcal{J}}_{0}\delta_{\tilde{x}}H(\tilde{x}(\cdot,t)),\delta_{\tilde{x}}H(\tilde{x}(\cdot,t))\rangle_{L^{2}}+\langle\delta_{\tilde{x}}H(\tilde{x}(\cdot,t)),\tilde{\mathcal{J}}_{0}\delta_{\tilde{x}}H(\tilde{x}(\cdot,t))\rangle_{L^{2}}\right)$
$\displaystyle=\langle
e_{\partial}(t),f_{\partial}(t)\rangle_{2}-e_{I}(t)f_{I}(t).$

This equation expresses the fact that the change of energy is given by the
power flow through the boundary minus the power flow at the interface
position. An interesting question is whether this class of systems even admits
a solution, and whether it is well-posed. We will postpone the answer to
questions of this kind to Section 19, as in this section, we only wanted to
lay the foundation for the upcoming segment.
In this section, we have formulated two systems of conservation laws that are
coupled by a fixed interface as a boundary port-Hamiltonian system with some
power flow at the interface. In the following, we repeat this procedure, but
we assume that the interface position is not fixed anymore, but moving over
time.

###  18 Port-Hamiltonian Systems Coupled by a Moving Interface