# Boundary feedback control for hyperbolic systems

Michael Herty111RWTH Aachen University, Templergraben 55, D-52056 Aachen,
Germany.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Ferdinand Thein111RWTH Aachen University,
Templergraben 55, D-52056 Aachen, Germany.
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

We are interested in the feedback stabilization of general linear multi-
dimensional first order hyperbolic systems in $\mathbb{R}^{d}$. Using a
Lyapunov function with a suited weight function dpending on the system under
consideration we show stabilization in $L^{2}$ for the studied system using a
suitable feedback control. Therefore the controlability of the studied system
is related to the feasibility of an associated linear matrix inequality. We
show the applicability discussing the barotropic Euler equations.

##  1 Introduction

The stabilization of spatially one–dimensional systems of hyperbolic balance
laws is a vivid subject attracting research interest in the mathematical as
well as in the engineering community and we refer to the monographs [4, 10,
41, 38] for further references. The mentioned references also provide a
comprehensive overview on related controllability problems. A particular focus
has been put on problems modeled by the barotropic Euler equations and the
shallow water equations which in one space dimension form a 2$\times$2
hyperbolic system to model the temporal and spatial evolution of fluid flows
including flows on networks. Analytical results concerning the boundary
control of such systems have been studied in several articles, cf. [29, 3, 21,
32] for gas flow and for water flow we refer to [9, 19, 30, 40, 15, 31]. One
key aspect in the analysis is the Lyapunov function which is introduced as a
weighted $L^{2}$ (or $H^{s}$) norm and which allows to estimate deviations
from steady states, see e.g. [4]. Under rather general dissipative conditions
the exponential decay of the Lyapunov function has been established for
various problem formulations and we exemplary refer to [13, 16, 11, 12]. For a
study on comparisons to other stability concepts we mention [14]. Stability
with respect to a higher $H^{s}$-norm $(s\geq 2)$ gives stability of the
nonlinear system [12, 4]. Without aiming at completeness, we mention that
recently the results have been extended to also deal with e.g. input-to-state
stability [45], numerical methods have been discussed in [25, 2, 28] and for
results concerning nonlocal hyperbolic partial differential equations (PDEs)
see [17].

However, to the best of our knowledge the presented results are limited to the
spatially one–dimensional case. Based on an example in metal forming
processes, see [1, 35] we extended results to multi–dimensional hyperbolic
balance laws. A specific system in two dimensions is discussed in [20] where a
control problem for the shallow water equations is studied. There the authors
take advantage of the structure of the system and show that the energy is non-
increasing upon imposing suited boundary conditions. Just very recently a
preprint was published where the boundary stabilization for two dimensional
systems is studied using a different Lyapunov function, see [51]. A further
recent result on multi–dimensional hyperbolic scalar conservation laws was
presented in [47]. However, the goal of the mentioned paper is different from
the one presented here. In [47] the author discusses scalar conservation laws
in $\mathbb{R}^{d}$ whereas we are concerned with systems in a bounded domain.
Moreover, we consider deviations from a reference state and thus study the
linearized system, whereas in [47] a situation is studied when two fixed
states at infinity are prescribed. It will be interesting and part of future
research to study the relation between the works [20, 51, 47] and the results
presented in this work.

Throughout this work we will fundamentally rely on the assumption that the
studied system is symmetric hyperbolic, i.e., the Jacobian matrices are
symmetric. Symmetry of the system, in particular in multi-dimensions, gives
well posed problems as it was shown in the pioneering work [23]. This is of
major importance since it is not possible to expect strict hyperbolicity,
another property of hyperbolic systems leading to well posedness, for relevant
systems in multi-dimensions. In [27] it is shown that the Euler equations can
be rewritten as a symmetric hyperbolic system. Later it was proven that all
hyperbolic systems which are equipped with an additional conservation law for
a quantity which is convex in terms of the conserved quantities can be put
into symmetric form, see [24]. Thus many, if not to say all, relevant systems
arising in applications can be symmetrized since they enjoy this additional
conservation law representing, e.g., the conservation of energy or entropy.
For some examples on symmetrizable systems we refer to [6]. Also,
well–posedness results related to symmetric hyperbolic systems are obtained in
several publications and without assuming completeness we exemplary refer [8]
for results in $L^{p}$ and [36] for the linear and quasi linear case. Initial
boundary value problems have been studied e.g. in [43] or [44] in the case of
constant coefficients. For a comprehensive presentation of results on initial
boundary value problems for Friedrichs symmetrizable multi-dimensional systems
we refer to [5]. Further insight on the properties of these large class of
systems are presented in [18, 46] where a brief historical review of this
topic is given in [46]. There are basically three major points where we make
use of the symmetry property. First, it ensures well-posedness and the
existence of sufficiently smooth solutions as already noted before. Second,
the symmetric Jacobians allow to use the product rule for a quadratic form
which is needed throughout the proof of the main result. The third point where
we make use of this property is the discussion of a _linear matrix inequality
(LMI)_ which is essential for a needed estimate and we exemplary refer to [7]
for a broad introduction and further references. The symmetric hyperbolicity
of the system induces a favorable algebraic structure which was already
investigated by Lax in [39]. This work stimulated further research and a
related conjecture was answered in [34]. We further recommend [49] for a
comprehensive overview and additional references.

In this work we will study the boundary stabilization of multi–dimensional
linear hyperbolic balance laws with variable coefficient matrices using an
extension of the Lyapunov function introduced in [4]. The particular structure
of a hyperbolic system is exploited to derive a condition on the feedback law,
such that the Lyapunov function decays exponentially fast, see Section 2 and
Section 3 for the proof. The studied results will be applied the barotropic
Euler equations in Section 4.

##  2 Stabilization of Multi-Dimensional Linear Hyperbolic Balance Laws

We are interested in the following initial boundary value problem (IBVP) for
the given system of hyperbolic PDEs

$\displaystyle\begin{dcases}\frac{\partial}{\partial
t}\mathbf{w}(t,\mathbf{x})+\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})\frac{\partial}{\partial
x_{k}}\mathbf{w}(t,\mathbf{x})+\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})&=0,\;(t,\mathbf{x})\in[0,T)\times\Omega\\\
\mathbf{w}(0,\mathbf{x})&=\mathbf{w}_{0}(\mathbf{x}),\;\mathbf{x}\in\Omega,\\\
\mathbf{w}(t,\mathbf{x})&=\mathbf{w}_{BC}(t,\mathbf{x}),\;(t,\mathbf{x})\in[0,T)\times\partial\Omega\end{dcases}$
(2.1)

Here
$\mathbf{w}(t,\mathbf{x})\equiv(w_{1}(t,\mathbf{x}),\dots,w_{n}(t,\mathbf{x}))^{T}$
is the vector of unknowns and $\Omega\subset\mathbb{R}^{d}$ a bounded domain
with piecewise $C^{1}$ smooth boundary $\partial\Omega$. Moreover,
$\mathbf{A}^{(k)}(\mathbf{x})$ and $\mathbf{B}(\mathbf{x})$ are sufficiently
smooth and bounded $n\times n$ real matrices. The
$\mathbf{A}^{(k)}(\mathbf{x})$ are in particular symmetric. Upon applying a
suited variable transformation we may transform any symmetric hyperbolic
system into the given form with the identity in front of the time derivative,
cf. [5]. The assumption of symmetry is no major restriction since it includes
all systems equipped with an additional conservation law, cf. [24, 18]. This
includes most systems relevant for applications, see [6, 46]. We define the
matrix

$\mathbf{A}^{\ast}(\mathbf{x},\mathbf{\nu}):=\sum_{k=1}^{d}\nu_{k}\mathbf{A}^{(k)}(\mathbf{x})$

with $\mathbf{\nu}=(\nu_{1},\dots,\nu_{d})^{T}\in\mathbb{S}^{d-1}$ being a
unit vector in $\mathbb{R}^{d}$ and by construction
$\mathbf{A}^{\ast}(\mathbf{x},\mathbf{\nu})$ is also symmetric. Adopting the
notion given in [42, 34] we refer to the matrix $\mathbf{A}^{\ast}$ as the
_pencil matrix_. Concerning the definition of hyperbolicity of system (2.1) we
follow [18]. System (2.1) is said to be hyperbolic if the matrix
$\mathbf{A}^{\ast}(\mathbf{x},\mathbf{\nu})$ has $n$ real eigenvalues
$\lambda_{i}=\lambda_{i}(\mathbf{x},\mathbf{\nu}),\,i=1,\dots,n$, and $n$
corresponding linearly independent right eigenvectors
$\mathbf{r}_{i}=\mathbf{r}_{i}(\mathbf{\mathbf{x},\nu}),i=1,\dots,n$ for all
$\mathbf{\nu}\in\mathbb{S}^{d-1}$. Note that by choosing

$\mathbf{\nu}=\mathbf{e}_{k}=(0,\dots,0,\underbrace{1}_{k},0,\dots,0)^{T}$

we have
$\mathbf{A}^{\ast}(\mathbf{x},\mathbf{e}_{k})=\mathbf{A}^{(k)}(\mathbf{x})$
and thus the individual Jacobians $\mathbf{A}^{(k)}(\mathbf{x})$ are also
diagonalizable with real eigenvalues. Further following [5] there exists a
well-conditioned orthogonal matrix $\mathbf{T}(\mathbf{x},\mathbf{\nu})$ such
that

$\displaystyle\mathbf{\Lambda}^{\ast}(\mathbf{x},\mathbf{\nu})=\mathbf{T}^{T}(\mathbf{x},\mathbf{\nu})\mathbf{A}^{\ast}(\mathbf{x},\mathbf{\nu})\mathbf{T}(\mathbf{x},\mathbf{\nu})$
(2.2)

is a diagonal matrix containing the eigenvalues of the system (2.1). The afore
mentioned properties are distinctive features of multi–dimensional hyperbolic
PDEs and we highly recommend the references [5, 18], cited above, for further
reading. A further delicate issue is the dependence of the eigenspaces and
eigenvalues on $\nu\in\mathbb{S}^{d-1}$. That this is in general not the case
is shown in [5] where a particular system fails the continuous dependence of
the eigenspaces with respect to $\nu$. However, for constantly hyperbolic
systems, i.e. systems with constant multiplicity of the eigenvalues, the
eigenspaces are continuous functions of $\nu\in\mathbb{S}^{d-1}$. Hence it is
possible to find locally an eigenbasis of $\mathbf{A}^{\ast}(\nu)$, which
depends continuously on $\nu$.

######  Remark 2.1.

Here we additionally assume that the transformation is continuous with respect
to each smooth boundary part and $\nu\in\mathbb{S}^{d-1}$. Following [5] for
systems with constant coefficients the class of constantly hyperbolic systems,
for example, enjoys this property.

The boundary $\partial\Omega$ will be separated in the controllable and
uncontrollable part, i.e. for $i=1,\dots,n$

$\displaystyle\Gamma_{i}^{+}$
$\displaystyle:=\left\\{\left.\mathbf{x}\in\partial\Omega\,\right|\,\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))\geq
0\right\\},$ $\displaystyle\Gamma_{i}^{-}$
$\displaystyle:=\left\\{\left.\mathbf{x}\in\partial\Omega\,\right|\,\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))<0\right\\}.$

We further need to discuss a LMI associated to the system (2.1) of the form

$\displaystyle\mathbf{A}(\mathbf{m}):=C\mathbf{Id}+\sum_{k=1}^{d}m_{k}\mathbf{A}^{(k)}\leq
0,\,C\in\mathbb{R}_{>0}$ (2.3)

which obviously may be reformulated to

$\displaystyle\overline{\mathbf{A}}(\mathbf{m}):=\mathbf{A}^{(0)}+\sum_{k=1}^{d}\overline{m}_{k}\mathbf{A}^{(k)}\geq
0,$ (2.4)

with $\mathbf{A}^{(0)}=-\mathbf{Id}$ and where the coefficients are given by
$\overline{m}_{k}=-m_{k}/C$. The matrices $\mathbf{A}^{(k)}$ are the Jacobians
of system (2.1). In the case that the Jacobians of system (2.1) are space
dependent the coefficients $\mathbf{m}$ of the LMI may also depend on
$\mathbf{x}$. Further it will turn out that for the boundary feedback control
we need $\mathbf{m}$ to be the gradient of a function and hence we have the
following result.

######  Lemma 2.2.

Assume there exists a function $\overline{\mu}(\mathbf{x})$ such that

$\displaystyle\overline{\mathbf{m}}:=\nabla\overline{\mu}(\mathbf{x})\quad\text{and}\quad\overline{\mathbf{A}}(\mathbf{m}):=-\mathbf{Id}+\sum_{k=1}^{d}\overline{m}_{k}\mathbf{A}^{(k)}\geq
0.$ (2.5)

Then system (2.1) has a _feasible Lyapunov potential_
$\mu(\mathbf{x})=-C\overline{\mu}(\mathbf{x})$ such that (2.3) holds.

Of course the space dependent case is more intricate since we not only need
any solution of the LMI, but also one which must be integrable. However, for
constant matrices we have for such coefficients $\mathbf{m}$ that

$\mu(\mathbf{x})=\sum_{k=1}^{d}m_{k}x_{k}$

is a suited potential.
Finally we want to recall the definition of exponential stability, cf. [4,
33].

######  Definition 2.3.

A solution $\mathbf{w}\in C^{1}(0,T;H^{s}(\Omega))^{n}$ for $s\geq
1+\frac{d}{2}$ of the initial boundary value problem (2.1) is called
_exponentially stable_ in the $L^{2}-$ sense, iff

$\displaystyle\|\mathbf{w}(t,.)\|_{L^{2}(\Omega)}\leq
C_{1}\exp(-C_{2}t)\|\mathbf{w}(0,.)\|_{L^{2}(\Omega)},\,C_{1},C_{2}\in\mathbb{R}_{>0},\,t\in[0,+\infty).$
(2.6)

The Lyapunov function we will introduce in Theorem 2.4 is equivalent to the
$L^{2}(\Omega)$ norm. For later use, we define

$\displaystyle\mathcal{A}(t,\mathbf{x})$
$\displaystyle:=\left(\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(1)}(\mathbf{x})\mathbf{w}(t,\mathbf{x}),\dots,\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(d)}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\right)^{T}\exp(\mu(\mathbf{x})),$
$\displaystyle m_{k}(\mathbf{x})$ $\displaystyle:=\frac{\partial}{\partial
x_{k}}\mu(\mathbf{x}),\,k=1,\dots,d.$

Now, we state the main result of this paper.

######  Theorem 2.4.

Let $\mathbf{w}(t,\mathbf{x})\in C^{1}\left((0,T),H^{s}(\Omega)\right)^{n}$,
$s\geq 1+d/2,$ be a solution to the IBVP (2.1). The problem (2.1) is boundary
diagonal and admits a feasible Lyapunov potential. Thus there exist
$\mu(\mathbf{x})\in H^{s}(\Omega)$ such that

$\displaystyle
C_{A}\mathbf{Id}+\sum_{k=1}^{d}m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\leq
0$ (2.7)

holds true for some value $C_{A}\in\mathbb{R}_{>0}$. Define the Lyapunov
function as follows

$\displaystyle
L(t)=\int_{\Omega}\mathbf{w}(t,\mathbf{x})^{T}\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}.$
(2.8)

The boundary condition for (2.1) is given by

$\displaystyle\mathbf{w}^{(i)}_{BC}(t,\mathbf{x})=u_{i}(t,\mathbf{x}),\,\mathbf{x}\in\Gamma_{i}^{-}$

where the imposed feedback controls $\mathbf{u}=(u_{1},\dots,u_{n})^{T}$
satisfy $\mathbf{u}=\mathbf{T}\tilde{\mathbf{u}}$

$\displaystyle\begin{split}&-\sum_{i=1}^{n}\int_{\Gamma_{i}^{-}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))\tilde{u}_{i}(t,\mathbf{x})^{2}\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}\leq\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))v_{i}(t,\mathbf{x})^{2}\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x},\end{split}$
(2.9)

with $\mathbf{v}=\mathbf{T}^{T}\mathbf{w}$ and (2.2). Then, it holds for the
Lyapunov function that

$\displaystyle\frac{\textup{d}}{\textup{d}t}L(t)\leq-
C_{L}L(t),\,C_{L}\in\mathbb{R}_{>0}.$

Further, the solution $\mathbf{w}$ of the IBVP (2.1) is exponentially stable
in the $L^{2}-$sense (2.6).

The proof is given in the following section for better readability. We refer
to Remark 3.3 for particular cases of interest and conditions under which
$u_{i}$ exists. Note that the choice of $s$ in the space $H^{s}$ guarantees
that $\mu$ and $\mathbf{w}$ are differentiable in $\mathbf{x}$ by Sobolev
embedding.

##  3 Proof of Theorem 2.4

Proof: We want to show that the Lyapunov function (2.8) decays exponentially
fast under the given conditions. We obtain for the time derivative of the
Lyapunov function

$\displaystyle\frac{\textup{d}}{\textup{d}t}L(t)$
$\displaystyle=\frac{\textup{d}}{\textup{d}t}\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}=2\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\frac{\partial}{\partial
t}\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}.$

Replacing the time derivative of $\mathbf{w}$ using the PDE system (2.1) gives

$\displaystyle\frac{\textup{d}}{\textup{d}t}L(t)$
$\displaystyle=2\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left[-\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})\frac{\partial}{\partial
x_{k}}\mathbf{w}(t,\mathbf{x})-\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\right]\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle=-2\int_{\Omega}\sum_{k=1}^{d}\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\frac{\partial}{\partial
x_{k}}\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))+\mathbf{w}^{T}(t,\mathbf{x})\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$

Now with using the product rule and introducing the abbreviations
$\mathcal{A}$ and $m_{k}(\mathbf{x})$ we obtain

$\displaystyle\frac{\textup{d}}{\textup{d}t}L(t)=\dots$
$\displaystyle=-\int_{\Omega}\sum_{k=1}^{d}\left[\frac{\partial}{\partial
x_{k}}\left(\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\right)-\mathbf{w}^{T}(t,\mathbf{x})\left(\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\right.$
$\displaystyle-\left.m_{k}(\mathbf{x})\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\right]+2\mathbf{w}^{T}(t,\mathbf{x})\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle=-\int_{\Omega}\nabla\cdot\mathcal{A}(\mathbf{x})\,\textup{d}\mathbf{x}+\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\sum_{k=1}^{d}\left(m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})+\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))$
$\displaystyle-2\mathbf{w}^{T}(t,\mathbf{x})\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle=-\underbrace{\int_{\partial\Omega}\mathcal{A}(t,\mathbf{x})\cdot\mathbf{n}(\mathbf{x})\,\textup{d}\mathbf{x}}_{=:\mathcal{B}(t)}$
$\displaystyle+\underbrace{\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left[\sum_{k=1}^{d}\left(m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})+\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})\right)-2\mathbf{B}(\mathbf{x})\right]\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}}_{=:\mathcal{I}(t)}.$

In the following we split the proof into two parts. First we show that the
boundary term $\mathcal{B}(t)$ is non-negative under the given conditions. In
the second part we estimate $\mathcal{I}(t)$ such that we can use Gronwalls
Lemma to show the decay of the Lyapunov function.

###  3.1 Estimating the Boundary Integral

In the following we want to discuss the boundary term. We have

$\displaystyle\mathcal{A}(t,\mathbf{x})\cdot\mathbf{n}(\mathbf{x})$
$\displaystyle=\sum_{k=1}^{d}\mathbf{w}^{T}(t,\mathbf{x})\mathbf{A}^{(k)}\mathbf{w}(t,\mathbf{x})n_{k}(\mathbf{x})\exp(\mu(\mathbf{x}))$
$\displaystyle=\mathbf{w}^{T}(t,\mathbf{x})\left(\sum_{k=1}^{d}n_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))$
$\displaystyle=\mathbf{w}^{T}(\mathbf{x})\mathbf{A}^{\ast}(\mathbf{x},\mathbf{n}(\mathbf{x}))\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x})).$

Since the system is symmetric we can transform the quadratic form according to
(2.2) and thus have

$\displaystyle\mathcal{B}(t)=\int_{\partial\Omega}\mathcal{A}(t,\mathbf{x})\cdot\mathbf{n}(\mathbf{x})\,\textup{d}\mathbf{x}=\int_{\partial\Omega}\mathbf{v}^{T}(t,\mathbf{x})\mathbf{\Lambda}^{\ast}(\mathbf{x},\mathbf{n}(\mathbf{x}))\mathbf{v}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}.$

Using the partitioning of the boundary introduced before we rewrite the
boundary term and obtain

$\displaystyle\mathcal{B}(t)$
$\displaystyle=\int_{\partial\Omega}\mathcal{A}(\mathbf{x})\cdot\mathbf{n}(\mathbf{x})\,\textup{d}\mathbf{x}=\int_{\partial\Omega}\mathbf{v}^{T}(t,\mathbf{x})\mathbf{\Lambda}^{\ast}(\mathbf{x},\mathbf{n}(\mathbf{x}))\mathbf{v}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle=\sum_{i=1}^{n}\int_{\Gamma_{i}^{-}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))\exp(\mu(\mathbf{x}))\tilde{u}_{i}(t,\mathbf{x})^{2}\,\textup{d}\mathbf{x}+\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))\exp(\mu(\mathbf{x}))v_{i}(t,\mathbf{x})^{2}\,\textup{d}\mathbf{x}$
$\displaystyle\geq 0.$

Due to $\mathbf{v}=\mathbf{T}^{T}\mathbf{w}$ we have
$\mathbf{u}=\mathbf{T}\tilde{\mathbf{u}}$.

###  3.2 Estimating the Volume Integral

We now want to study the expression

$\displaystyle\mathcal{I}(t)$
$\displaystyle:=\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left[\sum_{k=1}^{d}\left(m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})+\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})\right)-2\mathbf{B}(\mathbf{x})\right]\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle=\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left(\sum_{k=1}^{d}m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle+\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left(\sum_{k=1}^{d}\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})-2\mathbf{B}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$

We first want to discuss the term

$\displaystyle\mathbf{w}^{T}(t,\mathbf{x})\left(\sum_{k=1}^{d}\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})-2\mathbf{B}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x}).$
(3.1)

If (3.1) is less or equal to zero there is nothing to do and it can simply be
estimated by zero. Assume it is strict positive. In the case that $\mathbf{B}$
is not symmetric it can be replaced by its symmetric part
$\mathbf{B}^{sym}=(\mathbf{B}+\mathbf{B}^{T})/2$. Since the $\mathbf{A}^{(k)}$
are symmetric this property is inherited by the derivatives of these matrices.
Hence the following estimate can be established

$\displaystyle\mathbf{w}^{T}(t,\mathbf{x})\left(\sum_{k=1}^{d}\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})-2\mathbf{B}^{sym}(\mathbf{x})\right)\mathbf{w}(t,\mathbf{x})\leq
C_{B}\|\mathbf{w}(t,\mathbf{x})\|_{2}^{2},\,C_{B}\in\mathbb{R}_{>0}.$

Furthermore, by (2.7), $\mu(\mathbf{x})$ and thus the $m_{k}(\mathbf{x})$,
respectively, are chosen such that the linear matrix inequality

$\displaystyle
C_{A}\mathbf{Id}+\sum_{k=1}^{d}m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\leq
0$

holds. Thus we have for $C_{A}>C_{B}$ and $0<C_{L}\leq C_{A}-C_{B}$

$\displaystyle\mathcal{I}(t)$
$\displaystyle:=\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left[\sum_{k=1}^{d}\left(m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})+\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})\right)-2\mathbf{B}(\mathbf{x})\right]\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle\leq\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\left[C_{B}\mathbf{Id}+\sum_{k=1}^{d}m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\right]\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}$
$\displaystyle\leq-
C_{L}\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}.$

Altogether we yield for the Lyapunov function

$\displaystyle\frac{\textup{d}}{\textup{d}t}L(t)=-\mathcal{B}(t)+\mathcal{I}(t)\leq\mathcal{I}(t)\leq-
C_{L}\int_{\Omega}\mathbf{w}^{T}(t,\mathbf{x})\mathbf{w}(t,\mathbf{x})\exp(\mu(\mathbf{x}))\,\textup{d}\mathbf{x}=-C_{L}L(t).$

Applying Gronwall’s Lemma gives the claimed exponential decay and thus the
proof is complete. $\square$

######  Remark 3.1.

We want to remark that the obtained result can be extended to the exponential
stability in the corresponding $H^{s}$-norm. One just has to add other
appropriate terms in the definition of the Lyapunov function. The $H^{s}$-norm
of a function $u\in H^{s}(\Omega)$ is given by

$\|u\|_{H^{s}(\Omega)}:=\left(\sum_{|\alpha|\leq
s}\|\partial^{\alpha}u\|^{2}_{2}\right)^{\frac{1}{2}}.$

For vector-valued functions the norm is interpreted component wise. Now we
define the Lyapunov function accordingly

$\displaystyle L(t):=\sum_{|\alpha|\leq
s}\left\|\exp\left(\frac{1}{2}\mu^{(\alpha)}\right)\partial^{\alpha}\mathbf{u}\right\|^{2}_{2}.$
(3.2)

Note that by introducing the $\exp\left(\mu^{(\alpha)}\right)$ we want to
highlight that in principal it is possible to choose suited Lyapunov
potentials for every occurring multi-index $|\alpha|\leq s$. Once the Lyapunov
function is introduced the strategy is the same and due to the similar
structure inherited by the governing PDE (2.1) the estimates can be adopted
from the previous proof. We exemplary want to give the Lyapunov function in
the case $H^{1}$

$\displaystyle L(t)$
$\displaystyle:=\int_{\Omega}\mathbf{w}(t,\mathbf{x})^{T}\mathbf{w}(t,\mathbf{x})\exp\left(\mu^{(0)}(\mathbf{x})\right)\,\textup{d}\mathbf{x}$
$\displaystyle+\sum_{k=1}^{d}\int_{\Omega}\left(\partial_{x_{k}}\mathbf{w}(t,\mathbf{x})\right)^{T}\left(\partial_{x_{k}}\mathbf{w}(t,\mathbf{x})\right)\exp\left(\mu^{(k)}(\mathbf{x})\right)\,\textup{d}\mathbf{x}$

######  Remark 3.2.

During the proof we encounter the term (3.1) and in particular treat the
situation when it is strictly positive. It is further stated that under the
given assumptions, namely (2.7), it can simply be estimated by zero in the
case that it is less or equal to zero. However, there may be situations when
the estimate of the volume integral may benefit from this term. In the case
that the reminder (3.1) should be taken into account we introduce

$\mathcal{R}(\mathbf{x}):=\sum_{k=1}^{d}\frac{\partial}{\partial
x_{k}}\mathbf{A}^{(k)}(\mathbf{x})-2\mathbf{B}^{sym}(\mathbf{x})$

and demand

$\displaystyle\mathbf{A}(\mathbf{m}):=C\mathbf{Id}+\mathcal{R}(\mathbf{x})+\sum_{k=1}^{d}m_{k}(\mathbf{x})\mathbf{A}^{(k)}(\mathbf{x})\leq
0,\,C\in\mathbb{R}_{>0}.$ (3.3)

The LMI (3.3) then replaces (2.7).

######  Remark 3.3.

The inequality for the (transformed) control $\tilde{u}_{i}$ can be simplified
under additional assumptions.

  1. (i)

A possible simplifying assumption on the controls is that they are uniform in
space, i.e. $\tilde{u}_{i}=\tilde{u}_{i}(t)$. This leads to the following
condition for the controls

$\displaystyle-\sum_{i=1}^{n}\tilde{u}_{i}(t)^{2}\int_{\Gamma_{i}^{-}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}\leq\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}v_{i}(t,\mathbf{x})^{2}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}.$

  2. (ii)

Further the same control may be applied to all components which leads to
$\tilde{u}_{i}\equiv\tilde{u}(t,\mathbf{x}),\,i=1,\dots,n$. This leads to the
following condition for the controls

$\displaystyle-\sum_{i=1}^{n}\int_{\Gamma^{-}_{i}}\tilde{u}(t,\mathbf{x})^{2}\lambda(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}\leq\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}v_{i}(t,\mathbf{x})^{2}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}.$

  3. (iii)

Assume that both previous assumptions hold, i.e.
$ßtilde{u}_{k}\equiv\tilde{u}(t),\,i=1,\dots,n$. Then we can give an explicit
condition for the feedback control as

$\displaystyle\tilde{u}(t)^{2}\leq-\left(\sum_{i=1}^{n}\int_{\Gamma^{-}_{i}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}\right)^{-1}\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}v_{i}(t,\mathbf{x})^{2}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}.$
(3.4)

Thus a possible control would be

$\displaystyle\tilde{u}(t)=C\sqrt{-\left(\sum_{i=1}^{n}\int_{\Gamma^{-}_{i}}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}\right)^{-1}\sum_{i=1}^{n}\int_{\Gamma_{i}^{+}}v_{i}(t,\mathbf{x})^{2}\lambda_{i}(\mathbf{x},\mathbf{n}(\mathbf{x}))e^{\mu(\mathbf{x})}\,\textup{d}\mathbf{x}}$
(3.5)

for $C\in[-1,1]$.

##  4 Application to Hyperbolic Systems

In this section we discuss the application of the stabilization result. As a
first example we want to mention that recent results have shown the
applicability of this approach to Hamilton-Jacobi equations, see [1, 35]. The
study on the stabilization of Hamilton–Jacobi equation has been motivated by a
stabilization problem arising in a forming process and we refer to the afore
mentioned literature for further details.
We now want to study the general case and show that the linearization matches
(2.1). Let us consider the following hyperbolic system

$\displaystyle\begin{dcases}\frac{\partial}{\partial
t}\mathbf{U}(t,\mathbf{x})+\mathbf{F}(\mathbf{x},\mathbf{U}(t,\mathbf{x}),\mathbf{U}_{x_{1}}(t,\mathbf{x}),\dots,\mathbf{U}_{x_{d}}(t,\mathbf{x}))&=0,\;(t,\mathbf{x})\in(0,T)\times\Omega,\\\
\mathbf{U}(0,\mathbf{x})&=\mathbf{U}_{0}(\mathbf{x}),\;\mathbf{x}\in\Omega,\\\
\mathbf{U}(t,\mathbf{x})&=\mathbf{U}_{BC}(t,\mathbf{x}),\;(t,\mathbf{x})\in[0,T)\times\partial\Omega\end{dcases}$
(4.1)

Here we have introduced the function

$\displaystyle\mathbf{F}:\mathbb{R}^{d}\times\underbrace{\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}}_{d+1}\to\mathbb{R}^{n}$
$\displaystyle(\mathbf{x},\mathbf{u},\mathbf{p}_{1},\dots,\mathbf{p}_{d})\mapsto\mathbf{F}(\mathbf{x},\mathbf{u},\mathbf{p}_{1},\dots,\mathbf{p}_{d})\equiv\mathbf{F}(\mathbf{x},\mathbf{u},\mathbf{p}).$

We want to linearize system (4.1) near a steady state solution
$\overline{\mathbf{U}}(\mathbf{x})$ of (4.1). Thus we have to linearize
$\mathbf{F}$ near a reference state
$(\overline{\mathbf{u}},\overline{\mathbf{p}})\equiv(\overline{\mathbf{u}},\overline{\mathbf{p}}_{1},\dots,\overline{\mathbf{p}}_{d})$

$\displaystyle\mathbf{F}(\mathbf{x},\mathbf{u},\mathbf{p})$
$\displaystyle\approx\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})+\underbrace{\frac{\partial}{\partial\mathbf{u}}\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})}_{=:\mathbf{B}(\mathbf{x})}(\mathbf{u}-\overline{\mathbf{u}})+\frac{\partial}{\partial\mathbf{p}}\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})(\mathbf{p}-\overline{\mathbf{p}})$
$\displaystyle=\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})+\mathbf{B}(\mathbf{x})(\mathbf{u}-\overline{\mathbf{u}})+\sum_{k=1}^{d}\underbrace{\frac{\partial}{\partial\mathbf{p}_{k}}\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})}_{=:\mathbf{A}^{(k)}(\mathbf{x})}(\mathbf{p}_{k}-\overline{\mathbf{p}}_{k})$
$\displaystyle=\mathbf{F}(\mathbf{x},\overline{\mathbf{u}},\overline{\mathbf{p}})+\mathbf{B}(\mathbf{x})(\mathbf{u}-\overline{\mathbf{u}})+\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})(\mathbf{p}_{k}-\overline{\mathbf{p}}_{k}).$
(4.2)

We now consider deviations of this desired state

$\mathbf{U}(t,\mathbf{x})=\mathbf{\overline{U}}(\mathbf{x})+\mathbf{w}(t,\mathbf{x})$

and for some initial disturbance we assume
$\mathbf{w}(0,\mathbf{x})=(w_{1}(0,\mathbf{x}),\dots,w_{n}(0,\mathbf{x}))\neq
0$. By the following computation we obtain a linear system using (4.2)
starting from (4.1)

$\displaystyle 0$ $\displaystyle=\frac{\partial}{\partial
t}\mathbf{U}(t,\mathbf{x})+\mathbf{F}(\mathbf{x},\mathbf{U}(t,\mathbf{x}),\mathbf{U}_{x_{1}}(t,\mathbf{x}),\dots,\mathbf{U}_{x_{d}}(t,\mathbf{x}))$
$\displaystyle\approx\frac{\partial}{\partial
t}\mathbf{U}(t,\mathbf{x})+\mathbf{F}(\mathbf{x},\overline{\mathbf{U}}(\mathbf{x}),\overline{\mathbf{U}}_{x_{1}}(\mathbf{x}),\dots,\overline{\mathbf{U}}_{x_{d}}(\mathbf{x}))+\mathbf{B}(\mathbf{x})(\mathbf{U}(t,\mathbf{x})-\overline{\mathbf{U}}(\mathbf{x}))$
$\displaystyle+\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})(\mathbf{U}_{x_{k}}(t,\mathbf{x})-\overline{\mathbf{U}}_{x_{k}}(\mathbf{x}))$
$\displaystyle=\frac{\partial}{\partial
t}(\mathbf{\overline{U}}(\mathbf{x})+\mathbf{w}(t,\mathbf{x}))+\mathbf{F}(\mathbf{x},\overline{\mathbf{U}}(\mathbf{x}),\overline{\mathbf{U}}_{x_{1}}(\mathbf{x}),\dots,\overline{\mathbf{U}}_{x_{d}}(\mathbf{x}))+\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})$
$\displaystyle+\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})\frac{\partial}{\partial
x_{k}}\mathbf{w}(t,\mathbf{x})$ $\displaystyle=\frac{\partial}{\partial
t}\mathbf{w}(t,\mathbf{x})+\sum_{k=1}^{d}\mathbf{A}^{(k)}(\mathbf{x})\frac{\partial}{\partial
x_{k}}\mathbf{w}(t,\mathbf{x})+\mathbf{B}(\mathbf{x})\mathbf{w}(t,\mathbf{x})$

Thus the linearized system has the desired form to apply the above results,
highlighting the wide range of applicability of the present work.

###  4.1 Barotropic Euler Equations

The barotropic Euler system (isothermal or isentropic) is of huge interest in
the literature. The system reads in two dimensions

$\displaystyle\begin{split}\frac{\partial}{\partial
t}\rho+\nabla_{x}\cdot\left(\rho\mathbf{v}\right)&=0,\\\
\frac{\partial}{\partial t}(\rho v_{i})+\nabla_{x}\cdot\left(\rho
v_{i}\mathbf{v}+p\mathbf{e}^{(i)}\right)&=0,\;i\in\\{1,2\\}.\end{split}$ (4.3)

The fluxes are given by

$\displaystyle\mathbf{F}^{(1)}(\mathbf{U})=\begin{pmatrix}\rho v_{1}\\\ \rho
v_{1}^{2}+p\\\ \rho
v_{1}v_{2}\end{pmatrix},\quad\mathbf{F}^{(2)}(\mathbf{U})=\begin{pmatrix}\rho
v_{2}\\\ \rho v_{1}v_{2}\\\ \rho v_{2}^{2}+p\end{pmatrix}.$ (4.4)

Thus we can write the system in the following conservative form

$\frac{\partial}{\partial t}\mathbf{U}+\frac{\partial}{\partial
x_{1}}\mathbf{F}^{(1)}(\mathbf{U})+\frac{\partial}{\partial
x_{2}}\mathbf{F}^{(1)}(\mathbf{U})=0$

and hence obtain the following linearization

$\frac{\partial}{\partial
t}\mathbf{w}+\mathbf{A}^{(1)}(\overline{\mathbf{U}})\frac{\partial}{\partial
x_{1}}\mathbf{w}+\mathbf{A}^{(2)}(\overline{\mathbf{U}})\frac{\partial}{\partial
x_{2}}\mathbf{w}=0.$

Now we could apply the obtained results to the linearized conservative system.
However, since we assume some regularity of the solution and the deviation, it
is beneficial to use a suited set of primitive variables. Here we will use
$(\rho,\mathbf{v})$ which results in the following systems, cf. [50, 48],

$\displaystyle\begin{split}\frac{\partial}{\partial
t}\rho+\mathbf{v}\cdot\nabla_{x}\rho+\rho\nabla_{x}\cdot\mathbf{v}&=0,\\\
\frac{\partial}{\partial
t}v_{i}+\mathbf{v}\cdot\nabla_{x}v_{i}+\frac{a^{2}}{\rho}\frac{\partial}{\partial
x_{i}}\rho&=0,\;i\in\\{1,2\\}.\end{split}$ (4.5)

The Jacobians are given by

$\displaystyle\mathbf{A}^{(1)}=\begin{pmatrix}v_{1}&\rho&0\\\
\frac{a^{2}}{\rho}&v_{1}&0\\\
0&0&v_{1}\end{pmatrix}\quad\text{and}\quad\mathbf{A}^{(2)}=\begin{pmatrix}v_{2}&0&\rho\\\
0&v_{2}&0\\\ \frac{a^{2}}{\rho}&0&v_{2}\end{pmatrix}.$ (4.6)

When linearizing the system we simply fix the Jacobians at the desired
reference state. We want to study the half-space problem for the linearized
Euler equations in $\Omega:=\\{x_{1}>0\\}\times\mathbb{R}$. Thus we have the
boundary $\partial\Omega=\\{x_{1}=0\\}$ with the outward pointing unit normal
$\mathbf{n}=(-1,0)$. It is now possible to transform the system such that
$\mathbf{A}^{(1)}$ is a diagonal matrix. We consider the reference state
$\overline{\rho}\in\mathbb{R}_{>0},\overline{\mathbf{v}}\in\mathbb{R}^{2}\setminus\\{\mathbf{0}\\}$.
The Jacobians of the transformed system read

$\displaystyle\overline{\mathbf{A}}^{(1)}=\begin{pmatrix}\overline{v}_{1}-a&0&0\\\
0&\overline{v}_{1}&0\\\
0&0&\overline{v}_{1}+a\end{pmatrix}\quad\text{and}\quad\overline{\mathbf{A}}^{(2)}=\begin{pmatrix}\overline{v}_{2}&\dfrac{\overline{\rho}}{2}&0\\\
\dfrac{a^{2}}{\overline{\rho}}&\overline{v}_{2}&\dfrac{a^{2}}{\overline{\rho}}\\\
0&\dfrac{\overline{\rho}}{2}&\overline{v}_{2}\end{pmatrix}.$ (4.7)

The linearized system can be symmetrized by applying the following variable
transformation $\mathbf{w}\to\mathbf{w}^{\ast}=(r,\mathbf{u})^{T}$, see also
[26],

$\rho\to r:=\frac{a}{\overline{\rho}}\rho,\quad v_{1}\to
u_{1}:=\frac{v_{1}}{\sqrt{2}},\quad v_{2}\to
u_{2}:=\frac{a}{\overline{\rho}}v_{2}.$

We hence obtain

$\frac{\partial}{\partial
t}\mathbf{w}^{\ast}+\overline{\mathbf{A}}^{(1)}\frac{\partial}{\partial
x_{1}}\mathbf{w}^{\ast}+\overline{\mathbf{B}}^{(2)}\frac{\partial}{\partial
x_{2}}\mathbf{w}^{\ast}=0$

with the Jacobian

$\displaystyle\overline{\mathbf{B}}^{(2)}=\begin{pmatrix}\overline{v}_{2}&\dfrac{a}{\sqrt{2}}&0\\\
\dfrac{a}{\sqrt{2}}&\overline{v}_{2}&\dfrac{a}{\sqrt{2}}\\\
0&\dfrac{a}{\sqrt{2}}&\overline{v}_{2}\end{pmatrix}.$ (4.8)

We now discuss the LMI

$\displaystyle
C_{L}\mathbf{Id}+m_{1}\overline{\mathbf{A}}^{(1)}+m_{2}\overline{\mathbf{B}}^{(2)}\leq
0$ (4.9)

which, as stated before, is equivalent to

$\displaystyle\mathbf{A}(\overline{\mathbf{m}})$
$\displaystyle:=-\mathbf{Id}+\overline{m}_{1}\overline{\mathbf{A}}^{(1)}+\overline{m}_{2}\overline{\mathbf{B}}^{(2)}\geq
0,$ $\displaystyle\mathbf{A}(\overline{\mathbf{m}})$
$\displaystyle=\begin{pmatrix}-1+\overline{m}_{1}(\overline{v}_{1}-a)+\overline{m}_{2}\overline{v}_{2}&\overline{m}_{2}\dfrac{a}{\sqrt{2}}&0\\\
\overline{m}_{2}\dfrac{a}{\sqrt{2}}&-1+\overline{m}_{1}\overline{v}_{1}+\overline{m}_{2}\overline{v}_{2}&\overline{m}_{2}\dfrac{a}{\sqrt{2}}\\\
0&\overline{m}_{2}\dfrac{a}{\sqrt{2}}&-1+\overline{m}_{1}(\overline{v}_{1}+a)+\overline{m}_{2}\overline{v}_{2}\end{pmatrix}.$

Note that we have

$\displaystyle\begin{cases}-1+\overline{m}_{1}(\overline{v}_{1}-a)+\overline{m}_{2}\overline{v}_{2}<-1+\overline{m}_{1}\overline{v}_{1}+\overline{m}_{2}\overline{v}_{2}<-1+\overline{m}_{1}(\overline{v}_{1}+a)+\overline{m}_{2}\overline{v}_{2},\;\overline{m}_{1}a>0,\\\
-1+\overline{m}_{1}(\overline{v}_{1}-a)+\overline{m}_{2}\overline{v}_{2}>-1+\overline{m}_{1}\overline{v}_{1}+\overline{m}_{2}\overline{v}_{2}>-1+\overline{m}_{1}(\overline{v}_{1}+a)+\overline{m}_{2}\overline{v}_{2},\;\overline{m}_{1}a<0.\end{cases}$

Now we determine $\overline{\mathbf{m}}$ such that we have

$\displaystyle\begin{cases}-1+\overline{m}_{1}(\overline{v}_{1}-a)+\overline{m}_{2}\overline{v}_{2}=\sqrt{2}|\overline{m}_{2}a|>0,\;\overline{m}_{1}a>0,\\\
-1+\overline{m}_{1}(\overline{v}_{1}+a)+\overline{m}_{2}\overline{v}_{2}=\sqrt{2}|\overline{m}_{2}a|>0,\;\overline{m}_{1}a<0.\end{cases}$

With this choice we have positive diagonal entries and a strongly diagonal
dominant matrix and hence the matrix $\mathbf{A}(\overline{\mathbf{m}})$ is
positive definite. Therefore $-\mathbf{A}(\overline{\mathbf{m}})$ is negative
definite and with the choice $\mathbf{m}=-C_{L}\overline{\mathbf{m}}$ the LMI
(4.9) holds and we further set

$\mu(\mathbf{x})=m_{1}x_{1}+m_{2}x_{2}.$

The treatment of the boundary integral is also rather simple since we have

$\displaystyle\mathbf{A}^{\ast}(\nu):=\nu_{1}\overline{\mathbf{A}}^{(1)}+\nu_{2}\overline{\mathbf{B}}^{(2)}\quad\text{and}\quad\mathbf{A}^{\ast}(\mathbf{n})=-\overline{\mathbf{A}}^{(1)}.$

Hence it is also straight forward to determine the boundary control.

##  5 Summary

A novel Lyapunov function for $L^{2}-$control of multi–dimensional systems of
hyperbolic equations has been presented. We want to emphasize that up to our
best knowledge this is the first result on stabilizing boundary feedback
control applied to general multi–dimensional hyperbolic systems. The analysis
relies on the fact that the system can be diagonalized on the boundary and
based on the proper Lyapunov function an estimate involving the symmetric
Jacobians can be established. A stabilizing feedback control has been derived
and exponential decay of a weighted $L^{2}-$norm has been proven. Illustrating
examples have been presented to show the applicability of the presented
result. Future work concerns numerical studies, the application to complex
domains, the investigation and classification of problems which posess a
feasible Lyapunov potential. It is of further interest to study whether this
approach can also be generalized to the semilinear or the quasilinear case.

### Acknowledgments

This research is part of the DFG SPP 2183 _Eigenschaftsgeregelte
Umformprozesse_ , project 424334423.
M.H. thanks the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) for the financial support through 525842915/SPP2410,
525853336/SPP2410, 320021702/GRK2326, 333849990/IRTG-2379, CRC1481,
423615040/SPP1962, 462234017, 461365406, ERS SFDdM035 and under Germany’s
Excellence Strategy EXC-2023 Internet of Production 390621612 and under the
Excellence Strategy of the Federal Government and the Länder.
F.T. thanks the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) for the financial support through 525939417/SPP2410.

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