text:
\section*{Introduction}

Systems of hyperbolic partial differential equations model fluid flow, chemotaxis  and viscoplastic material dynamics~\cite{O1,Leveque,Kac1974,Inelasticity}.
Boundary stabilization of these problems has been  studied intensively in the past years~\cite{O1,O2}.
An underlying tool for the study of these problems are Lyapunov functions that yield upper bounds on the deviation from steady states in suitable norms.
The virtue of this approach are control rules that  do not require the solution of the whole system, but  take  only measurements at the boundaries into account.
So-called~\emph{dissipative} boundary conditions~\cite{L1,L2,L4,L5}
ensure exponential decay of a continuous Lyapunov function, which in turn guarantees that the solution  converges exponentially fast to a desired steady state.

More precisely, a general theory for the stabilization of linear conservation laws  with respect to the~$L^2$-norm  is available~\cite[Sec.~3]{O1}.
For nonlinear systems, however, results are still partial. A problem is posed by the fact that Lyapunov's indirect method~\cite{Khalil} does not necessarily hold for hyperbolic systems. Furthermore,  solutions to systems of conservation laws exist in the classical sense only for a finite time due to formation of shocks~\cite{Riemann1859}.
To this end, stability results are typically stated in terms of the Sobolev ${H^2}$-norm or in the $C^1$-norm~\cite{L5,L4,Hayat2019,Hu2019} and
restrictive smoothness assumptions on the ${H^2}$-norm of the initial data may be needed~\cite[Sec.~4]{O1}.
Furthermore,
most analytical results are based on the assumption that the influence of the source term is  small or in intuitive terms, the
considered balance laws are viewed as perturbations of conservation laws~\cite{L1}.
On the other hand, if the destabilizing effect of the source term is sufficiently large, the system may be not stabilizable~\cite{Bastin2011,Gugat2019,GugatHerty2020}.

Recently, interest has increased in studying the stabilizability of semilinear hyperbolic systems, when the advection part is linear, but the source term is nonlinear~\cite[Sec.~10]{Petit}.
In particular, analytical results are available for semilinear Euler equations~\cite{GugatHerty2020,Gugat2021,GugatSemiLinear2022}.
Assumptions on initial data are typically imposed with respect to the ${H^1}$-Sobolev norm and, hence, are  less restrictive than for  general nonlinear systems.
The assumption of a Lipschitz continuous source term even allows to establish estimates in terms of the $L^2$-norm~\cite{Prieur2018,Hayat2021}, which is desirable, since also discontinuous initial and boundary data can be treated. Analytical results for general semilinear systems, however, will always come along with restrictions on initial data. In particular, initial data must be sufficiently close to a steady state, where the distance is measured in a suitable norm, for instance in terms of the~$L^2$-~or~$H^1$-norm in space. Otherwise, a blow up of the solution in finite time may occur~\cite[Sec.~2]{Prieur2018}.

These restrictions motivate a computational approach.
Indeed, the development and analysis of numerical schemes that preserve continuous
stability results is an active field of research. In particular,  first-order upwind  discretizations of linear balance laws are used to construct discretized Lyapunov functions that decay exponentially  fast~\cite{D2,Gerster2019,Weldegiyorgis2020,Weldegiyorgis2021}. Moreover, a second-order scheme applied to scalar nonlinear conservation laws with dissipative feedback boundary conditions is analyzed in~\cite{Dus2022}. \\

\noindent
The main contribution of this work is a computational framework that is specifically taylored to semilinear boundary value problems
without any smoothness assumptions on initial data.
In contrast to most analytical results in the~$H^1$-setting, we measure the distance to a desired state by the  $L^2$-norm and use a Lyapunov function as upper bound, which allows to consider discontinuous solutions.
Since no systematic procedure exists for deriving Lyapunov functions in this setting, we use a candidate Lyapunov function that is typically applied in the  linear case~\cite[Sec.~3]{O1}.  We parameterize it  up to a constant~$\mub>0$ that is computed numerically.
Two approaches are presented and analyzed for determining an appropriate constant. Those are based on a weighted Rayleigh coefficient and eigenvalue estimates.
The computational framework  is consistent with existing theoretical results, but also allows to investigate numerically problems which are beyond the current state of research on analytical control rules.
The proposed method is based on high-order CWENO reconstructions~\cite{Semplice2019,Visconti2018,Cravero2018,Elena} that are  high-order accurate in smooth regions, but can  resolve discontinuities in an essentially nonoscillatory~(ENO) fashion.
CWENO reconstructions  consist of weighted combinations of local reconstructions on different stencils.
Furthermore, they also allow reconstructing  source terms and  boundary conditions at high order, a crucial feature when solving semilinear boundary value problems for balance laws. \\

\noindent
This paper is structured as follows. Section~\ref{SecSemiLinear} reviews semilinear hyperbolic boundary value problems. Section~\ref{SectionConditions} is devoted to their control.
In particular, Lyapunov functions and their estimates as well as benchmark problems are introduced.
Section~\ref{ComputationalFramework} describes the computational framework which is based on high-order  CWENO discretizations.
Finally, numerical results are presented in Section~\ref{SecNumericalResults}.

\section{Semilinear hyperbolic boundary value problems}\label{SecSemiLinear}

We consider semilinear hyperbolic balance laws of the form
\begin{equation}\label{SemiLinear}
\partial_t \u(t,x)
+
A(x) \partial_x \u(t,x)
=-
h\big(\u(t,x);x\big)
\end{equation}
that  are defined on a finite space interval~$[0,L]$.
We consider~$2 \times 2$ systems, where the advection part~$A(x)=T(x) \Lambda(x) T^{-1}(x) $ is diagonalizable with distinct eigenvalues
$$
\Lambda^-(x)
<0<
\Lambda^+(x)
\quad
\text{for all}
\quad
x\in[0,L]
\quad
\text{and}
\quad
\Lambda(x)
=
\diag\big\{
\Lambda^+(x),
\Lambda^-(x)
\big\}.
$$
Under the assumptions
$
A(x)\in C^2\big(
[0,L];\mathbb{R}^{2\times 2}
\big)
$
and
$
h\in C^2\big(
\mathbb{R}^2 \times [0,L];\mathbb{R}^2
\big)
$
the semilinear system~\eqref{SemiLinear} admits a classical smooth solution provided that initial data are differentiable~\cite{BRESSAN,O1}. Hence, it can be equivalently written in \textbf{Riemann invariants}
$
\R(t,x)
\coloneqq
T^{-1}(x)
\u(t,x)
$
satisfying
\begin{equation}\label{DiagonalizedSystem}
\begin{aligned}
&\partial_t \R(t,x)
+
\Lambda(x)
\partial_x \R(t,x)
=-
G\big( \R(t,x);x \big) \\
&\text{for}
\quad
G\big( \R(t,x);x \big)
=
T^{-1}(x)
h\big(
T(x)\R(t,x);x
\big)
+
T^{-1}(x)\Lambda(x) \partial_x T(x)  \R(t,x).
\end{aligned}
\end{equation}
The diagonalized system~\eqref{DiagonalizedSystem} is endowed with possibly nonlinear feedback boundary conditions~$
B\in C^2\big(
\mathbb{R}^2;\mathbb{R}^2
\big)
$.
The \textbf{initial boundary value problem (IBVP)} with  initial values $\IR\in L^2\big( (0,L); \mathbb{R}^2 \big)$, which satisfy boundary conditions,  reads as
\begin{align}
\partial_t \R(t,x) + \Lambda(x) \partial_x \R(t,x) &=  -G\big( \R(t,x);x \big)
& & \text{for} \ \ t \in (0,T), \ x \in (0,L),  \label{CauchyProblem1} \\
\begin{pmatrix} \R^+(t,0) \\ \R^-(t,L) \end{pmatrix}
&=
{B}
\begin{pmatrix} \R^+(t,L) \\ \R^-(t,0) \end{pmatrix}
& & \text{for} \ \ t \in [0,T),  \label{CauchyProblem2} \\
\R(0,x)&= \IR(x)
& & \text{for} \ \ x \in [0,L].
\label{CauchyProblem3}
\end{align}
Riemann invariants that come along with positive  speeds~$\lambda^+(x)>0$ are denoted as~$\R^+(t,x)$ and those with negative characteristic speeds as~$\R^-(t,x)$, respectively.
Typical examples, see~e.g.~\cite[Sec.~1.11]{O1}, are the
\textbf{Kac-Goldstein equations}, which explain the spatial pattern formations in chemosensitive populations.  The unknowns~$\u=(\rho,q)^\T$ are the  density~$\rho=\R^++\R^-$ and the mass flux~$q=\gamma(\R^+-\R^-)$
of right~{($\R^+$)} and left-moving~{($\R^-$)} cells,
where  the velocity of cell motion is described by the parameter~$\gamma>0$
and
$
\ST(\R^+,\R^-)
$
is a turning function.
\begin{align*}
&\textbf{Kac-Goldstein equations}
&
&\ \ \textbf{diagonalized form} \\
&
\begin{aligned}
\rho_t
+
q_x
&=
0, \\
q_t
+
\gamma^2
\rho_x
&=
2\ST\bigg(
\frac{\rho}{2}+\frac{q}{2\gamma},
\frac{\rho}{2}-\frac{q}{2\gamma}
\bigg)q
\end{aligned}
& & \ \
\begin{aligned}
\R^+_t
+
\gamma \R^+_x
&=
\ST(\R^+,\R^-) (\R^+ - \R^-), \\
\R^-_t
-
\gamma
\R^-_x
&=
\ST(\R^+,\R^-) (\R^- - \R^+)
\end{aligned}
\end{align*}

\noindent
\textbf{Steady states} are denoted by  $\bar{\u}=(\bar{\rho},\bar{q})^\T$, ~$\bar{\R}=(\bar{\R}^+,\bar{\R}^-)^\T$. Those satisfy the conditions~$\partial_t \bar{\u}(x)=\partial_t \u(t,x)=0$,
$\partial_t \bar{\R}(x)=\partial_t \R(t,x)=0$
 and are typically space-dependent. In the special case of Kac-Goldstein equations, however, the steady states are constant and read as
$$
\bar{\R}^\pm
=
\frac{1}{2}
\int_0^L
\rho(0,x) \d x.
$$
\textbf{Boundary conditions} are specified by
\begin{equation} \label{BCKAC}
\begin{pmatrix} \R^+(t,0) \\ \R^-(t,L) \end{pmatrix}
=
\begin{pmatrix}
& \hspace{-2mm}\kappa \\ \kappa
\end{pmatrix}
\begin{pmatrix} \R^+(t,L) \\ \R^-(t,0) \end{pmatrix}
\qquad
\text{for}
\qquad
|\kappa| \in [0,1].
\end{equation}
In particular the choice~$\kappa=1$ models the case, when cells are confined within  the spatial domain,
since it holds~$q(t,0)=0$ and~$q(t,L)=0$.

\noindent
According to~\cite[Th.~10.1]{Petit},  there exists  the following wellposedness result.
Provided that initial data are sufficiently smooth and close to a steady state, i.e.
\begin{equation}\label{IVsmooth}
\text{there exists}
\quad
\delta>0
\quad
\text{such that}
\quad
\big\lVert
\u_0 - \bar{\u}
\big\rVert_{H^1\big((0,L);\mathbb{R}^{2}  \big)}
<\delta,
\end{equation}
the IBVP~\eqref{CauchyProblem1} -- \eqref{CauchyProblem3}
has a unique maximal classical solution satisfying
$$
\u
\in
C^0\Big(
[0,T);
H^1\big( (0,L);\mathbb{R}^{2} \big)
\Big).
$$
Under the assumption~\eqref{IVsmooth}, there are
conditions  available,  see~e.g.~\cite[Th.~10.2]{Petit}, that stabilize the dynamics at a steady state.
This assumption, however, is relatively restrictive for hyperbolic systems, which may involve discontinuous solutions, e.g.~in the case of time-dependent boundary controls and for initial perturbations that are  away from a steady state.
To this end, we consider in Section~\ref{SectionConditions} stabilization concepts with respect to the $L^2$-norm, which are typically applied to linear systems~\cite[Sec.~5]{O1}, and use them to establish a computational framework for semilinear problems~in~Section~\ref{ComputationalFramework}.

\section{Sufficient conditions for stability}\label{SectionConditions}

We are interested in  boundary controls that make the system converge exponentially fast to a steady state. As  in~\cite[Ch.~10]{Petit}, we introduce the distance to this steady state by~$
\Riemann(t,x) \coloneqq
\R(t,x)-\bar{\R}(x)
$. Then, the IBVP~\eqref{CauchyProblem1} -- \eqref{CauchyProblem3} reads as
\begin{align}
	\partial_t \Riemann(t,x) + \Lambda(x) \partial_x \Riemann(t,x) &=-  \mathcal{G}\big( \Riemann(t,x);x \big)
	& & \text{for} \ \ t \in (0,T), \ x \in (0,L),  \label{IBVP1} \\

	\begin{pmatrix} \Riemann^+(t,0) \\ \Riemann^-(t,L) \end{pmatrix}
	&=
	\mathcal{B}
	\begin{pmatrix} \Riemann^+(t,L) \\ \Riemann^-(t,0) \end{pmatrix}
	& & \text{for} \ \ t \in [0,T),  \label{IBVP2} \\

	\Riemann(0,x)&= \IR(x)-\bar{\R}(x)
	& & \text{for} \ \ x \in [0,L]. \label{IBVP3}
\end{align}
The source term and the boundary conditions are defined by
\begin{align*}
	\mathcal{G}\big( \Riemann;x \big)
	&\coloneqq
	G\big( \Riemann + \bar{\R}(x) ;x \big)
	-
	G\big( \bar{\R}(x) ;x \big), \label{SourceTerm}  \\

	\mathcal{B}
	\begin{pmatrix} \Riemann^+(t,L) \\ \Riemann^-(t,0) \end{pmatrix}
	&\coloneqq
	{B}
	\begin{pmatrix} \Riemann^+(t,L)+\bar{R}^+(L) \\ \Riemann^-(t,0)+\bar{R}^-(0) \end{pmatrix}
	-
	\begin{pmatrix}
		\bar{R}^+(0) \\
		\bar{R}^-(L)
	\end{pmatrix}.
\end{align*}

\noindent
To specify boundary conditions, we
introduce a Lyapunov function candidate
\begin{equation}\label{analyticalLF}
	\L(t;\mub) \coloneqq \intL \Riemann(t,x)^{\T} W(x;\mub) \Riemann(t,x) \d x
\end{equation}
with the weights~$W(x;\mub)\coloneqq \diag \big\{ W^+(x;\mub),W^-(x;\mub)  \big\}$ that are defined by
\begin{equation}\label{Weights}
	\begin{aligned}
		W^+(x;\mub)
		&\coloneqq
		\frac{ 1 }{\Lambda^+(x)}
		\textup{exp} \bigg(
		-\mub \int_0^x
		\frac{1}{\Lambda^+(s)} \d s
		\bigg), \\

		W^-(x;\mub)
		&\coloneqq
		\frac{ 1 }{|\Lambda^-(x)|}
		\textup{exp} \bigg(
		\mub \int_x^L
		\frac{1}{\Lambda^-(s)} \d s
		\bigg).
	\end{aligned}
\end{equation}

\begin{remark}
		As described in~\cite[Sec.~3.5]{O1} and as illustrated in Figure~\ref{FigureNetwork},  the $2\times 2$~system~{\eqref{CauchyProblem1}~--~\eqref{CauchyProblem3}} is extendable to a network with   $n$~arcs  by specifying appropriate coupling conditions.
		Then, the weights~\eqref{Weights} must be replaced by those in~\cite[Th.~3.16]{O1}.
		In the sequel, this article is concerned with~$2 \times 2$ systems.
	\begin{figure}[H]

\tikzstyle{block} = [thick,draw, fill=black!10, rectangle,
minimum height=3em, minimum width=3em]
\tikzstyle{Kreis} = [thick,draw, fill=blue!20, circle,
minimum height=3em, minimum width=3em]
\tikzstyle{sum} = [draw, fill=darkgreen!20, rectangle,
minimum height=2em, minimum width=3em]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{pinstyle} = [ pin edge={to-,thick,black}]
\tikzstyle{vecArrow} = [thick,double distance=23pt]

\hspace*{-8mm}
\begin{tikzpicture}[scale=0.082]

	\draw [vecArrow] (5,0) -- (50,0);
	\draw [vecArrow] (145,15) -- (80,2.6);
	\draw [vecArrow] (145,-15) -- (80,-2.6);

	\node [block,align=center] at (70,0) {\textbf{coupling conditions}\\
		$
		\begin{pmatrix}\color{blue} \R^+(t,0) \\ \color{red} \R^-(t,L) \end{pmatrix}
		=
		B
		\begin{pmatrix} \color{blue} \R^+(t,L) \\ \color{red} \R^-(t,0) \end{pmatrix}
		$
	};

	\node at (25,0) {arc 1};
	\node at (120,10.5) {arc 2};
	\node at (120,-10.5) {arc $n$};

	\node at (15.9,9) {\color{blue}$\R^{+,(1)}(t,x)$};
	\node at (28,-9) {\color{red}$\R^{-,(1)}(t,x)$};

	\node at (118,-21) {\color{blue}$\R^{+,(n)}(t,x)$};
	\node at (143,3.5) {\color{red}$\R^{-,(2)}(t,x)$};

	\node at (118,21) {\color{blue}$\R^{+,(2)}(t,x)$};
	\node at (143,-3.5) {\color{red}$\R^{-,(n)}(t,x)$};

	\node at (68,16) {\color{blue}$
		{
			\R^+= \Big(\R^{+,(1)},\ldots,\R^{+,(n)} \Big)^\T
		}$};

	\node at (68,-16) {\color{red}$
		{
			\R^-= \Big(\R^{-,(1)},\ldots,\R^{-,(n)} \Big)^\T
		}
		$};

	\draw [ultra thick, blue, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}] (	28,9) -- (38,9);
	\draw [ultra thick, red, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}] (15,-9) -- (5,-9);

	\draw [ultra thick, blue, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}]  (132,-21.4) -- (145,-23.5);

	\draw [ultra thick, red, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}] (130,-3.8) -- (120,-2);

	\draw [ultra thick, red, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}] (130,3.8) -- (120,2);

	\draw [ultra thick, blue, decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}]  (132,21.4) -- (145,23.5);

\end{tikzpicture}

		\caption{Network with~$n$ arcs and coupling conditions~$B:\mathbb{R}^{2n}\rightarrow \mathbb{R}^{2n}$.}
		\label{FigureNetwork}
	\end{figure}
\end{remark}

\noindent
We introduce the notations~$
W^\pm_b\coloneqq W^\pm(b,\mub)$,
$
\Lambda^\pm_b\coloneqq \Lambda^\pm(b)$,
	$
	\Lambda_b  \coloneqq
	\diag\big\{
	\Lambda_b^+, \Lambda_b^-
	\big\}
	$,
	$
	W_b  \coloneqq
	\diag\big\{
	W_b^+, W_b^-
	\big\}
	$,
$\Riemann^\pm_b(t)\coloneqq \Riemann^\pm(t,b)$ with ${b\in\{0,L\}}$
and, for now, we assume
\begin{equation} \label{ToStrongRegularity}
	\Riemann \in C^1\Big([0,\infty) \times [0,L];\mathbb{R}^{2}\Big).
\end{equation}
According to~\cite[Sec.~10.2]{Petit},
there exists a matrix~$\bar{\mathcal{G}}$ satisfying~$
\mathcal{G}(\Riemann;x)
=
\bar{\mathcal{G}}(\Riemann;x)
\Riemann
$ and
$
\bar{\mathcal{G}}(0;x)
=
\D_{\Riemann} \mathcal{G}(\Riemann;x) \big|_{\Riemann=0}
$.
This allows to define the matrices
\begin{align*}
	\Mtilde(\Riemann;x,\mub)
	&\coloneqq
	\mub\,
	W(x;\mub)
	+
	W(x;\mub)\bar{\mathcal{G}}(\Riemann;x)
	+
	\bar{\mathcal{G}}(\Riemann;x)^\T W(x;\mub),\\

	\Mequivalent(\Riemann;x,\mub)
	& \coloneqq
	W^{-\nicefrac{1}{2}}(x,\mub)
	\Mtilde(\Riemann;x,\mub)
	W^{-\nicefrac{1}{2}}(x,\mub) , \\
	\H\big(\Riemann^+_L,\Riemann^-_0;\mub\big)
	&\coloneqq
	\mathcal{B}
	\begin{pmatrix} \Riemann^+_L \\ \Riemann^-_0 \end{pmatrix}^{\T}
	\begin{pmatrix}
		W^+_0(\mub)\Lambda^+_0 \\ &\hspace{-6mm}  W^-_L(\mub) |\Lambda^-_L|
	\end{pmatrix}
	\mathcal{B}
	\begin{pmatrix} \Riemann^+_L \\ \Riemann^-_0 \end{pmatrix} \\

	&\quad \, -
	\begin{pmatrix} \Riemann^+_L \\ \Riemann^-_0 \end{pmatrix}^{\T}
	\begin{pmatrix}
		W^+_L(\mub)\Lambda^+_L \\ &\hspace{-6mm}  W^-_0(\mub)|\Lambda^-_0|
	\end{pmatrix}
	\begin{pmatrix} \Riemann^+_L \\ \Riemann^-_0 \end{pmatrix}.
\end{align*}
Then, the time derivative of the Lyapunov function is
\begin{align}
	\L'(t;\mub)
	=& \hspace{4.6mm}
	2 \intL \Riemann(t,x)^{\T}W(x;\mub) \partial_t\Riemann(t,x) \d x \nonumber \\
	=&
	-2 \intL \Riemann(t,x)^{\T}W(x;\mub)\Lambda(x) \partial_x\Riemann(t,x) \,
	+  \Riemann(t,x)^{\T}W(x;\mub)\mathcal{G}(\Riemann;x) \d x \nonumber \\
	=&
	\intL
	\Riemann(t,x)^{\T} \Big[
	\partial_x \big( W(x;\mub)\Lambda(x) \big)
	-
	2\, W(x;\mub)\bar{\mathcal{G}}(\Riemann;x)
	\Big]
	\Riemann(t,x)
	\d x \nonumber \\
	&- \Big[
	\Riemann(t,L)^{\T} W_L(\mub)\Lambda_L \Riemann(t,L)
	-
	\Riemann(t,0)^{\T} W_0(\mub)\Lambda_0 \Riemann(t,0)
	\Big] \nonumber \\
	=&
	-\intL\Riemann(t,x)^\T \Mtilde\big(\Riemann(t,x);x,\mub\big)\Riemann(t,x) \d x
	+
	\H\big(\Riemann^+_L,\Riemann^-_0;\mub\big).
	\label{DerivativeLyapunov}
\end{align}

\noindent
We observe from~equation~\eqref{DerivativeLyapunov} that a \emph{sufficient, but not necessary} condition to make the Lyapunov function decay exponentially fast is to choose boundary conditions and a parameter~$\mub\geq 0$ such that
\begin{align}
	&\text{the inequality }
	\H\big(\Riemann^+_L,\Riemann^-_0;\mub\big) \leq 0
	\text{ holds and }
	\label{A2}\tag{\textup{A1}} \\

	&\text{the matrix }
	\Mtilde(\Riemann;x,\mub)
	\text{ is strictly positive definite for all }
	x\in[0,L]. \label{A1tilde}\tag{$\textup{A2}$}
\end{align}
Furthermore, we introduce the~\textbf{weighted Rayleigh quotient}
\begin{equation}\label{Rayleigh}
	\begin{aligned}
		\Rayleigh\big[
		\Riemann,\mub
		\big](t)
		&\coloneqq
		\big\lVert \Riemann^{(\mub)} \big\rVert_{L^2}^{-2}
		\left\langle \Riemann^{(\mub)}, \Mequivalent (\Riemann;\cdot,\mub) \Riemann^{(\mub)} \right\rangle_{L^2}
		\quad \ \text{for}\\

		\Riemann^{(\mub)}(t,x)
		&\coloneqq
		W(x;\mub)^{\nicefrac{1}{2}}  \Riemann(t,x)

	\end{aligned}
\end{equation}
where the~$L^2$-norm satisfies~$
\big\lVert \Riemann^{(\mub)} \big\rVert_{L^2}^{2}
\coloneqq
\big\lVert \Riemann^{(\mub)} \big\rVert_{L^2( (0,L);\mathbb{R}^{2} )}^{2}
=
\L(t;\mub)
$.
Then,
the time derivative~\eqref{DerivativeLyapunov}  fulfills
\begin{equation*
	\L'(t;\mub)
	\leq
	-
	\Rayleigh\big[
	\Riemann,\mub
	\big](t)
	\L(t;\mub)
\end{equation*}
provided that~assumption~\eqref{A2} holds.
If the weighted Rayleigh quotient~\eqref{Rayleigh} remains strictly positive, i.e.
\begin{equation} \label{A1}\tag{\textup{A3}}
	\Rayleigh\big[
	\Riemann,\mub
	\big](t)
	>\mu>0
	\quad
	\text{for all}
	\quad
	t \geq 0,
\end{equation}
the solution converges to the steady state exponentially fast. More precisely,
the norm equivalence~$
\big\lVert \Riemann(t,x) \big\rVert_2^2
\sim
\big\lVert \Riemann(t,x) \big\rVert_{W(x)}^2
\coloneqq
\Riemann(t,x)^\T W(x) \Riemann(t,x)
$ and the estimate~$\L'(t)\leq - \mu\L(t) $ imply for a \emph{fixed} parameter~$\mub \geq 0$ the bound
$$
\big\lVert
\Riemann(t,\cdot)
\big\rVert_{
	L^2
}
\lesssim
e^{-\mu t}
\big\lVert
\Riemann(0,\cdot)
\big\rVert_{
	L^2
}
\quad
\text{for all}
\quad
t \in \mathbb{R}^+_0.
$$

\begin{remark}\label{Remark1}
	The conditions~\eqref{A2}~--~\eqref{A1} are coupled by the parameter~$\mub\geq 0$, which enters the weights of the Lyapunov function. More precisely,
	it has been shown in~\cite[Th.~2.3.5]{StephansDiss} for linear boundary controls, imposed by a matrix~$\bar{\mathcal{B}} \in\mathbb{R}^{2n\times 2n}$,
	that the inequality~\eqref{A2} is satisfied if  the condition
	\begin{equation}\label{StephansDiss}
		\textup{exp} \left(  \frac{\mub L}{2 \lambda_{\min} }  \right)
		\big\lVert \bar{\mathcal{B}} \big\rVert_2
		\leq
		1

		\quad\text{holds for}\quad

		\lambda_{\min} \coloneqq

		\min_{x \in [0,L]}
		\Big\{ \big| \Lambda^\pm(x) \big| \Big\}.
	\end{equation}
	Hence, a \emph{small} value of~$\mub\geq 0$ is desirable. On the other hand, a \emph{large} value may be necessary to make the conditions~\eqref{A1tilde} and~\eqref{A1} hold. This ambiguity reflects the fact that some systems are not even stabilizable unless the length~$L>0$ is sufficiently small~\cite{Bastin2011,Gugat2019}.
\end{remark}

	\begin{remark}[{Stabilization of~$H^1$-solutions according to~\cite[Th.~10.2]{Petit}}]
		Provided that initial data are sufficiently close to a steady state, i.e.~condition~\eqref{IVsmooth} is satisfied,
		exponential stability holds with respect to the $H^1$-norm if the matrix
		$$

		\mub\,
		W(x;\mub)
		+
		W(x;\mub) \boldsymbol{\bar{\mathcal{G}}}(x)
		+
		\boldsymbol{ \bar{\mathcal{G}}}(x)^\T W(x;\mub)
		$$
		is strictly positive definite for all~$x\in[0,L]$ and the matrix
		$$

		\boldsymbol{\bar{\mathcal{B}}}^{\T}
		\begin{pmatrix}
			W^+_0(\mub) \Lambda^+_0 \\ & W^-_L(\mub) \big| \Lambda^-_L \big|
		\end{pmatrix}
		\boldsymbol{\bar{\mathcal{B}}}
		-
		\begin{pmatrix}
			W^+_L(\mub) \Lambda^+_L \\ & W^-_L(\mub) \big| \Lambda^-_L \big|
		\end{pmatrix}
		$$
		is negative semidefinite,
		where~$
		\boldsymbol{\bar{\mathcal{B}}}
		=
		\D_{\Riemann} \mathcal{B}(\Riemann) \big|_{\Riemann=0}
		$
		and
		$

		\boldsymbol{\bar{\mathcal{G}}}(x)
		=
		\bar{\mathcal{G}}(0;x)
		$
		denote linearizations at steady  state.

		Hence, assumptions~\eqref{A2} and~\eqref{A1tilde} are \emph{more restrictive} as they must be satisfied also apart from the steady state.

	\end{remark}

	\noindent
	The stabilization concept with respect to the $L^2$-norm, which we follow in this work, does not serve as a general stability result.  However, it
	comes along with a computational framework that allows for an efficient numerical verification of the conditions~\eqref{A2}~--~\eqref{A1} and hence  allows to investigate numerically problems where initial data may vary widely from steady states.

	It is justified analytically  in the special cases considered in Section~\ref{SubSectionI} and  Section~\ref{SecLipschitz}.

\subsection{Linearized case}\label{SubSectionI}
In the linear case, when the source term and the boundary conditions are given by matrices, i.e.~$\bar{\mathcal{G}}\in\mathbb{R}^{2n\times 2n}$ and
$
\bar{\mathcal{B}}\in\mathbb{R}^{2n\times 2n}
$,
the Lyapunov function decays exponentially fast if the the matrix~$\Mlinear(x,\mub)=\Mtilde(\Riemann;x,\mub)$
is strictly positive definite for all~$x\in[0,L]$ and the matrix
\begin{equation}\label{HLyapunov}
	\Hlinear(\mub) \coloneqq
	\bar{\mathcal{B}}^{\T}
	\begin{pmatrix}
		W^+_0(\mub) \Lambda^+_0 \\ & W^-_L(\mub) \big| \Lambda^-_L \big|
	\end{pmatrix}
	\bar{\mathcal{B}}
	-
	\begin{pmatrix}
		W^+_L(\mub) \Lambda^+_L \\ & W^-_L(\mub) \big| \Lambda^-_L \big|
	\end{pmatrix}
\end{equation}
is negative semidefinite.
More precisely, the derivative~\eqref{DerivativeLyapunov} is estimated by
\begin{align*}
	&\L'(t;\mub)
	\leq
	-\intL  \Riemann(t,x)^\T \Mlinear(x;\mub) \Riemann(t,x) \d x
	\leq
	-\mu \L(t;\mub)
	\quad
	\Leftrightarrow
	\quad
	\L(t) \leq e^{-\mu t} \L(0) \\

	&
	\text{with decay rate}\quad
	\mu \coloneqq \min_{x \in [0,L]}\Big\{ \smallestEV \Big\{ W^{-\nicefrac{1}{2}}(x;\mub) \Mlinear(x;\mub)W^{-\nicefrac{1}{2}}(x;\mub)  \Big\} \Big\}>0,
\end{align*}
where~$\smallestEV$ denotes the smallest eigenvalue of a matrix.
An example, used in the following as a benchmark problem, is as follows:

\begin{proposition}\label{Corollary1}
	The linear boundary value problem
	$$
	\begin{aligned}
		\partial_t
		\begin{pmatrix} \Riemann^+(t,x) \\ \Riemann^-(t,x) \end{pmatrix}
		+
		\partial_x
		\begin{pmatrix} \hspace{2.6mm}\Riemann^+(t,x) \\ -\Riemann^-(t,x) \end{pmatrix}
		&=
		\frac{\ST(x)}{2} \begin{pmatrix}
			\hspace{2.5mm} 1 & \hspace{-2.5mm}-1 \\ -1 & 1
		\end{pmatrix}
		\begin{pmatrix} \Riemann^+(t,x) \\ \Riemann^-(t,x) \end{pmatrix},\\

		\begin{pmatrix} \Riemann^+(t,0) \\ \Riemann^-(t,L) \end{pmatrix}
		&=
		\begin{pmatrix}
			&\hspace{-2mm} \kappa \\ \kappa
		\end{pmatrix}
		\begin{pmatrix} \Riemann^+(t,L) \\ \Riemann^-(t,0) \end{pmatrix}
	\end{aligned}
	$$
	is exponentially stable provided that the inequality
	\begin{equation}\label{ConditionStable}
		\Big|
		\kappa \ln\big(|\kappa|\big)
		\Big| >{  \alpha L}
		\quad\text{holds for}\quad
		\alpha \coloneqq \max\limits_{x\in[0,L]}
		\Big\{ \big|
		\ST(x)
		\big|  \Big\}
		\quad\text{and}\quad
		|\kappa|\in(0,1).
	\end{equation}
	Furthermore, there exists a value~$|\kappa|\in(0,1)$ in the case~$\alpha L<\nicefrac{1}{e}$.
\end{proposition}

\begin{proof}
	According to~\cite[Th.~2.3.5]{StephansDiss}, the matrix~\eqref{HLyapunov} is negative semidefinite for
	$$
	1\geq \textup{exp} \Big( \frac{\mub L}{2}  \Big)
	\big\lVert \bar{\mathcal{B}} \big\rVert_2
	=
	\textup{exp} \Big( \frac{\mub L}{2}  \Big) |\kappa|
	\quad\Leftrightarrow\quad
	\mub \leq \bar{\mu}\coloneqq
	-\frac{2}{L} \ln\big( {|\kappa|} \big),
	\quad
	|\kappa|\in(0,1).
	$$
	\noindent
	The choice~$\mub=\bar{\mu}$ yields the weights
	$$
	w^+(x;\bar{\mu})
	=
	e^{-\bar{\mu} x}
	=
	\kappa^{\frac{2x}{L}}
	\quad\text{and}\quad
	w^-(x;\bar{\mu})
	=
	e^{-\bar{\mu} (L-x)}
	=
	\kappa^{\frac{2(L-x)}{L}}.
	$$
	Furthermore, the maximum of the sum~$w^+(x;\bar{\mu})^p+w^-(x;\bar{\mu})^p$, $p\in\mathbb{N}$ is obtained at~$x=\nicefrac{L}{2}$, which yields the upper  bounds~$w^\pm(x;\bar{\mu})\leq 1$,~$
	w^+(x;\bar{\mu})+w^-(x;\bar{\mu})\leq 2|\kappa|
	$ and~$w^+(x;\bar{\mu})^2+w^-(x;\bar{\mu})^2\leq 2\kappa^2
	$.
	This gives the  eigenvalue estimate
	\begin{align}
		&\begin{aligned}
			\smallestEV\big\{
			\Mlinear(x;\bar{\mu})
			\big\}
			&\geq
			\bar{\mu} W(x;\bar{\mu})
			-

			\largestEV
			\Big\{
			W(x;\bar{\mu})\bar{\mathcal{G}}(x)+\bar{\mathcal{G}}(x)^{\T}W(x;\bar{\mu})
			\Big\}
			\nonumber \\

			&\geq
			\bar{\mu} W(x;\bar{\mu})
			-
			\alph
			\sqrt{ \frac{ w^+(x;\bar{\mu})^2+w^-(x;\bar{\mu})^2}{2} }
			+\alpha
			\frac{ w^+(x;\bar{\mu})+w^-(x;\bar{\mu})}{2}
		\end{aligned}
		\nonumber \\

		&\Longrightarrow \quad
		\min_{x \in [0,L]}\Big\{
		\smallestEV\big\{
		\Mlinear(x;\bar{\mu})
		\big\}
		\Big\}
		\geq
		-\frac{2}{L} \ln\big(|\kappa|\big) \kappa^2
		- 2\alpha |\kappa|, \label{BoundStable}
	\end{align}
	where~$\largestEV$ denotes the spectral radius. The bound~\eqref{BoundStable} is strictly positive if the condition~\eqref{ConditionStable} holds.

\end{proof}

\subsection{Semilinear case with Lipschitz continuous source term}\label{SecLipschitz}
Similarly to~\cite{Hayat2021}, we consider Lipschitz continuous source terms.
Then, results of the linearized case can be partially extended
as shown in the following proposition.

\begin{proposition}\label{Corollary2}
	The semilinear boundary value problem
	$$
	\begin{aligned}
		\partial_t \Riemann(t,x)
		+
		\Lambda(x)
		\partial_x \Riemann(t,x)
		&=-
		\mathcal{G} \big( \Riemann(t,x);x \big) \\

		\begin{pmatrix} \Riemann^+(t,0) \\ \Riemann^-(t,L) \end{pmatrix}
		&=
		\begin{pmatrix}
			&\hspace{-2mm} \kappa \\ \kappa
		\end{pmatrix}
		\begin{pmatrix} \Riemann^+(t,L) \\ \Riemann^-(t,0) \end{pmatrix}
	\end{aligned}
	$$
	with Lipschitz continuous source term
	$$
	\big\lVert
	\mathcal{G}(\Riemann;x)
	\big\rVert_2
	\leq C_{\mathcal{G}} \big\lVert
	\Riemann
	\big\rVert_2
	\quad\text{for all}\quad
	x\in[0,L]
	$$
	is exponentially stable provided that 	$\big|\kappa \ln(|\kappa|)\big|
		>
		L C_{\mathcal{G}}$  and~$|\kappa|\in(0,1)$ holds.
\end{proposition}

\begin{proof}
		As shown in~Proposition~\ref{Corollary1},   the boundary conditions~\eqref{HLyapunov} require the bound ${
		\mub \leq \bar{\mu}\coloneqq
		-\nicefrac{2}{L} \ln\big( {|\kappa|} \big).
	}$
	Hence, the weights read as~$w^{\pm}(x;\bar{\mu})
	=
	\kappa^{\frac{2x}{L}}
	\in
	\big[ \kappa^2,1 \big]
	$
	and the Lipschitz continuity of the source term implies
	$$
	\left\lVert
	\mathcal{G}\big(\Riemann(t,x);x\big)
	\right\rVert_{W(x)}

	\leq
	C_{\mathcal{G}}
	\big\lVert
	\Riemann(t,x)
	\big\rVert_{2}
	\leq
	\frac{ C_{\mathcal{G}} }{|\kappa|}
	\big\lVert
	\Riemann(t,x)
	\big\rVert_{W(x)}.
	$$
	Then, the claim follows from the assumption~$\big|\kappa \ln(|\kappa|)\big|
		>
		L C_{\mathcal{G}}$ and the estimate
		\begin{align*}
			\Riemann(t,x)^\T
			\Mtilde\big(\Riemann(t,x);x,\bar{\mu}\big)
			\Riemann(t,x)
			&=
			\mub\,
			\big\lVert \Riemann(t,x) \big\rVert_{W(x)}
			+2
			\left\langle
			\Riemann(t,x),
			\mathcal{G}\big(\Riemann(t,x);x\big)
			\right\rangle_{W(x)} \\

			&\geq
			\left[\bar{\mu}- \frac{ 2C_{\mathcal{G}} }{|\kappa|}  \right]
			\big\lVert
			\Riemann(t,x)
			\big\rVert_{W(x)}^2
			>0
			\ \ \text{for}\ \
			\Riemann(t,x)\neq 0.
	\end{align*}
\end{proof}

\noindent
Finally, we~remark that the regularity assumption~\eqref{ToStrongRegularity}, which has been used to deduce the previous results, can be stated in terms of~$L^2$-solutions for general linear balance laws~\cite[Sec.~2.1.3]{O1} and for  semilinear systems with locally Lipschitz continuous source term~\cite[Th.~1]{Prieur2018}. The following computational framework is based on these $L^2$-solutions, i.e.~$
\u
\in
C^0\left(
[0,T);
L^2\big( (0,L);\mathbb{R}^{2n} \big)
\right)
$,
which allow to consider discontinuities and initial data that may vary widely from steady states.

\section{Computational framework}\label{ComputationalFramework}
Since analytical results are \emph{in general} not available for   semilinear systems, we introduce a computational framework that  is based on a central, weighted, essentially non-oscillatory (CWENO) reconstruction.  The aim is to find a control law such that there exists a parameter~$\mub\geq 0$  that satisfies the conditions~\eqref{A2},~\eqref{A1tilde} and~\eqref{A1}, respectively.

\subsection{High-order discretization inside the spatial domain}\label{SecCWENO}
A desirable numerical scheme should approximate at high-order not only the semilinear system~\eqref{IBVP1} with space-depending source term and the boundary conditions~\eqref{IBVP2}, but also the conditions~~\eqref{A2},~\eqref{A1tilde} and~\eqref{A1}.
To this end, we use a finite-volume based CWENO reconstruction.
The spatial domain~$[0,L]$ is divided into $N$~cells~$\mathbb{C}_j\coloneqq\big[x_{j-\nicefrac{1}{2}}, x_{j+\nicefrac{1}{2}} \big]$ for ${j=1,\ldots,N}$ by a space discretization~${\Delta x > 0}$ satisfying ${\Delta x N = L}$. The cell centers are~${x_j \coloneqq (j-\nicefrac{1}{2}) \Delta x}$ and the  cell edges are~${x_{j+{\nicefrac{1}{2}}} \coloneqq j \Delta x}$.
The evolution of cell averages
$$
\Rbar_j(t)\coloneqq
\frac{1}{\Delta x}
\int\limits_{x_{j-{\nicefrac{1}{2}}}}^{x_{j+{\nicefrac{1}{2}}}}
\Riemann(t,x)
\d x
$$
for a general balance law
${\Riemann_t + \tilde{f}(\Riemann)_x=-\tilde{\mathcal{G}}(\Riemann;x)}$
is described by the ordinary differential equation
\begin{equation*}
	\frac{\textup{d}}{\textup{d} t}\Rbar_j(t)=
	- \frac{1}{\Delta x} \bigg[ \tilde{f}\Big( \Riemann(t,x_{j+\nicefrac{1}{2}}) \Big) - \tilde{f}\Big( \Riemann(t,x_{j-\nicefrac{1}{2}})\Big) \bigg]
	-
	\frac{1}{\Delta x}
	\int\limits_{x_{j-{\nicefrac{1}{2}}}}^{x_{j+{\nicefrac{1}{2}}}}
	\tilde{\mathcal{G}}\Big( \Riemann(t,x);x \Big)
	\d x .
\end{equation*}

\noindent
Here, the linear PDE~\eqref{IBVP1} is written in conservative form by defining
$$
\tilde{f}\Big( \Riemann(t,x) \Big)
\coloneqq
\Lambda(x)  \Riemann(t,x)
\quad
\text{and}
\quad
\tilde{\mathcal{G}}\Big( \Riemann(t,x);x \Big)
\coloneqq
\mathcal{G}\Big( \Riemann(t,x);x \Big)
-
\partial_x
\Lambda(x)  \Riemann(t,x).
$$
Furthermore, the  central, weighted, essentially non-oscillatory (CWENO) reconstruction from~\cite{Visconti2018} is applied for the interior cells~$j=2,\ldots,N-1$.
A third-order reconstruction is of the form
\begin{equation}\label{CWENO}
	\textup{CWENO}
	\ : \
	\big[ \Rbar_{j-1},\Rbar_{j},\Rbar_{j+1} \big]
	\ \mapsto \
	\poly_j(x),
\end{equation}
where $\poly_j(x)$ denotes a  reconstruction polynomial defined for~$x\in\mathbb{C}_j$.
The reconstruction for the semi-discretization at the right ($r$) side of a cell interface is denoted by~${
	\caSD^r_{j-\nicefrac{1}{2}}(t)
	\coloneqq
	\poly_j(x_{j-\nicefrac{1}{2}})
}$,
at the left~($\ell$) side by~${
	\caSD^\ell_{j+\nicefrac{1}{2}}(t)
	\coloneqq
	\poly_j(x_{j+\nicefrac{1}{2}})
}$ and at the cell center~($c$) by~${
	\caSD^c_{j}(t)
	\coloneqq
	\poly_j(x_{j})
}$.
The source term is discretized by the Gauss-Lobatto rule with three quadrature nodes.
Then, the resulting semi-discretization of the balance law with the upwind flux~$\NumFlux$ reads as
\begin{align*}
	\frac{\textup{d}}{\textup{d} t}\Rbar_j(t)=
	&- \frac{1}{\Delta x} \bigg[ \NumFlux \Big( \caSD^\ell_{j+\nicefrac{1}{2}}(t),\caSD^r_{j+\nicefrac{1}{2}}(t) \Big)
	-
	\NumFlux\Big(
	\caSD^\ell_{j-\nicefrac{1}{2}}(t),\caSD^r_{j-\nicefrac{1}{2}}(t)
	\Big)  \Bigg] \\
	&-
	\frac{1}{6}
	\Big[
	\mathcal{G} \big( \caSD^r_{j-\nicefrac{1}{2}}(t) ;x_{j-\nicefrac{1}{2}}\big)
	+ 4
	\mathcal{G} \big( \caSD^c_{j}(t) ;x_j\big)
	+
	\mathcal{G} \big( \caSD^\ell_{j+\nicefrac{1}{2}}(t) ;x_{j+\nicefrac{1}{2}}\big)
	\Big] \\
	&+\  \mathcal{O}\big(\Delta x^3\big).
\end{align*}
It is approximated in time with a strong stability preserving (SSP) Runge-Kutta
method with three stages~\cite{Jiang1996}.

\begin{figure}[H]
	\begin{center}

\tikzstyle{block} = [thick,draw, fill=blue!20, rectangle,
minimum height=3em, minimum width=3em]
\tikzstyle{Kreis} = [thick,draw, fill=blue!20, circle,
minimum height=3em, minimum width=3em]
\tikzstyle{sum} = [draw, fill=darkgreen!20, rectangle,
minimum height=2em, minimum width=3em]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{pinstyle} = [ pin edge={to-,thick,black}]
\tikzstyle{vecArrow} = [ultra thick,double distance=60pt]

\hspace*{-1.4cm}
\begin{tikzpicture}[scale=1]

	\draw [vecArrow] (0,0) -- (4,0);
	\draw [vecArrow] (8,0) -- (12,0);

	\begin{scope}[xshift=-0.4cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=-0.5:0.5,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=blue, ultra thick] {      x^2 + 0.5*x + 0.2};
			\addplot [color=red, ultra thick] {      0.5*x^2 + -0.5*x + -1};
		\end{axis}

	\end{scope}

	\begin{scope}[xshift=7.6cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=-0.5:0.5,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=blue, ultra thick] { -x^2 + 0.5*x + 0.6};
			\addplot [color=red, ultra thick] { 1*x^2 - 0.2*x - 1.2};
		\end{axis}

	\end{scope}

	\filldraw[fill=gruen!20]  (4,0.5) -- (4,2) -- (8,2) -- (8,0.5);
	\filldraw[fill=black!10]  (4,0.5) -- (4,-1.3) -- (8,-1.3) -- (8,0.5);
	\filldraw[fill=gruen!20]  (0,+1.1) -- (0,+2) -- (4,+2) -- (4,+1.1);
	\filldraw[fill=gruen!20]  (8,+1.1) -- (8,+2) -- (12,+2) -- (12,+1.1);

	\node [align=center] at (6,-0.4) {\textbf{coupling conditions}\\
		$
		\begin{pmatrix} \Riemann^+_0 \\ \Riemann^-_L \end{pmatrix}
		=
		\mathcal{B}_t
		\begin{pmatrix} \Riemann^+_L \\ \Riemann^-_0 \end{pmatrix}
		$};

	\node [align=center] at (6,1.26) {\textbf{verification of}\\
		$
		\H\big(\Riemann^+_L,\Riemann^-_0;\mut(t)\big) \leq 0
		$};

	\node [align=center] at (2,1.5) {\textbf{verify }
		\eqref{A1tilde}~or~\eqref{A1}};

	\node [align=center] at (1.8,0.8) {\color{blue} $\Riemann^+(t,x)$ };
	\node [align=center] at (1.8,-0.3) {\color{red} $\Riemann^-(t,x)$ };

	\node [align=center] at (10.8,0.5) {\color{blue} $\Riemann^+(t,x)$ };
	\node [align=center] at (9,-0.3) {\color{red} $\Riemann^-(t,x)$ };

	\node [align=center] at (10,1.5) {\textbf{verify }
		\eqref{A1tilde}~or~\eqref{A1}};

	\draw[thick] (4,2) -- (4,-1.3)		node[below] {$x=L$};
	\draw[thick] (8,2) -- (8,-1.3)		node[below] {$x=0$};

\end{tikzpicture}

		\caption{Continuous setting for the computation of stabilizing boundary controls~$\mathcal{B}_t$.}
		\label{FigContinuous}
	\end{center}
\end{figure}

\subsection{High-order discretization at boundaries}
Figure~\ref{FigContinuous} illustrates the continuous setting to obtain stabilizing boundary controls.
As  mentioned in~Remark~\ref{Remark1},
the parameter~$\mub\geq 0$ should be chosen as \emph{small} as possible, since a higher value restricts the choice of boundary conditions. To ensure a  given decay rate~$\mu>0$ we define the possibly time-depending parameter ${
	\mut(t) \in \big\{
	\mut_{\M}(t),\mut_{\Rayleigh}(t)
	\big\}
}$
by
\begin{equation}\label{Defmut}
	\begin{aligned}
		\mut_{\M}(t)
		&\coloneqq
		\inf
		\Big\{
		\mub>0 \ \Big| \
		\smallestEV\left\{
		\M\big(\Riemann(t,x);x;\mub\big)
		\right\}
		\geq \mu>0 \text{ for all }x\in[0,L]
		\Big\}, \\

		\mut_{\Rayleigh}(t)
		&\coloneqq
		\inf
		\Big\{
		\mub>0 \ \Big| \
		\Rayleigh\big[
		\Riemann,\mub
		\big](t)
		\geq
		\mu> 0
		\Big\}.

	\end{aligned}
\end{equation}

\noindent
Since this parameter enters the weights $W(x;\mut(t))$ of the Lyapunov function, the condition~$\H\big(\Riemann^+_L,\Riemann^-_0;\mut(t)\big)\leq 0$ is time-dependent as well. This may require also time-dependent coupling conditons~$\mathcal{B}_t$.

\begin{figure}[H]
	\begin{center}

\tikzstyle{block} = [thick,draw, fill=blue!20, rectangle,
minimum height=3em, minimum width=3em]
\tikzstyle{Kreis} = [thick,draw, fill=blue!20, circle,
minimum height=3em, minimum width=3em]
\tikzstyle{sum} = [draw, fill=darkgreen!20, rectangle,
minimum height=2em, minimum width=3em]
\tikzstyle{input} = [coordinate]
\tikzstyle{output} = [coordinate]
\tikzstyle{pinstyle} = [ pin edge={to-,thick,black}]
\tikzstyle{vecArrow} = [ultra thick,double distance=60pt]

\hspace*{-7mm}
\begin{tikzpicture}[scale=1]

	\draw [vecArrow] (0,0) -- (3.5,0);
	\draw [vecArrow] (8.5,0) -- (12,0);

	\begin{scope}[xshift=-0.4cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=-0.25:0.45,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			xmin=-0.5,xmax=0.5,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=black, ultra thick, dashed] {      1*x^2 + 0.1*x + 0.45};
			\addplot [color=black, ultra thick,dashed] {      0.6*x^2 + -0.5*x + -1};
		\end{axis}

	\end{scope}

	\begin{scope}[xshift=-0.4cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=-0.45:-0.25,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			xmin=-0.5,xmax=0.5,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=black, ultra thick, dashed] {       + 0.1*x + 0.35};
			\addplot [color=black, ultra thick,dashed] {      0.6*x^2 + 1*x + -0.6};
		\end{axis}

	\end{scope}

	\begin{scope}[xshift=7.6cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=-0.45:0.25,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			xmin=-0.5,xmax=0.5,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=black, ultra thick,dashed] { -1.5*x^2 + 0.6*x + 0.6};
			\addplot [color=black, ultra thick,dashed] { 3*(x+0.049)^2 - 0*x - 1.2};
		\end{axis}

	\end{scope}

	\begin{scope}[xshift=7.6cm,yshift=-1.7cm]

		\begin{axis}[scale=0.7,
			domain=0.25:0.45,
			xticklabels={,,},
			yticklabels={,,},
			samples=100,
			ymin=-2,ymax=2,
			xmin=-0.5,xmax=0.5,
			hide x axis,
			hide y axis,
			axis line style={draw opacity=0}],

			\addplot [color=black, ultra thick,dashed] { -1.5*x^2 + 0.6*x + 0.4};
			\addplot [color=black, ultra thick,dashed] {  0.4*x - 1.2};
		\end{axis}

	\end{scope}

	c
	\filldraw[fill=gruen!20]  (3.5,0.5) -- (3.5,2) -- (8.5,2) -- (8.5,0.5);
	\filldraw[fill=black!10]  (3.5,0.5) -- (3.5,-1.3) -- (8.5,-1.3) -- (8.5,0.5);
	\filldraw[fill=gruen!20]  (0,+1.1) -- (0,+2) -- (3.5,+2) -- (3.5,+1.1);
	\filldraw[fill=gruen!20]  (8.5,+1.1) -- (8.5,+2) -- (12,+2) -- (12,+1.1);

	\node [align=center] at (6,-0.4) {\textbf{coupling conditions}\\
		$
		\begin{pmatrix} \Riemann^+_0 \\ \Riemann^-_L \end{pmatrix}
		=
		\mathcal{B}_t
		\begin{pmatrix}{ \color{blue} \caSD^{+,\ell}_{N+\nicefrac{1}{2}}} \\ {\color{red} \caSD^{-,r}_{\nicefrac{1}{2}} } \end{pmatrix}
		$};

	\node [align=center] at (6,1.26) {\textbf{verification of}\\
		$
		\H\Big({\color{blue} \caSD^{+,\ell}_{N+\nicefrac{1}{2}}},{\color{red} \caSD^{-,r}_{\nicefrac{1}{2}} }, \mud(t) \Big)\leq 0
		$};

	\node [align=center] at (2,1.5) {\textbf{compute}
		$\mud(t)$};

	\node [align=center] at (2.5,0.1) {\color{blue} $\caSD^{+,\ell}_{N+\nicefrac{1}{2}}$ };

	\node [align=center] at (11.8,0) {\color{black} $\poly_{2}(x)$ };
	\node [align=center] at (0.1,0) {\color{black} $\poly_{N-1}(x)$ };

	\node [align=center] at (1.4,-0.3) {\color{black} $\poly_N(x)$ };
	\node [align=center] at (10,0.5) {\color{black} $\poly_1(x)$ };

	\node [align=center] at (10,-0.47) {\color{red} $\caSD^{-,r}_{\nicefrac{1}{2}}$ };

	\draw[thick,red,decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}]  (9.4,-0.47) -- (8.6,-0.47);

	\draw[thick,blue,decoration={markings,mark=at position 1 with	{\arrow[scale=2,>=stealth]{>}}},postaction={decorate}]  (2.8,0.4) -- (3.4,0.78);

	\node [align=center] at (10,1.5) {\textbf{compute}
		$\mud(t)$};

	\draw[thick] (3.5,2) -- (3.5,-1.3)		node[below] {$x=L$};
	\draw[thick] (8.5,2) -- (8.5,-1.3)		node[below] {$x=0$};
	\draw[thick] (0.8,1.1) -- (0.8,-1.3)		node[below] {\ \; \quad $x=L-\Delta x$};
	\draw[thick] (11.2,1.1) -- (11.2,-1.3)		node[below] {$x=\Delta x$  \quad};

\end{tikzpicture}

		\caption{Semi-discrete setting for the computation of stabilizing boundary controls~$\mathcal{B}_t$.}
		\label{FigDiscrete}
	\end{center}
\end{figure}

\noindent
Figure~\ref{FigDiscrete} illustrates the discretized setting. Therein, the reconstruction polynomials~$\poly_j(x)$ are black dotted. Since the CWENO reconstruction~\eqref{CWENO} requires the central stencil~$\mathbb{C}_{j-1},\ldots,\mathbb{C}_{j+1}$ to reconstruct the polynomial~$\poly_j(x)$ it can be only used for interior cells. The cells~$\mathbb{C}_0, \mathbb{C}_N$, which are adjacent to the boundary, need a special treatment. Therein,   reconstructions of the form
\begin{alignat*}{8}
	&\textup{CWENOb}^{(r)}
	\ : \
	\big[ \Rbar_{N-2},\Rbar_{N-1},\Rbar_{N} \big]
	&&\ \mapsto \
	\poly_N(x), \\

	&\textup{CWENOb}^{(\ell)}
	\ : \
	\big[ \Rbar_{1},\Rbar_{2},\Rbar_{3} \big]
	&&\ \mapsto \
	\poly_1(x)
\end{alignat*}
are applied~that have been recently introduced by~\cite[Semplice, Travaglia,  Puppo]{Elena}.
A crucial property of these  reconstruction are polynomials that  are defined within the whole cell. This  allows to determine the expressions~\eqref{Defmut} at high order, i.e.
\begin{align*}
	\mud_{\M}(t)
	&\coloneqq
	\inf
	\Big\{
	\mub>0 \ \Big| \
	\smallestEV\left\{
	\Mtilde\big(\poly_j(x_k);x_k;\mub\big)
	\right\} \geq \mu \\

	&\qquad\quad
	\text{ for all } \ j=1,\ldots,N, \
	\text{ and }
	k\in\{j\pm\nicefrac{1}{2},j \}
	\Big\}, \\

	\mud_{\Rayleigh}(t)
	&\coloneqq
	\inf
	\bigg\{
	\mub>0 \ \bigg| \
	\sum\limits_{j=1}^N
	\RayleighQ\big[
	\poly_j,\mub
	\big](t)
	\geq
	\mu> 0
	\bigg\},

\end{align*}
where~$\RayleighQ\big[
\poly_j,\mub
\big]$ denotes the discretized weighted Rayleigh quotient~\eqref{Rayleigh} that is obtained by the Gauss-Lobatto  rule with the quadrature nodes~$
\poly_j(x_{j\pm \nicefrac{1}{2} })
$ and~$
\poly_j(x_{j })
$.

Furthermore, the boundary values~$\caSD^{+,\ell}_{N+\nicefrac{1}{2}}  $ and~$\caSD^{-,r}_{\nicefrac{1}{2}}  $ are reconstructed at high order. This is crucial to verify the condition~$\H\big( \caSD^{+,\ell}_{N+\nicefrac{1}{2}},\caSD^{-,r}_{\nicefrac{1}{2}};\mud(t) \big)\leq 0$.
The  inflow reconstructed values are given by the coupling conditions, i.e.~$
\caSD^{+,\ell}_{\nicefrac{1}{2}}
\coloneqq \Riemann_0^+$ and
$
\caSD^{-,r}_{N+\nicefrac{1}{2}}
\coloneqq \Riemann_L^-.
$

This framework guarantees  parameters~$\mud(t)$ that satisfy the properties~\eqref{A1tilde},~\eqref{A1} and specify  the condition~\eqref{A2}, which ensures dissipative boundary controls. Those can be specified problem and time-dependent, which ensures stabilizing boundary conditions. However, we note  that
	there do not necessarily exist
boundary controls with the property~\eqref{A2}.
 Systems that are not stabilizable, see~e.g.~\cite{Bastin2011,Gugat2019,GugatHerty2020}, serve as examples.
Hence, only sufficient conditions are obtained, but not necessarily a control rule that steers the system exponentially fast to a desired state.

\begin{remark}\label{RemarkII}
	The presented computational framework is consistent with the previously mentioned theoretical state of research apart from the fact that   the choices~\eqref{Defmut} lead to a numerical Lyapunov function whose parameters are time-dependent, in contrast to the analytical Lyapunov function~\eqref{analyticalLF}. Therefore, the numerical Lyapunov function may be~\emph{not strictly decreasing}, but still leads to an exponential decay of perturbations. Analytical results are recovered by setting~$\mud(t)=\mub$~for all~$t\in\mathbb{R}^+_0$.

	Furthermore, the parameter~$\mud_{\Rayleigh}(t)$, which is based on the weighted Rayleigh quotient, leads to a weaker estimate of the parameter~$\mub$ that enters the Lyapunov function, i.e.~a smaller parameter~$\mub$ is obtained. Hence, the disadvantage of this choice is a control rule that may not be sufficient to establish exponential decay. On the other hand, perturbations from steady states are still damped and  the computational cost is reduced significantly. Namely, the choice~$\mud_{\M}(t)$ requires to solve an optimization problem for each point in space, whereas the choice~$\mud_{\Rayleigh}(t)$ requires the solution of only~\emph{one} optimization problem.
\end{remark}

\section{Numerical results}\label{SecNumericalResults}

The  computational framework is applicable for general systems of semilinear hyperbolic balance laws and will be illustrated by means of the Kac-Goldstein equations for chemotaxis with boundary conditions~\eqref{BCKAC}. Since these systems are in general not stabilizable~\cite{Bastin2011,Gugat2019,GugatHerty2020}, we consider the benchmark problems derived in Section~\ref{SectionConditions}.
More precisely, we consider the Kac-Goldstein equations with the following source terms:
\begin{alignat*}{8}
&\textup{Section~\ref{SecTurningI}:}\quad
&&
\mathcal{G} \big( \Riemann(t,x);x \big)
=
\frac{1}{2e}
\begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}
\Riemann(t,x) \\
&\textup{Section~\ref{SecTurningII}:}\quad
&&
\mathcal{G} \big( \Riemann(t,x);x \big)
=
\frac{1}{2e}
\cos\Big(
\Riemann^+(t,x)^2
+
\Riemann^-(t,x)^2
\Big)
\begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}
\Riemann(t,x)\\
&\textup{Section~\ref{SecTurningIII}:}\quad
&&
\mathcal{G} \big( \Riemann(t,x);x \big)
=
\frac{1}{2e}
\begin{pmatrix}
\Riemann^-(t,x) & \Riemann^+(t,x) \\
\Riemann^+(t,x) & \Riemann^-(t,x)
\end{pmatrix}
\begin{pmatrix}
1 & -1 \\
-1 & 1
\end{pmatrix}
\Riemann(t,x) \\
\end{alignat*}

\noindent
The  nonlinear source term  requires the computational approach of  Section~\ref{ComputationalFramework}. It is a relevant practical example~to model the alignment  of animals
and cells~\cite[Sec.~2]{Lutscher}. Although a small value~$|\kappa|\ll 1$ may damp perturbations faster, a large value is typically of interest. Namely,  larger values can lead to a smaller net flux at the boundary, which in turn results in a more economical control.
The discretizations
$\Delta x=2^{-5}$ and $\textup{CFL}=0.45$ for~$L=1$ are used in all simulations.  Initial conditions read as~$\Riemann^+(0,x)=\sin(\pi x)$ and~$\Riemann^-(0,x)=\cos(\pi x)$.
Furthermore, parameters for the CWENO reconstructions are chosen as $Z$-nonlinear weights~as described in~\cite{Elena}.

\subsection{Linear source term}\label{SecTurningI}
The linear source term is a special case of Proposition~\ref{Corollary1} for~${ \theta(x)=\nicefrac{1}{e}}$.
 The largest parameter for the boundary conditions that is guaranteed in Proposition~\ref{Corollary1} is~$\kappa=\nicefrac{1}{e}$, which is obtained for~$
\mub=
-\nicefrac{2}{L} \ln\big( {|\kappa|} \big)=2
$.

Figure~\ref{Corollary_1_Solution} shows the deviation~$\rho(t,x)-\bar{\rho}$ of the density for~$t\in[0,10]$. If the time-dependent parameter~$\mud_{{\M}(t)}$ is used~(left panels), the obtained boundary control is able to steer the system towards the equilibrium for both decay rates~$\mu=0.1$ and~$\mu=1$. In contrast, the parameter~$\mud_{\mathcal{Q}}(t)$, which is based on the weighted Rayleigh quotient only stabilizes the system provided that the larger desired decay rate~$\mu=1$ is used.
Figure~\ref{Corollary_1_Lyapunov}  shows the corresponding Lyapunov functions that yield upper bounds for the $L^2$-norm of deviations. More precisely, the scaled Lyapunov function
\begin{equation}\label{LFtilde}
	\widetilde{\L}(t)
	\coloneqq
	W^{-1}_{\min}(t) \L(t)
	\geq \big\lVert \Riemann \big\rVert_{L^2}
	\ \text{ for } \
	W_{\min}(t)
	\coloneqq
	\min_{x\in[0,L]} \bigg\{
	\smallestEV\Big\{
	W\big(x,\mud(t)\big)
	\Big\}
	\bigg\}
\end{equation}
is shown as blue line and serves as upper bound for the $L^2$-norm that is obtained by simulations with the parameter~$\mud_{{\M}(t)}$, which is shown in blue. They decay exponentially fast with a rate that is larger than the desired  rate~$\mu=0.1$~(left panel) and $\mu=1$~(right panel).
The $L^2$-norm for deviations that are based on the~parameter~$\mud_{{\mathcal{Q} }(t)}$ is shown in red.
Deviations even increase for the smaller desired decay rate and decrease for~$\mu=1$, however, with a lower rate.

\begin{figure}[H]

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mud_{\M}(t)}$\textbf{  and }$\boldsymbol{\mu=0.1}$
		\end{center}
		\vspace{0mm}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_muHatI_muI}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mud_{Q}(t)}$\textbf{  and }$\boldsymbol{\mu=0.1}$
		\end{center}
		\vspace{0mm}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_muHatII_muI}}

	\end{minipage}

	\bigskip\bigskip

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mud_{\M}(t)}$\textbf{  and }$\boldsymbol{\mu=1}$
		\end{center}
		\vspace{0mm}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_muHatI_muII}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mud_{\mathcal{Q}}(t)}$\textbf{  and }$\boldsymbol{\mu=1}$
		\end{center}
		\vspace{0mm}	\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_muHatII_muII}}

	\end{minipage}

	\caption{Deviations of the density to the steady state. The upper panels show a simulation with desired decay rate~$\mu=0.1$ where the parameters~${\mud_{\M}(t)}$~(left) and ${\mud_{\mathcal{Q}}(t)}$ (right) are used. The lower panels show simulations for the decay rate~$\mu=1$, respectively. }
	\label{Corollary_1_Solution}

\end{figure}

\begin{figure}[H]
	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=0.1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_Lyapunov_muI}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_Lyapunov_muII}}

	\end{minipage}

	\caption{The~$L^2$-norm~$\big\lVert \Riemann(t,\cdot) \big\rVert^2_{L^2}$
		and the scaled Lyapunov function~\eqref{LFtilde}
		obtained by simulations with the parameter~$\mud_{{\M }(t)}$ are plotted in blue. Simulations that   are based on the parameter~$\mud_{{\mathcal{Q} }(t)}$ are shown in red, respectively. The desired exponential decay is black dashed. }
	\label{Corollary_1_Lyapunov}
\end{figure}

\begin{figure}[H]
	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=0.1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_Parameter_muI}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_1_Parameter_muII}}

	\end{minipage}

	\caption{Left~$y$-axis shows the obtained boundary control~$\kappa^*(t)$ for the parameters~$\mud_{{\mathcal{Q} }(t)}$ and $\mud_{{\mathcal{M} }(t)}$ which are plotted against the right axis as dashed line.}
	\label{Corollary_1_BC}

\end{figure}

Figure~\ref{Corollary_1_BC} shows the obtained boundary control~$\kappa^*(t)$ and the parameter~$\mud(t)$, which enters the Lyapunov function.
	More precisely,
	the parameters $\mud_{{\M }(t)}$ and $\mud_{{\mathcal{Q} }(t)}$ are dashed plotted against the right~$y$-axis.
	The resulting control~$\kappa^*(t)$ is shown with respect to the left~$y$-axis. Namely, the control corresponding to the parameter~$\mud_{{\M }(t)}$
	is plotted as blue line and  that one corresponding to~$\mud_{{\mathcal{Q} }(t)}$ as red line, respectively.
We observe that simulations that are based on the weighted Rayleigh quotient yield smaller values~$
\mud_{{\mathcal{Q} }(t)}
\leq
\mud_{{\M }(t)}
$
which in turn result in a larger control parameter. Likewise, a smaller desired decay rate leads to larger control parameters. Furthermore, we note that the control based on the Rayleigh quotient, which does not steer the system to the steady state, still counteracts fluctuations. In particular, it actively reduces the control parameters when instabilities increase.
	A zoom in the left panel of Figure~\ref{Corollary_1_Lyapunov} highlights the non-monotonic behaviour of the scaled Lyapunov function and the $L^2$-norm of deviations when the control parameter~$\mud_{\mathcal{Q} }(t)$ is used. These oscillations arise from fluctuations of the control, which are illustrated in  the left panel of Figure~\ref{Corollary_1_Lyapunov}. In contrast, the~control parameter~$\mud_{\mathcal{M} }(t)$ is almost constant and the resulting Lyapunov function decays monotonically.

\subsection{Lipschitz continuous source term}\label{SecTurningII}

The distance to the equilibrium state is shown in Figure~\ref{Corollary_2_Solution} for the boundary control that is obtained by the parameters~$\mud_{\M}(t)$~(left panel) and~$\mud_{\Q}(t)$~(right panel).
As expected, the parameter~$\mud_{\M}(t)$ leads to a boundary control that steers the system faster to the steady state.

\begin{figure}[H]

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			\textbf{perturbations for }$\boldsymbol{\mud_{\M}(t)}$
		\end{center}
		\vspace{0mm}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_2_muHatI_muII}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			\textbf{perturbations for }$\boldsymbol{\mud_{\mathcal{Q}}(t)}$
		\end{center}
		\vspace{0mm}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_2_muHatII_muII}}

	\end{minipage}

	\caption{Deviations of the density to the steady state  with desired decay rate~$\mu=1$, where the parameter~${\mud_{\M}(t)}$~(left) and ${\mud_{\mathcal{Q}}(t)}$ (right) are used for the boundary control. }
	\label{Corollary_2_Solution}

\end{figure}

\begin{figure}[H]

	\begin{minipage}{0.45\textwidth}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_2_Lyapunov_muII}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_2_Parameter_muII}}

	\end{minipage}

	\caption{The left panel shows the scaled Lyapunov function~\eqref{LFtilde} and  the~$L^2$-norm  obtained by simulations with the parameter~$\mud_{{\M }(t)}$ in blue, while those corresponding to the  parameter~$\mud_{{\mathcal{Q} }(t)}$ are shown in red.  The desired decay is black dashed. The right panel states the control~$\kappa^*(t)$ at the left~$y$-axis and the corresponding parameters~$\mud_{{\Q }(t)}$ and $\mud_{{\M }(t)}$  at the right axis.}
	\label{Corollary_2_Control}

	\color{blue}

\end{figure}

The decay is shown in the left panel of Figure~\ref{Corollary_2_Control}. Therein, the $L^2$-norm of the distance to the steady state is shown as blue line for the parameter~$\mud_{\M}(t)$ and in red for the control that is based on the weighted Rayleigh quotient. The scaled~Lyapunov function  yields an upper bound on the $L^2$-norm and decays at least with the desired rate (black, dashed) if the parameter~$\mud_{\M}(t)$ is used. The~$L^2$-norm of deviations that are based on the weighted Rayleigh quotient decays more slowly, but still exponentially fast.

The right panel of Figure~\ref{Corollary_2_Control} shows the boundary control parameter in the scale of the left~$y$-axis. The weighted Rayleigh quotient leads to a smaller time-dependent value $\mud_{\Q}(t)\leq \mud_{\M}(t)$ and hence to a larger control parameter. Furthermore, the control remains  stable, while large changes in the control occur if the parameter~$\mud_{\M}(t)$ is used.

\subsection{General source term}\label{SecTurningIII}

Figure~\ref{Corollary_3_solution} shows the solution to the nonlinear source term for the decay rate~$\mu=0.1$~(left panel) and~$\mu=1$~(right panel), where the control  based on the parameter~$\mud_{{\M }(t)}$ is used. We observe that the control is able to stabilize the system. More precisely, Figure~\ref{Corollary_3_LF} shows the scaled Lyapunov function~\eqref{LFtilde} and  the $L^2$-norm of deviations.
	In this particular example,
	simulations for both $\mud_{{\M }(t)}$ and $\mud_{{\Q }(t)}$
	only slightly differ and cannot be distinguished in the plot.
We observe that the control steers the system to the equilibrium with the desired decay, which is black dashed.  However, the Lyapunov function is~\emph{not strictly decreasing}.

	Figure~\ref{Shock} considers discontinuous initial data~$\Riemann^\pm(0,x)=2\, \textup{sign}(x)$ and a control based on the weighted Rayleigh quotient, i.e.~the 	parameter~$\mud_{{\mathcal{Q} }(t)}$ is used.
	For comparison, a simulation without source term is included. The~$L^2$-norm of deviations, described by the resulting conservation law, is shown as blue line.
	The scaled Lyapunov function~\eqref{LFtilde} yields an upper bound that decays exponentially fast. The observed decay is within the desired rates~$\mu=0.1$ (left panel) and $\mu=1$ (right panel), which are plotted as dashed, black lines.
	Deviations that result from the semilinear system are shown in red. The scaled Lyapunov function, which serves as upper bound for the $L^2$-norm, decreases over time and makes the $L^2$-norm decay with an asymptotic rate that is similar to the desired decay rate.
	However, the decay is non-monotone and deviations may be larger than desired.

Hence,  observations from~Figure~\ref{Corollary_3_LF} and Figure~\ref{Shock}
reflect the fact that the computational framework is beyond the theoretical stabilization concept for~$L^2$-solutions if it is applied to nonlinear source terms.

\begin{figure}[H]
	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=0.1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_3_muHatI_muI}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_3_muHatI_muII}}

	\end{minipage}

	\caption{Deviations of the density from steady state for the nonlinear systems with the control based on the parameter~$\mud_{{\M }(t)}$.}
	\label{Corollary_3_solution}

\end{figure}

\begin{figure}[H]
	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=0.1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_3_Lyapunov_muI}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{Corollary_3_Lyapunov_muII}}

	\end{minipage}

	\caption{Deviations from steady state are stated in terms of the
		scaled Lyapunov function~\eqref{LFtilde} as dashed line, which is an upper bound of the $L^2$-norm. Here, simulations for both parameters differ in a magnitude less than~$\sim10^{-5}$.}
	\label{Corollary_3_LF}

\end{figure}

\begin{figure}[H]
	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=0.1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{SHOCK_I}}
	\end{minipage}
	\hfil

	\begin{minipage}{0.45\textwidth}
		\begin{center}
			$\boldsymbol{\mu=1}$
		\end{center}
		\scalebox{1}{\includegraphics[width=\linewidth]{SHOCK_II}}

	\end{minipage}

	\caption{Discontinuous initial data~$
		\Riemann^\pm(0,x)
		=
		2\,
		\textup{sign}(x)
		$ and feedback control based on the weighted Rayleigh quotient.
		Simulations for the conservation law without source term are shown in blue. The scaled Lyapunov function~\eqref{LFtilde} yields an upper bound on the $L^2$-norm that  decays at least with the desired rate~$\mu=0.1$ (left panel) and $\mu=1$ (right panel).
		Simulations for the semilinear system are shown in red. The corresponding deviations decrease non-monotonically.}
	\label{Shock}

\end{figure}

\subsection{Conclusions from numerical experiments}
The numerical results show that the computational framework is able to stabilize semilinear hyperbolic balance laws.
The control that is based on the weighted Rayleigh quotient is computationally less expensive and leads to a larger control parameter. However, the obtained control may be not sufficient to establish exponential decay in general.

In contrast,
the control that uses the parameter~$\mud_{{\M}(t)}$ is computationally more expensive and yields a smaller control parameter. However, this choice can ensure  exponential decay to a desired state for problems where the control based on the weighted Rayleigh quotient only counteracts instabilities.

Finally, we recall that the numerical Lyapunov function is not necessarily strictly decreasing, since the involved parameters are time-dependent. Still, the presented framework is consistent with theoretical results when a fixed parameter~$\mud(t)=\mub$ is used.

\section*{Summary}
	We have considered the numerical treatment of stabilization problems for semilinear hyperbolic boundary value problems. A Lyapunov function that yields an upper bound on the~$L^2$-norm of the distance to a desired state
is used as an analytical  tool. It is defined up to a parameter that is calculated numerically by a high-order CWENO reconstruction.
The computational framework gives sufficient conditions on a feedback control  to steer the system to a desired state. Numerical experiments illustrate the applicability of the presented approach and its consistency to previous theoretical findings.

\section*{Acknowledgments}

This work is supported by the PRIME programme of the German Academic Exchange Service (DAAD).
The authors acknowledge support
from ``National Group for Scientific Computation (GNCS-INDAM)'' and by MUR
(Ministry of University and Research) PRIN2017 project number 2017KKJP4X.\\

\noindent
Furthermore, we would like to offer special thanks to the anonymous reviewers for their  valueable feedback.

\section*{Conflict of interest}

The authors declare that they have no conflict of interest.

\bibliographystyle{AIMS}

redpajama_set_name: RedPajamaArXiv


text:
U.K. Kodak Pension Plan Completes Acquisition of Eastman Kodak Company's Personalized Imaging and Document Imaging Businesses
Posted on September 4, 2013 by Web Desk
– KPP Creates Kodak Alaris to Ensure Long-Term Growth of Valuable Businesses
LONDON, Sept. 3, 2013 / PRNewswire / – The U.K. Kodak Pension Plan (KPP) has completed its acquisition of the Kodak Personalized Imaging and Document Imaging businesses from Eastman Kodak Company and created a new company, which will be known as Kodak Alaris. The new company and its name preserve the heritage and legacy of the Kodak brand, while embodying greater speed and agility to meet market needs and changes. Kodak Alaris, which has a perpetual license to use the Kodak brand, will focus on strategic, ongoing investments for these businesses to ensure long-term growth and success.
"Our excitement around the acquisition of these businesses comes not just from their market strength but from what we see as long-term, highly successful growth opportunities," said Steven Ross, independent chairman of KPP. "Today starts the new chapter of a storied brand and we're thrilled with the potential the new company holds for our plan members, our customers, and our employees."
Dennis Olbrich, president of Kodak Alaris' Personalized Imaging business, and Dolores Kruchten, president of Kodak Alaris' Document Imaging business, will provide leadership for Kodak Alaris.
"Innovation has always been critical to our success and it will continue to be a cornerstone of our businesses," said Kruchten. "Under new ownership, we will increase our commitment to pioneering new products and solutions. We will continue to deliver the industry leading services and support that our customers know and trust. And we will be a nimble, innovative company, capable of achieving greater speed to market."
"Our new owner shares the same high commitment to our Kodak Alaris' Personalized Imaging and Document Imaging customers, which is what makes the future of Kodak Alaris so exciting," said Olbrich. "As we celebrate this critical step on our journey, the priority is to continue to meet and exceed the needs of customers and consumers. This we will do by continuing our solid tradition of customer service, all made possible by our committed and talented employees."
When the transaction completes globally, Kodak Alaris will have more than 4,700 employees in approximately 30 countries with expected revenue of more than $1.3 billion.
About KPP:
KPP Trustees Ltd. (KPP), trustee of the Kodak Pension Plan of the United Kingdom (the Plan), is the owner of Kodak Alaris into which the Personalized Imaging and Document Imaging businesses of Eastman Kodak Company are being transferred. The Plan has approximately 15,000 members and assets in excess of £1 billion.
Eastman Kodak Company has licensed the KODAK trademark for use on Personalized Imaging and Document Imaging products as well as in the name of Kodak Alaris to the KPP.
About Kodak Alaris' Personalized Imaging Business
This business combines a unique heritage, expertise and ongoing innovation to market high-quality imaging products and services online, in store and on mobile devices. Kodak Alaris' Personalized Imaging business consists of Retail Systems Solutions, the world leader in retail photo kiosks and dry lab systems, offering retailers a competitive advantage in the photo services market; Paper & Output Systems, offering photo specialty retailers, professional and wholesale labs, and photographers the broadest portfolio of traditional photographic paper and workflow solutions; Film Capture, offering consumers and professionals an award-winning range of still-camera film products; and Event Imaging Solutions, providing digital souvenir photography services and solutions at theme parks, iconic destinations, resorts and other markets worldwide.
For more information, please visit www.kodak.com/go/PI and our blog, 1000Words. Follow us on Twitter @kodakCB, on Facebook and on Instagram @kodakcb.
About Kodak Alaris' Document Imaging Business
Kodak Alaris' Document Imaging business enables customers to effectively collect data from information flows, route it to the right people within an organization, understand its meaning, and extract more value from its contents through the use of superior systems and solutions. Our offerings include award-winning scanners and capture software, information management platforms, an expanding range of professional services, and industry-leading service and support. From small offices to global operations, Kodak Alaris' Document Imaging business intelligently delivers the solutions that enterprises need, automating business processes, enhancing customer experience management and enabling better business decisions.
For more information, please visit kodak.com/go/dinews. Follow us on Twitter at @kodakdi and visit our blog, Info Over Drive.

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Home » What Does the High School Core Curriculum Include?
As a prospective secondary education teacher, you may be wondering exactly what classes you can expect to teach once you start your career—and what the course content of those classes will be. The answer is contained in your middle and high school curriculum.
While schools, districts and states all have varying standards, there are some general principles you can expect to find in virtually every secondary school curriculum. Read on to find out what kids are learning in secondary schools today.
With the relative youth of their students and the integrated approach they take toward education, middle schools have the flexibility to create effective teaching units that cross subject-matter barriers and help students learn across educational disciplines.
When they use this collaborative freedom to its fullest extent, teachers can carry out a middle school curriculum that engages young minds to explore subjects beyond the common terms of math, science, social studies, and English.
For example, an integrated middle school curriculum unit might explore both the students' awareness of self-identity and the different types and levels of cultural awareness in different countries.
Elements of history, politics, religion and diversity could be discussed in social studies. Aspects of human development can be covered in both science and creative writing, and for additional perspective, art or drama could be studied and physical education could explore and test the new capabilities their growing bodies gain at that age.
If you are interested in working in a collaborative, team-focused environment and enjoy thinking outside the box of conventional subject-matter blocks, you might enjoy working within a middle school curriculum.
High schools in the U.S. serve our oldest secondary students in grades 9 through 12. Since high schools are rated based on test results (which can affect funding) and college admissions become critical as graduation approaches, high schools place a great deal of emphasis on academic performance.
Therefore, teachers have the opportunity to teach challenging, high level courses for college-bound students who take their studies seriously. At the same time, high schools also offer a wide variety of classes for students of all abilities and interests.
Because of this course variety, your high school curriculum can vary widely as well. As a teacher with a math endorsement, you might teach a class on consumer mathematics for those who struggle with the subject and follow it up with an advanced class in calculus, statistics or trigonometry.
If you have a history or social studies endorsement, you might be asked to teach world history, U.S. history, civics, or even sociology or economics.
Of course, these core classes just scratch the surface of high school offerings. Art, theater, public speaking and academic electives such as marine biology, geology, astronomy, statistics, poetry and psychology, as well as hands-on classes like driver's training and wood shop all contribute to the wealth of classes you can teach as part of a comprehensive high school curriculum.
As any conscientious teacher knows, preparation is crucial to a good lesson plan. And the product of a year's lesson plans is the high school curriculum that you teach.
So research different curricular variations and use all of your secondary curriculum knowledge to plan out which subjects and age-levels you would like to focus on in your career. Many schools offer specific degree programs that lead to specialized teacher certification that can help you get the job you want. For example, you can specialize in teaching science in high school or English in middle school.


We have had a very productive day at Pontispool amazingly with all five horses getting a place. The dressage overall went pretty well with all five horses being in the 20's.
Ludo FH had his last run before Osberton and his second Intermediate. He did a much better test and was nice and relaxed to score 26, he then followed that with a pleasing double clear. I took him quite quietly cross country so he got a few time faults, and he finished in 6th place.
The Graduate was having his first Intermediate run this season and he also did a pretty pleasing test of 28. He rolled one unlucky pole show jumping and then gave me a very happy confident ride cross country. He finished in 7th place.
Grafennacht was having her second Intermediate outing and she did another very nice test of 24. She rolled one pole show jumping, just getting a little bit long in the double. And she was very happy and genuine cross country and she also finished 6th.
Atlantic Vital Spark went very pleasingly on the flat, he is really beginning to come together and he scored 27. He rolled one pole and actually jumped very decently, and he flew around the cross country feeling full of confidence. It is only his third Intermediate run so he is still quite green but he made everything feel good and I am hoping we have now done enough to cope with Osberton.
I was having my second Novice run on Valmy Biats, and he felt much more together this time. He did a nice test but only managed 25 (having had 16 at Gatcombe). He show jumped an easy clear and was clear cross country with 8 time faults to drop down to 5th.
Pontispool had tried very hard with the ground and they had luckily also put in some very good landings. All the aggrevating work did break the ground up and take the worst jarring out of it, but it was obviously still firm in parts so it was not a day of speed for me. However it was great to be able to run them so at least they are getting some form of preparation.
I am off to Waregem this week for the CIC3*, I haven't been there since the 1990's, and I am taking Little Fire and Yes I Can. I know it has also been very dry in Belgium but I am hoping they will have done everything in their power to make the going acceptable.


EDOTCO Group, an end to end integrated telecommunications infrastructure services company, and JTOWER Inc, a company specialising in in-building mobile infrastructure sharing services based in Japan, on Jan 22 entered into a strategic collaboration to advance the capabilities of their next-generation connectivity offerings in the areas of small cells telecommunications infrastructures, in-building network solutions (IBS) and macro towers.
edotco Group chief executive officer Suresh Sidhu, said, "By partnering with an established IBS firm like JTOWER, we are able to merge our industry knowledge and further boost our next-generation technology portfolio.
JTOWER chief executive officer Atsushi Tanaka, said, "It is a great honour to collaborate with edotoco, the largest tower company in Southeast Asia. Working together with edotco, we would like to expand our technology footprint across Southeast Asia region.


I live in a house that my roommates refer to as the Monastery. It is not an actual monastery, but a spacious house on a hill with a magnificent view of Diamond Head where a bunch of single, Christian males spend a significant amount of their free time in prayer and quiet introspection.
On any given day, I can see whether or not Cliffs - about three to five miles away - is breaking. Right now it's not. This morning is cloudy and the streets are slick black, but the sheen should disappear at around noon, along with the clouds, to reveal the sunny splendor of East Honolulu.
This town is very special to me. Waialae Avenue, just a mile down the hill, has its own distinct character. Not that other streets don't -- Kapahulu has the feel of a small town, and then a beach city as you travel from mauka to makai. Kalakaua, though rife with upscale tourist traps, has retained some classic integrity with cobblestone and some very expensive street light fixtures. These avenues stand in contrast to the streets coursing through new cities like Kapolei. This town looks like it's trying to prove to the other towns that it's legit. I don't mean to discount its potential to be the economic hub of West Oahu, but it's like young America, prior to World War I, trying to find its place among longer-standing world powers. It's like a pretentious middle manager whose ego might be the proper size for him ten years from now ... maaaayybe. But until then, he seeks to prove what he suspects - but does not yet fully know - about himself. He reminds me of Kapolei.
And Kapolei sometimes reminds me of ... me.
I guess I love Waialae because it's all the things I want to be. Old school, yet hip. Busy, yet laid back. Professional, yet approachable. Upscale, yet attainable and generally fun to be around. In spite of all its paradoxes, what you see is what you get. It's an honest town, tried and true. Genuine. At least, that's the way I see it. And in time, that's the way I'd like people to see me.


Allow access to an object if a condition is true for at least one record of a tabular section.
Allow access to an object if a condition is true for all the records of a tabular section.
Allow access to an object if a condition is true for all the records of a tabular section or if a tabular section is empty.
The access restrictions script allows you to implement any of the listed access restrictions. Suppose that the UserRights register contains the mapping between users and the objects they have rights to access. The TabularSection1 tabular section of the Catalog1 catalog contains the Object field, which references objects that define data separation between users. The CurrentUser session parameter contains a reference to the Users catalog item, which corresponds to the current infobase user. Below you will find sample restrictions that deal with each of the above tasks.
This restriction allows the current user to access only those items of the Catalog1 catalog whose tabular sections contain at least one record with its Object attribute referencing an object allowed for the user.
This restriction allows to access only those items of the Catalog1 catalog where all tabular section records reference allowed objects.
This restriction allows to access only those items of the Catalog1 catalog whose tabular sections are empty or only have references to allowed objects.


At Professional Insurance Associates, we help you find low rates from Mercury Insurance. What will you do with the money you save?
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text:
» X-Rite/Pantone launch PantoneLIVE
X-Rite/Pantone launch PantoneLIVE
Color management specialist X-Rite has unveiled PantoneLIVE, a cloud-based color service that provides instant access to brand color standards.
X-Rite said there have been notable recent counterfeiting cases tied to the ubiquitous colors that uniquely identify brands. This includes Cadbury and Veuve Clicquot, who were involved in high-profile legal battles to own their brand colors, while Christian Louboutin fought to trademark its signature red soles, X-Rite said.
'With color so critically tied to brand identity, inconsistent brand color can lead to a lack of consumer confidence and lost sales'
In a recent survey conducted by the Pantone Color Institute, more than 70 percent of creatives noted that brand color definitions, accuracy and consistency in creating products or packaging are important to their business, while 42 percent indicated that color-related challenges have a negative impact on their company.
Brand color standards are the principal component of PantoneLIVE and are derived from real ink on real substrates using real printing processes. This allows brand owners to predict how corporate spot colors will reproduce on a wide variety of substrates including brown corrugated, clear film and white polypropylene. A brand's color assets, analogous to a brand's color DNA, are managed and maintained in a secure cloud-based data repository to ensure accurate color communication – to any supplier, around the world.
PantoneLIVE is connected to a large portfolio of software, containing real-world color data for hundreds of thousands of colors, and is supported by the latest color measurement technology. This is combined with professional services including workflow and color rationalization audits, and customized operating procedures from Pantone and X-Rite. Custom and bespoke spectral data, as well as metadata, are used to digitize brand colors. Digitized palettes are then expanded to create independent color standards to allow for accurate color reproduction on a variety of substrates.
Brand color data, equivalent to a digital color swatch, is stored in a secure, cloud-based portal that lets brand owners and other approved members of the supply chain manage digital rights and facilitate color communication across all materials in the production process. This centralized color communication process promotes consistency and helps achieve speed to market efficiencies from initial design to final production, X-Rite said. The portal also provides direction to suppliers to meet brand requirements related to color quality.
'PantoneLIVE represents a transformational change in color management for brand owners across their entire supply chain'
Heinz's director of corporate and government affairs Nigel Dickie said: 'The benefits of using PantoneLIVE are clear. The digital tools gave us unprecedented control and consistency from different print processes and materials. Across all of our packaging formats we saw a reduction in color variance of 50 percent and saved time by establishing one color target that can be applied to all our Heinz Beanz designs.
'The results with our Beanz packaging have been so remarkable that we plan to extend PantoneLIVE to additional product lines, including Heinz soups and Spaghetti Hoops.'
While accurate color is important to the brand identity of consumer packaged goods, protecting brand integrity in the pharmaceutical industry is crucial as counterfeit drugs put the health of consumers at risk, where the slightest variation in packaging color can make a product suspect and the brand vulnerable to counterfeiting, X-Rite said. It highlighted Chesapeake, a global producer of consumer packaging for many of the world's leading pharmaceutical companies, who started using PantoneLIVE to increase consistency in its customers' packaging.
Chesapeake was able to reduce color variation by 84 percent in one case study and improve process controls, which led to zero rejections from the print run and 100 percent client approval. Chesapeake has recorded an extra benefit, as where it previously stocked as many as 3,000 different inks in its Leicester, UK plant it can now store just 537 without reducing color choices.
X-Rite/Pantone has named Sun Chemical and EskoArtwork as preferred partners supporting PantoneLIVE. Both companies worked closely with Pantone and X-Rite to develop PantoneLIVE. Sun Chemical's technology and color data are the foundations for PantoneLIVE, and this technology is integrated into EskoArtwork's solutions.
'PantoneLIVE delivers an end-to-end solution that, for the first time, addresses the needs and requirements of every single discipline within the packaging supply chain. This means that the brand manager will see an accurate representation of on-press color early in the design process, while converters can be assured of first-time-right color proofing and easy-to-match color on press. This sets the right expectations from the beginning of the process and has the potential to significantly reduce time to market.'
In addition, Windmöller & Hölscher, a supplier of flexographic central impression and rotogravure printing presses, is recognized as the PantoneLIVE technology partner, serving the flexible packaging industry. In this unique capacity, Windmöller & Hölscher will extend the capability of their EASY COL on-press color matching solution to incorporate access to the PantoneLIVE ecosystem, thereby allowing converters to reduce press set-up times and in turn assure the quality of important brand colors on press.
'PantoneLIVE digitizes the process, taking it from visual and subjective to consistent and repeatable'
Pantone and X-Rite said they are continuing to work with other vendors to integrate and enhance their solutions with PantoneLIVE.
Access to the PantoneLIVE database starts at US$99 (£63, €76) annually for a designer, US$1,150 (£730, €885) annually for pre-production and from US$2,000 (£1,275, €1,540) to US$2,650 (£1,690, €2,040) annually for production.
A color audit for a brand owner starts at US$4,500 (£2,870, €3,460). Additional fees apply depending on services and scope required. PantoneLIVE solutions will be available June 15.
Ron Potesky, senior vice president and general manager of Pantone, said: 'Nearly 50 years ago, Pantone brought consistency and a common language to an industry that lacked standardization. Historically an analog process, reliant on centuries' old color alchemy, printing and production have advanced with technology in the digital age.
'PantoneLIVE digitizes the process, taking it from visual and subjective to consistent and repeatable – significantly reducing production timelines and improving the bottom line.'
Tom Vacchiano, president and CEO of X-Rite, said: 'PantoneLIVE represents a transformational change in color management for brand owners across their entire supply chain.'
PantoneLIVE is the first service under the Pantone Digital Business unit, a newly created division of X-Rite that is the byproduct of Pantone's expertise color and X-Rite's color science and technology experience. X-Rite's Dr. Sonia Megert is to lead the new Pantone Digital Business unit, and said: 'Globally consistent color standards are essential to brand identity. With supply chains made up of hundreds of different facilities scattered around the world, corporations struggle to control and maintain color consistency. PantoneLIVE is a dynamic ecosystem, open to all supply chain participants, which delivers consistent color across the entire packaging workflow – from design concept to retail store shelves.'
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