# Feedback stabilization for entropy solutions of a $2\times 2$ hyperbolic
system of conservation laws at a junction

Giuseppe Maria Coclite  Polytechnic University of Bari, Department of
Mechanics, Mathematics and Management, Via E. Orabona 4, 70125 Bari, Italy.
<EMAIL_ADDRESS>,  Nicola De Nitti  EPFL, Institut de
Mathématiques, Station 8, 1015 Lausanne, Switzerland<EMAIL_ADDRESS>,  Mauro Garavello  Università di Milano Bicocca, Dipartimento di Matematica e
Applicazioni, via R. Cozzi 55, 20125 Milano, Italy<EMAIL_ADDRESS>and  Francesca Marcellini  Università di Brescia, Dipartimento di Ingegneria
dell’Informazione, via Branze 38, 25123 Brescia, Italy.
<EMAIL_ADDRESS>
###### Abstract.

We consider the $p$-system in Eulerian coordinates on a star-shaped network.
Under suitable transmission conditions at the junction and dissipative
boundary conditions in the exterior vertices, we show that the entropy
solutions of the system are exponentially stabilizable. Our proof extends the
strategy by Coron et al. (2017) and is based on a front-tracking algorithm
used to construct approximate piecewise constant solutions whose BV norms are
controlled through a suitable exponentially-weighted Glimm-type Lyapunov
functional.

###### Key words and phrases:

Gas transport, hyperbolic systems of conservation laws, networks, junction
conditions, BV exponential stabilization, entropy solutions.

###### 2020 Mathematics Subject Classification:

35L65, 35L45, 35B25.

##  1\. Introduction

We aim to stabilize the entropy solutions to the $p$-system (or Euler
equations) in a star-shaped network around a given equilibrium by using
boundary feedback controls. The $p$-system models a fluid in a rectilinear 1D
pipe through the evolution of some macroscopic quantities, namely its density
and linear momentum; see [18, 32] and the references therein for a more
complete description.

Let us consider $N\in\mathbb{N}\setminus\\{0,1\\}$ rectilinear tubes, modeled
by the real interval $I\coloneqq(0,1)$, exiting a junction $J$, which is
located at the position $x=0$. For $\ell\in\\{1,\ldots,N\\}$, the direction
and section of the $\ell$-th tube are described, respectively, by the
direction and the norm of a vector
$\nu_{\ell}\in\mathbb{R}^{3}\backslash\\{0\\}.$ All tubes are filled with the
same compressible, inviscid and isentropic (or isothermal) fluid, and we
assume that friction along the walls is neglected. Hence the fluid dynamics
can be modeled through $n$ copies of the one-dimensional $p$-system in
Eulerian coordinates:

(1.1)
$\displaystyle\begin{cases}\partial_{t}\rho_{\ell}+\partial_{x}q_{\ell}=0,&t>0,\,x\in[0,1),\\\
\partial_{t}q_{\ell}+\partial_{x}\left(\frac{q_{\ell}^{2}}{\rho_{\ell}}+p\left(\rho_{\ell}\right)\right)=0,\qquad\qquad&t>0,\
x\in{[0,1)},\end{cases}\qquad\ell\in\\{1,\ldots,N\\}.$

Here, $t$ is the time, and, along the $\ell$-th tube, $x$ is the abscissa,
$\rho_{\ell}=\rho_{\ell}(t,x)$ is the fluid density, and
$q_{\ell}=q_{\ell}(t,x)$ is its linear momentum density. We assume that the
pressure law $p=p(\rho)$ is the same for all the tubes; it plays the role of
the equation of state of the fluid under consideration and it is assumed to
satisfy the following hypothesis:

(P) $\displaystyle p\in C^{2}(\mathbb{R}^{+};\mathbb{R}^{+}),\quad
p^{\prime}>0,\quad p^{\prime\prime}>0.$

We supplement 1.1 with a set of initial conditions, of coupling conditions at
the junction $J$, and of boundary conditions at the position $x=1$ of each
pipe. As initial conditions, we consider

(1.2)
$\left(\rho_{\ell},q_{\ell}\right)(0,x)=\left({\rho}_{0,\ell},{q}_{0,\ell}\right)(x),\quad
x\in I,\quad\text{$\ell\in\\{1,\ldots,N\\}$},$

where ${\rho}_{0,\ell}$ and ${q}_{0,\ell}$ are given functions in
$L^{1}\left(I\right)$ with finite total variation.

In this paper, at the junction $J$, we consider the notion of _P-solutions_ ,
introduced in [13], which prescribes that

  1. (1)

mass is conserved;

  2. (2)

the trace of the linear momentum flux (sometimes called _dynamic pressure_) at
$J$ is the same for all tubes;

  3. (3)

entropy may not decrease at the junction.

This concept of solution yields the well-posedness in $L^{1}$; see [14, 15]
for further details. Finally, at the exterior boundary we impose a feedback
control, described in terms of the Riemann invariants of the $p$-system.

The main result of the present paper is the existence of a feedback type
control, acting at the exterior boundary of the star-shaped network, which
asymptotically stabilizes the $p$-system 1.1 around an equilibrium
configuration. Roughly speaking, with an equilibrium over the network we mean
a vector

(1.3)
$\left(\left(\bar{\rho}_{1},\bar{q}_{1}\right),\ldots,\left(\bar{\rho}_{N},\bar{q}_{N}\right)\right)\in\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{N}$

of constant states, which provide an equilibrium solution for 1.1. The
stabilization’s proof is done through a Lyapunov-type functional, similar to
the one considered in [17], which is decreasing in time, on the wave-front
tracking construction, and eventually vanishes. We strongly use the fact that,
locally around the equilibrium, one eigenvalue of the linearized system is
strictly negative and one strictly positive. It is important to notice that,
in this paper, we do not work with strong solutions, but with weak solutions,
which possibly have discontinuities.

Existence and well-posedness of entropy admissible solutions for the Cauchy
problem at junctions was established in [15], and, with a similar technique,
can be also extended in the case of the initial-boundary value problem. The
problem of the stabilization is, instead, more difficult. While our Lyapunov
functional is similar to the one considered in [17], we have to face
additional challenges. Indeed, in [17], the authors considered a $2\times 2$
hyperbolic system of conservation laws with positive characteristic speeds on
a bounded segment; as a consequence, the feedback boundary control acts on the
left boundary according to the trace of the solution at the right boundary. On
the other hand, in our work, the presence of a strictly positive eigenvalue
and a strictly negative one complicates the analysis of the feedback
mechanism.

Further recent results inspired by [17] are contained in [21, 22]. Previous to
[17], the stabilization of a scalar conservation law through a stationary
feedback mechanism was achieved in [33]; in the case of open-loop controls,
results on asymptotic stabilization and on controllability for a general
hyperbolic system of conservation laws with either genuinely nonlinear or
linearly degenerate characteristic fields (in the sense of Lax) and
characteristic speeds strictly separated from $0$, were obtained in [10] by
suitably acting on both sides of the interval or in [5] by acting on a single
boundary point. In [7], an entropy-based Lyapunov functional was used for the
stability analysis of equilibria in networks of scalar conservation laws. For
the problem of exact controllability of both classical and entropy solutions
for systems of conservation laws on an interval, we refer to [3, 4, 28, 29,
30, 31]. Instead, few results about the stabilization of hyperbolic systems on
networks in the context of entropy solutions are currently available in the
literature (see for example [19, 34] for networked wave equations).

On the other hand, for the stabilization of classical solutions, many more
works have appeared throughout the years (for both scalar equations and
systems, on a single segment or on networks). The exponential stabilization of
gas flow governed by the isothermal Euler equations in fan-shaped networks in
the $L^{2}$-sense has been studied in [24]. For a single pipe, a strict
$H^{1}$-Lyapunov functional and feedback stabilization for the quasilinear
isothermal Euler equations with friction have been studied in [20]. Similar
results have been obtained for $H^{2}$-Lyapunov functions in [25] and for
$C^{1}$-Lyapunov functions in [16]. The finite-time stabilization of a network
of strings is studied in [1, 26]. For the problem of exact controllability of
classical solutions on networks, we refer to [23]. Finally, we refer to the
book [8] for further results.

The paper is organized as follows. In Section 2, we collect several
preliminary notions on the solutions of the $p$-system at a junction. In
Section 3, we present our main stabilization result. The proof, based on a
front-tracking approximation and on the study of an exponentially weighted
Glimm functional, is contained in Section 4.

##  2\. Basic definitions

In this section, we recall the key features of the $p$-system in Eulerian
coordinates, we introduce the concept of solution of 1.1 with initial
conditions 1.2 and with coupling and feedback type boundary conditions, and we
state the main result of this paper.

###  2.1. Quantities of interest for the $p$-system

Following [12, Section 6.1], let us recall here some quantities of interest
for the $p$-system (1.1). First, let us introduce the notation

(2.1) $u\coloneqq(\rho,q)^{\top},\qquad
f(u)\coloneqq\left(q,\frac{q^{2}}{\rho}+p(\rho)\right)^{\top},$

so that the $p$-system 1.1 can be written in the compact form
$\partial_{t}u+\partial_{x}f(u)=0$. The Jacobian matrix of flux function $f$
is

$\displaystyle\nabla f(u)=\left(\begin{array}[]{cc}0&1\\\
-\frac{q^{2}}{\rho^{2}}+p^{\prime}(\rho)&\frac{2q}{\rho}\end{array}\right)$

and so, by assumption P, the system is strictly hyperbolic and its eigenvalues
$\lambda_{1}$, $\lambda_{2}$ and the corresponding right eigenvectors
$r_{1},r_{2}$ are given by

$\displaystyle\begin{array}[]{ll}\lambda_{1}(\rho,q)=\frac{q}{\rho}-c(\rho),&\lambda_{2}(\rho,q)=\frac{q}{\rho}+c(\rho),\vspace{.2cm}\\\
r_{1}(\rho,q)=\left[\begin{array}[]{c}-\rho\\\ \rho
c(\rho)-q\end{array}\right],&r_{2}(\rho,q)=\left[\begin{array}[]{c}\rho,\\\
q+\rho c(\rho)\end{array}\right],\end{array}$

where $c(\rho):=\sqrt{p^{\prime}(\rho)}$ is the sound speed. Moreover, we have
that

(2.2) $\nabla\lambda_{1}(\rho,q)\cdot r_{1}(\rho,q)=c(\rho)+\rho
c^{\prime}(\rho)>0,\qquad\nabla\lambda_{2}(\rho,q)\cdot
r_{2}(\rho,q)=c(\rho)+\rho c^{\prime}(\rho)>0,$

so that both characteristic fields are genuinely nonlinear. The _Riemann
invariants_ $v_{1}$ and $v_{2}$ of the $p$-system (1.1) are

(2.3)
$v_{1}\left(\rho,q\right):=\frac{q}{\rho}+\int_{1}^{\rho}\frac{\sqrt{p^{\prime}(r)}}{r}\,\mathrm{d}r,\qquad
v_{2}\left(\rho,q\right):=\frac{q}{\rho}-\int_{1}^{\rho}\frac{\sqrt{p^{\prime}(r)}}{r}\,\mathrm{d}r.$

Note that

(2.4) $\begin{split}\nabla v_{1}(\rho,q)\cdot r_{1}(\rho,q)&=\nabla
v_{2}(\rho,q)\cdot r_{2}(\rho,q)=0,\\\ \nabla v_{1}(\rho,q)\cdot
r_{2}(\rho,q)&=\nabla v_{2}(\rho,q)\cdot
r_{1}(\rho,q)=2\sqrt{p^{\prime}(\rho)},\end{split}$

for every $(\rho,q)\in\mathbb{R}^{+}\times\mathbb{R}$; $v_{1}$ is constant
along the rarefaction curves of the first family, while $v_{2}$ is constant
along the rarefaction curves of the second family. Moreover, $v_{1}$ is
monotonically increasing along the rarefaction curves of the second family and
$v_{2}$ is monotonically increasing along the rarefaction curves of the first
family.

The shock and the rarefaction curves through a state
$u\in\mathbb{R}^{+}\times\mathbb{R}$ can be parameterized as

(2.5) $\displaystyle\sigma\mapsto S_{k}(\sigma)(u),\quad\sigma\mapsto
R_{k}(\sigma)(u),$

where $k\in\left\\{1,2\right\\}$ and the parametrization is chosen so that

$\displaystyle\lambda_{k}\left(S_{k}(\sigma)(u)\right)=\lambda_{k}\left(R_{k}(\sigma)(u)\right)=\lambda_{k}(u)+\sigma;$

see Figure 1 (left). Consequently we use the following parametrization for the
Lax curve

$\sigma\mapsto\mathcal{L}_{k}(\sigma)(u)=\left\\{\begin{array}[]{ll}R_{k}(\sigma)(u),&\qquad\textrm{
if }\sigma\geq 0,\\\ S_{k}(\sigma)(u),&\qquad\textrm{ if
}\sigma<0.\end{array}\right.$

We introduce the regions

(2.6) $\displaystyle\begin{aligned}
&A_{-}=\left\\{\mathbb{R}^{+}\times\mathbb{R}:\lambda_{2}(\rho,q)<0\right\\},\\\
&A_{0}^{-}=\left\\{\mathbb{R}^{+}\times\mathbb{R}:\lambda_{2}(\rho,q)\geq
0,q\leq 0\right\\},\\\
&A_{0}^{+}=\left\\{\mathbb{R}^{+}\times\mathbb{R}:\lambda_{1}(\rho,q)\leq
0,q\geq 0\right\\},\\\
&A_{+}=\left\\{\mathbb{R}^{+}\times\mathbb{R}:\lambda_{1}(\rho,q)>0\right\\},\\\
&A_{0}=A_{0}^{-}\cup A_{0}^{+}.\end{aligned}$

Here $A_{0}$ is known as _subsonic region_ ; see Figure 1 (right).

Finally, we introduce the functions

(2.7) $\displaystyle P(\rho,q)$ $\displaystyle=\frac{q^{2}}{\rho}+p(\rho),$
(2.8) $\displaystyle E(\rho,q)$
$\displaystyle=\frac{q^{2}}{2\rho}+\rho\int_{1}^{\rho}\frac{p(r)}{r^{2}}\,\mathrm{d}r,$
(2.9) $\displaystyle F(\rho,q)$
$\displaystyle=\frac{q}{\rho}(E(\rho,q)+p(\rho)),$

which are respectively the dynamic pressure, the energy, and the energy flow;
see Figure 1 (right) for the plot of a level curve of $P$.

$R_{1}(\sigma)(u)$ $q$$\rho$$u=(\rho,q)$$S_{1}(\sigma)(u)$ $S_{2}(\sigma)(u)$
$R_{2}(\sigma)(u)$

$q$$A_{0}^{-}$ $A_{0}^{+}$ $\rho$$\lambda_{1}(\rho,q)=0$$A_{-}$
$\lambda_{2}(\rho,q)=0$$A_{+}$ $P(\rho,q)=k$

Figure 1. Left: The Lax curves in (2.5) through a point. More precisely,
$R_{1}$ and $R_{2}$ denote the rarefaction curves of the first and second
family. Instead $S_{1}$ and $S_{2}$ denote the shock curves of the first and
second family. Right: The regions $A_{-}$, $A_{0}^{\pm}$, $A_{+}$, defined in
(2.6), the sonic curves $\lambda_{1}(\rho,q)=0$ and $\lambda_{2}(\rho,q)=0$,
and a level curve of the dynamic pressure, defined in (2.7).

###  2.2. Initial-Boundary Valued Problem

Assigning at time $t=0$ an initial state
$\left({\rho}_{o,\ell},{q}_{o,\ell}\right)\in
L^{1}(I;\mathbb{R}^{+}\times\mathbb{R})$ in each of the $N$ ducts exiting the
junction $(\ell\in\\{1,\ldots,N\\})$ and $N$ boundary data $B_{\ell}\in
L^{1}_{loc}(\mathbb{R}^{+};\mathbb{R})$, we consider with the following
initial-boundary value problem:

(2.10)
$\displaystyle\begin{cases}\partial_{t}\rho_{1}+\partial_{x}q_{1}=0,&t>0,\,x\in
I,\\\ \partial_{t}q_{1}+\partial_{x}P(\rho_{1},q_{1})=0,&t>0,\,x\in I,\\\
\vdots\\\ \partial_{t}\rho_{N}+\partial_{x}q_{N}=0,&t>0,\,x\in I,\\\
\partial_{t}q_{N}+\partial_{x}P(\rho_{N},q_{N})=0,&t>0,\,x\in I,\\\
(\rho_{1},q_{1})(0,x)=\left(\rho_{0,1},q_{0,1}\right)(x),&x\in I,\\\ \vdots\\\
(\rho_{N},q_{N})(0,x)=\left(\rho_{0,N},q_{0,N}\right)(x),&x\in I,\\\
v_{2}(\rho_{1}(t,1),q_{1}(t,1))=B_{1}(t),&t>0,\\\ \vdots\\\
v_{2}(\rho_{N}(t,1),q_{N}(t,1))=B_{N}(t),&t>0.\end{cases}$

######  Definition 2.1 (Entropy solution of 2.10).

The tuple $((\rho_{1},q_{1}),\ldots,(\rho_{N},q_{N}))$ is an entropy solution
to the Cauchy problem 2.10 if the following conditions hold.

  1. (1)

For every $\ell\in\\{1,\ldots,N\\}$, $(\rho_{\ell},q_{\ell})\in
C^{0}([0,+\infty);L^{1}(I;\mathbb{R}^{+}\times\mathbb{R}))$.

  2. (2)

For every $\ell\in\\{1,\ldots,N\\}$ and for a.e. $t>0$, the map
$x\mapsto(\rho_{\ell}(t,x),q_{\ell}(t,x))$ has finite total variation.

  3. (3)

For every $\ell\in\\{1,\ldots,N\\}$ and for every $\Phi\in
C^{\infty}_{c}\left((0,+\infty)\times I;\mathbb{R}\right)$, it holds

$\int_{0}^{+\infty}\int_{0}^{1}\left[\partial_{t}\Phi(t,x)\left(\begin{array}[]{c}\rho_{\ell}(t,x)\\\
q_{\ell}(t,x)\end{array}\right)+\partial_{x}\Phi(t,x)f(\rho_{\ell}(t,x),q_{\ell}(t,x))\right]\,\mathrm{d}x\,\mathrm{d}t=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$

  4. (4)

For every $\ell\in\\{1,\ldots,N\\}$ and for every $\Phi\in
C^{\infty}_{c}\left((0,+\infty)\times I;\mathbb{R}^{+}\right)$, it holds

$\int_{0}^{+\infty}\int_{0}^{1}\left[\partial_{t}\Phi(t,x)E\left(\rho_{\ell}(t,x),q_{\ell}(t,x)\right)+\partial_{x}\Phi(t,x)F(\rho_{\ell}(t,x),q_{\ell}(t,x))\right]\,\mathrm{d}x\,\mathrm{d}t\geq
0.$

  5. (5)

For every $\ell\in\\{1,\ldots,N\\}$,
$(\rho_{\ell}(0,x),q_{\ell}(0,x))=(\rho_{0,\ell}(x),q_{0,\ell}(x))$ for a.e.
$x\in I$.

  6. (6)

For a.e. $t>0$, the following coupling conditions hold:

    1. (a)

$\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|q_{\ell}(t,0^{+})=0$;

    2. (b)

there exists $P_{\ast}=P_{\ast}(t)>0$ (possibly depending on $t$) such that,
for every $\ell\in\\{1,\ldots,N\\}$,

$P(\rho_{\ell}(t,0^{+}),q_{\ell}(t,0^{+}))=P_{\ast}(t);$

    3. (c)

$\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|F(\rho_{\ell}(t,0^{+}),q_{\ell}(t,0^{+}))\leq
0$.

  7. (7)

For every $\ell\in\left\\{1,\ldots,N\right\\}$ and for a.e. $t>0$,

$v_{2}\left(\rho_{\ell}(t,1^{-}),q_{\ell}(t,1^{-})\right)=B_{\ell}(t).$

Given $N$ constant states
$\bar{u}_{1},\ldots,\bar{u}_{N}\in\mathbb{R}^{+}\times\mathbb{R}$ and the
vector $\vec{k}=\left(k_{1},\ldots,k_{N}\right)\in[0,1]^{N}$, we consider the
system with feedback boundary control:

(2.11)
$\displaystyle\begin{cases}\partial_{t}\rho_{1}+\partial_{x}q_{1}=0,&t>0,\,x\in
I,\\\ \partial_{t}q_{1}+\partial_{x}P(\rho_{1},q_{1})=0,&t>0,\,x\in I,\\\
\vdots\\\ \partial_{t}\rho_{N}+\partial_{x}q_{N}=0,&t>0,\,x\in I,\\\
\partial_{t}q_{N}+\partial_{x}P(\rho_{N},q_{N})=0,&t>0,\,x\in I,\\\
(\rho_{1},q_{1})(0,x)=\left(\rho_{0,1},q_{0,1}\right)(x),&x\in I,\\\ \vdots\\\
(\rho_{N},q_{N})(0,x)=\left(\rho_{0,N},q_{0,N}\right)(x),&x\in I,\\\
v_{2}\left(\rho_{1}(t,1),q_{1}(t,1)\right)=k_{1}v_{1}\left(\rho_{1}(t,1^{-}),q_{1}(t,1^{-})\right)\\\
\qquad\qquad\qquad\qquad\qquad-
k_{1}v_{1}\left(\bar{u}_{1}\right)+v_{2}\left(\bar{u}_{1}\right),&t>0,\\\ \\\
\vdots\\\
v_{2}\left(\rho_{N}(t,1),q_{N}(t,1)\right)=k_{N}v_{1}\left(\rho_{N}(t,1^{-}),q_{N}(t,1^{-})\right)\\\
\qquad\qquad\qquad\qquad\qquad-
k_{N}v_{1}\left(\bar{u}_{N}\right)+v_{2}\left(\bar{u}_{N}\right),&t>0.\end{cases}$

Formally system 2.10 reduces to 2.11 with the position

$\left\\{\begin{array}[]{l}B_{1}(t)=k_{1}v_{1}\left(\rho_{1}(t,1^{-}),q_{1}(t,1^{-})\right)-k_{1}v_{1}\left(\bar{u}_{1}\right)+v_{2}\left(\bar{u}_{1}\right),\\\
\vdots\\\
B_{N}(t)=k_{N}v_{1}\left(\rho_{N}(t,1^{-}),q_{N}(t,1^{-})\right)-k_{N}v_{1}\left(\bar{u}_{N}\right)+v_{2}\left(\bar{u}_{N}\right).\end{array}\right.$

We now define first the notion of solution and then that of equilibrium
solution for the system with feedback boundary control.

######  Definition 2.2 (Entropy solution of 2.11).

The tuple $((\rho_{1},q_{1}),\ldots,(\rho_{N},q_{N}))$ is an entropy solution
to the problem 2.11 if the following conditions hold.

  1. (1)

For every $\ell\in\\{1,\ldots,N\\}$, $(\rho_{\ell},q_{\ell})\in
C^{0}([0,+\infty);L^{1}(I;\mathbb{R}^{+}\times\mathbb{R}))$.

  2. (2)

For every $\ell\in\\{1,\ldots,N\\}$ and for a.e. $t>0$, the map
$x\mapsto(\rho_{\ell}(t,x),q_{\ell}(t,x))$ has finite total variation.

  3. (3)

For every $\ell\in\\{1,\ldots,N\\}$ and for every $\Phi\in
C^{\infty}_{c}\left((0,+\infty)\times I;\mathbb{R}\right)$, it holds

$\int_{0}^{+\infty}\int_{0}^{1}\left[\partial_{t}\Phi(t,x)\left(\begin{array}[]{c}\rho_{\ell}(t,x)\\\
q_{\ell}(t,x)\end{array}\right)+\partial_{x}\Phi(t,x)f(\rho_{\ell}(t,x),q_{\ell}(t,x))\right]\,\mathrm{d}x\,\mathrm{d}t=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$

  4. (4)

For every $\ell\in\\{1,\ldots,N\\}$ and for every $\Phi\in
C^{\infty}_{c}\left((0,+\infty)\times I;\mathbb{R}^{+}\right)$, it holds

$\int_{0}^{+\infty}\int_{0}^{1}\left[\partial_{t}\Phi(t,x)E\left(\rho_{\ell}(t,x),q_{\ell}(t,x)\right)+\partial_{x}\Phi(t,x)F(\rho_{\ell}(t,x),q_{\ell}(t,x))\right]\,\mathrm{d}x\,\mathrm{d}t\geq
0.$

  5. (5)

For every $\ell\in\\{1,\ldots,N\\}$,
$(\rho_{\ell}(0,x),q_{\ell}(0,x))=(\rho_{0,\ell}(x),q_{0,\ell}(x))$ for a.e.
$x\in I$.

  6. (6)

For a.e. $t>0$, the following coupling conditions hold:

    1. (a)

$\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|q_{\ell}(t,0^{+})=0$;

    2. (b)

there exists $P_{\ast}=P_{\ast}(t)>0$ (possibly depending on $t$) such that,
for every $\ell\in\\{1,\ldots,N\\}$,

$P(\rho_{\ell}(t,0^{+}),q_{\ell}(t,0^{+}))=P_{\ast}(t);$

    3. (c)

$\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|F(\rho_{\ell}(t,0^{+}),q_{\ell}(t,0^{+}))\leq
0$.

  7. (7)

For every $\ell\in\left\\{1,\ldots,N\right\\}$ and for a.e. $t>0$,

$v_{2}\left(\rho_{\ell}(t,1),q_{\ell}(t,1)\right)=k_{\ell}v_{1}\left(\rho_{\ell}(t,1^{-}),q_{\ell}(t,1^{-})\right)-k_{\ell}v_{1}\left(\bar{u}_{\ell}\right)+v_{2}\left(\bar{u}_{\ell}\right).$

######  Definition 2.3 (Equilibrium solution).

We say that the tuple
$\left(\bar{\rho},\bar{q}\right)=((\bar{\rho}_{1},\bar{q}_{1}),\ldots,(\bar{\rho}_{N},\bar{q}_{N}))\in\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{N}$
is a equilibrium solution to the Cauchy problem 2.11 if the tuple
$((\rho_{1},q_{1}),\ldots,(\rho_{N},q_{N}))$, defined by
$\left(\rho_{\ell}(t,x),q_{\ell}(t,x)\right)=\left(\bar{\rho}_{\ell},\bar{q}_{\ell}\right)$
for every $\ell\in\left\\{1,\ldots,N\right\\}$, $x\in I$, and $t>0$, provides,
in the sense of Definition 2.2, a solution to the Cauchy problem 2.11 with
initial conditions

$\left(\rho_{0,\ell}(x),q_{0,\ell}(x)\right)=\left(\bar{\rho}_{\ell},\bar{q}_{\ell}\right)\quad\text{
for $x\in I$.}$

##  3\. Main results: well-posedness and feedback stabilization via a Lyapunov
functional

The main result of this paper deals with the well posedness result for the
Cauchy problem 2.11 and with a stabilization result for the solutions of the
same Cauchy problem 2.11 with feedback control acting at the external
boundary, according to Definition 2.2. More precisely, following [15, Theorem
3.3], we prove existence of entropy-admissible solutions; then we consider a
suitable feedback mechanism to achieve exponential stability in a similar way
as in [17].

######  Theorem 3.1 (Exponential stabilization for entropy solutions).

Fix an $N$-tuple of subsonic states
$(\bar{\rho},\bar{q})=\left(\left(\bar{\rho}_{1},\bar{q}_{1}\right),\ldots,\left(\bar{\rho}_{N},\bar{q}_{N}\right)\right)\in(\mathring{A}_{0})^{N}$,
giving a equilibrium solution to the Cauchy problem 2.11 in the sense of
Definition 2.3 and such that
$\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|F\left(\bar{\rho}_{\ell},\bar{q}_{\ell}\right)<0$.

Then, there exist the constants $\bar{k}>0$, $\delta_{0}>0$, $L>0$, $C>0$,
$\nu>0$, a domain $\mathcal{D}$, and, for every
$\vec{k}=(k_{1},\ldots,k_{N})\in[0,\bar{k}]^{N}$, a semigroup
$S:[0,+\infty[\times\mathcal{D}\rightarrow\mathcal{D}$, with the following
properties.

  1. (i)

_Domain:_
$\mathcal{D}\supseteq\left\\{(\rho,q)\in(\bar{\rho},\bar{q})+{L}^{1}\left(I;\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{N}\right):\,\operatorname{TV}(\rho,q)\leq\delta_{0}\right\\}$.

  2. (ii)

_Semigroup property:_ For $(\rho,q)\in\mathcal{D},S_{0}(\rho,q)=(\rho,q)$ and,
for $s,t\geq 0,S_{s}S_{t}(\rho,q)=S_{s+t}(\rho,q)$.

  3. (iii)

_Lipschitz type estimate:_ For
$(\rho,q),\left(\rho^{\prime},q^{\prime}\right)\in\mathcal{D}$ and $s,t\geq
0$, it holds

$\displaystyle\left\|S_{t}(\rho,q)-S_{s}\left(\rho^{\prime},q^{\prime}\right)\right\|_{{L}^{1}}\leq
L\cdot\left(\left\|(\rho,q)-\left(\rho^{\prime},q^{\prime}\right)\right\|_{{L}^{1}}+|t-s|\right).$

  4. (iv)

_Entropy admissibility:_ For every $(\rho,q)\in\mathcal{D}$, the map
$(t,x)\mapsto S_{t}(\rho,q)(x)$ is a weak entropy solution to the Cauchy
problem 2.11 in the sense of Definition 2.2.

  5. (v)

_Exponential stabilization:_ For every $(\rho_{0},q_{0})\in\mathcal{D}$ and
$t\geq 0$,

$\mathrm{TV}\left(S_{t}\left(\rho_{0},q_{0}\right)\right)\leq Ce^{-\nu
t}\mathrm{TV}(\rho_{0},q_{0}).$

##  4\. Proof of Theorem 3.1

This section contains the proof of Theorem 3.1, which is based on the wave-
front tracking technique with the use of a specific weighted Glimm-type
functional, inspired by the one introduced in [17].

First, in Section 4.1, we construct a Riemann solver. Then, in Section 4.2, we
introduce a suitable Glimm-type functional. These preliminaries allow us to
construct an approximate wave-front tracking solution in Section 4.4.

The interaction estimates in Section 4.2 yield the existence of an entropy
admissible solution in Section 4.5. Moreover, in Section 4.6, we actually show
that there exist a $L^{1}$-contracting semigroup of solutions.

Finally, in Section 4.7, we obtain the exponential stabilization result; thus
concluding the proof of Theorem 3.1.

###  4.1. Wave-front tracking approximation and Riemann solvers

In this subsection, we construct piecewise constant approximations via the
wave-front tracking method; see [9, 18, 27] for the general theory. Note that
here we can avoid the use of non-physical waves; see [2, Lemma 2.5] or [6].

At first, we give the following definition of an $\varepsilon$-approximate
wave-front tracking solution to 2.11.

######  Definition 4.1 ($\varepsilon$-approximate wave-front tracking
solution).

Given $\varepsilon>0$, the map
$\mathbf{u}_{\varepsilon}=\left(u_{1,\varepsilon},\ldots,u_{N,\varepsilon}\right)$
is an $\varepsilon$-approximate wave-front tracking solution to 2.11 if the
following conditions hold.

  1. (1)

For every $\ell\in\left\\{1,\ldots,N\right\\}$,
$u_{\ell,\varepsilon}=\left(\rho_{\ell,\varepsilon},q_{\ell,\varepsilon}\right)\in
C^{0}\left([0,+\infty[;L^{1}(I;\mathbb{R}^{+}\times\mathbb{R})\right)$.

  2. (2)

For every $\ell\in\left\\{1,\ldots,N\right\\}$,
$\left(\rho_{\ell,\varepsilon},q_{\ell,\varepsilon}\right)$ is piecewise
constant, with discontinuities along finitely many straight lines in
$(0,+\infty)\times I$. Moreover, the jumps can be of the first family or of
the second family.

  3. (3)

For $\ell\in\left\\{1,\ldots,N\right\\}$, along each jump $x=x(t)$ of the
first family (resp. second family), the values
$u^{-}=u_{\ell,\varepsilon}\left(t,x(t)^{-}\right)$ and
$u^{+}=u_{\ell,\varepsilon}\left(t,x(t)^{+}\right)$ are related by

$u^{+}=\mathcal{L}_{1}\left(\sigma_{1}\right)\left(u^{-}\right)\quad\left(\textrm{resp.
}u^{+}=\mathcal{L}_{2}\left(\sigma_{2}\right)\left(u^{-}\right)\right)$

for some wave size $\sigma_{1}$ (resp. $\sigma_{2}$).

Moreover, if $\sigma_{1}<0$ (resp. $\sigma_{2}<0$), then the discontinuity is
a shock wave and

$\left|\dot{x}(t)-\lambda(u^{-},u^{+})\right|\leq\varepsilon$

where $\lambda(u^{-},u^{+})$ denotes the velocity associated to the Rankine-
Hugoniot condition.

Finally, if $\sigma_{1}>0$ (resp. $\sigma_{2}>0$), then
$\sigma_{1}\leq\varepsilon$ (resp. $\sigma_{2}\leq\varepsilon$), the
discontinuity is a part of a rarefaction fan and

$\left|\dot{x}(t)-\lambda_{1}(u^{-})\right|\leq\varepsilon\quad\left(\left|\dot{x}(t)-\lambda_{2}(u^{-})\right|\leq\varepsilon\right).$

  4. (4)

For every $\ell\in\left\\{1,\ldots,N\right\\}$, it holds that

$\left\\{\begin{array}[]{l}\left\|\left(\rho_{\ell,\varepsilon}(0,\cdot),q_{\ell,\varepsilon}(0,\cdot)\right)-\left(\rho_{0,\ell},q_{0,\ell}\right)\right\|_{L^{1}(I_{\ell})}<\varepsilon,\vspace{.2cm}\\\
\mathrm{TV}\left(\rho_{\ell,\varepsilon}(0,\cdot),q_{\ell,\varepsilon}(0,\cdot)\right)\leq\mathrm{TV}\left(\rho_{0,\ell},q_{0,\ell}\right).\end{array}\right.$

  5. (5)

For a.e. $t\in\mathbb{R}^{+}$,

(4.1)
$\left\\{\begin{array}[]{l}\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|q_{\ell,\varepsilon}(t,0^{+})=0,\\\
P\left(\rho_{1,\varepsilon}(t,0^{+}),q_{1,\varepsilon}(t,0^{+})\right)=P\left(\rho_{2,\varepsilon}(t,0^{+}),q_{2,\varepsilon}(t,0^{+})\right),\\\
\vdots\\\
P\left(\rho_{N-1,\varepsilon}(t,0^{+}),q_{N-1,\varepsilon}(t,0^{+})\right)=P\left(\rho_{N,\varepsilon}(t,0^{+}),q_{N,\varepsilon}(t,0^{+})\right),\\\
\displaystyle\sum_{\ell=1}^{N}\left\|\nu_{\ell}\right\|F\left(\rho_{\ell,\varepsilon}(t,0^{+}),q_{\ell,\varepsilon}(t,0^{+})\right)\leq
0.\end{array}\right.$

  6. (6)

For every $\ell\in\left\\{1,\ldots,N\right\\}$ and for a.e. $t>0$

$v_{2}\left(\rho_{\ell,\varepsilon}(t,1),q_{\ell,\varepsilon}(t,1)\right)=k_{\ell}v_{1}\left(\rho_{\ell,\varepsilon}(t,1^{-}),q_{\ell,\varepsilon}(t,1^{-})\right)-k_{\ell}v_{1}\left(\bar{u}_{\ell}\right)+v_{2}\left(\bar{u}_{\ell}\right).$

We briefly review how to construct a wave-front tracking approximate solution
in the sense of Definition 4.1.

Define $\bar{u}_{\ell}=\left(\bar{\rho}_{\ell},\bar{q}_{\ell}\right)$ for
every $\ell=1,\ldots,N$ and choose $\bar{\delta}>0$ such that
$B\left(\bar{u}_{\ell},\bar{\delta}\right)\subseteq A_{0}$ for every
$\ell=1,\ldots,N$. Given $\varepsilon>0$, for every $\ell$, approximate the
initial condition with piecewise constant functions
$\left(\rho_{0,\ell,\varepsilon},q_{0,\ell,\varepsilon}\right)$ with a finite
number of discontinuities such that

$\left\\{\begin{array}[]{l}\left\|\left(\rho_{0,\ell,\varepsilon},q_{0,\ell,\varepsilon}\right)-\left(\rho_{0,\ell},q_{0,\ell}\right)\right\|_{L^{1}(0,1)}<\varepsilon,\vspace{.2cm}\\\
\mathrm{TV}\left(\rho_{0,\ell,\varepsilon},q_{0,\ell,\varepsilon}\right)\leq\mathrm{TV}\left(\rho_{0,\ell},q_{0,\ell}\right).\end{array}\right.$

Then, at the junction, at the exterior boundary, and at each point of jump
along the pipes we solve the corresponding Riemann problems.

  1. (1)

At each discontinuity inside a pipe we use the accurate Riemann solver,
described in Section 4.1.1.

  2. (2)

At the boundary $x=1$, we use the Riemann solver $\mathcal{RS}^{e}$, described
in Section 4.1.3.

  3. (3)

At the junction $x=0$, we use the Riemann solver $\mathcal{RS}^{J}$, described
in Section 4.1.2.

We approximate each rarefaction wave by means of rarefaction fans.

This construction can be extended up to a first time $\bar{t}_{1}$ at which
two waves interact in a duct or a wave hits the junction or the external
boundary. Since at time $\bar{t}_{1}$ the approximation functions are
piecewise constant with a finite number of discontinuities we can repeat the
previous construction up to e second time $\bar{t}_{2}$ of interaction and so
on. In this construction we impose that any rarefaction fan is not split any
further and, without loss of generality, we assume that no more of two waves
interact at the same point in a pipe and no more of one wave interacts at same
time with the junction or with the external boundary.

####  4.1.1. The classical Riemann problem

Assuming, without loss of generality, that a pipe is modeled by the real line
$\mathbb{R}$, we consider the Riemann problem

(4.2)
$\left\\{\begin{array}[]{ll}\partial_{t}\rho+\partial_{x}q=0,&t>0,\,x\in\mathbb{R},\\\
\partial_{t}q+\partial_{x}P(\rho,q)=0,&t>0,\,x\in\mathbb{R},\\\
\left(\rho(0,x),q(0,x)\right)=\left\\{\begin{array}[]{ll}\left(\rho_{l},q_{l}\right),&x<0,\\\
\left(\rho_{r},q_{r}\right),&x>0;\end{array}\right.\end{array}\right.$

see [9, 18] for more details. We denote with

(4.3)
$\begin{array}[]{rccc}{\mathcal{RS}}^{\textrm{acc}}:&\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{2}&\longrightarrow&\mathbb{R}^{2}\\\
&\left(\left(\rho_{l},q_{l}\right),\left(\rho_{r},q_{r}\right)\right)&\longmapsto&\left(\sigma_{1},\sigma_{2}\right)\end{array}$

the accurate Riemann solvers, which is well defined provided
$(\rho_{r},q_{r})$ sufficiently closed to $(\rho_{l},q_{l})$; see [9, Chapter
7.2] for a complete construction.

####  4.1.2. The Riemann problem at the junction

Assuming that we have $N$ pipes, modeled by the semiline line $(0,+\infty)$,
we consider the Riemann problem

(4.4)
$\left\\{\begin{array}[]{ll}\partial_{t}\rho_{1}+\partial_{x}q_{1}=0,&t>0,\,x>0,\\\
\partial_{t}q_{1}+\partial_{x}P(\rho_{1},q_{1})=0,&t>0,\,x>0,\\\ \vdots\\\
\partial_{t}\rho_{N}+\partial_{x}q_{N}=0,&t>0,\,x>0,\\\
\partial_{t}q_{N}+\partial_{x}P(\rho_{N},q_{N})=0,&t>0,\,x>0,\\\
\left(\rho_{1}(0,x),q_{1}(0,x)\right)=\left(\rho_{0,1},q_{0,1}\right),&x>0,\\\
\vdots\\\
\left(\rho_{N}(0,x),q_{N}(0,x)\right)=\left(\rho_{0,N},q_{0,N}\right),&x>0.\end{array}\right.$

We denote with

(4.5)
$\begin{array}[]{rccc}\mathcal{RS}^{J}:&\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{N}&\longrightarrow&\mathbb{R}^{N}\\\
&\left(\left(\rho_{0,1},q_{0,1}\right),\ldots,\left(\rho_{0,N},q_{0,N}\right)\right)&\longmapsto&\left(\sigma_{2,1},\ldots,\sigma_{2,N}\right)\end{array}$

the Riemann solver for 4.4. More precisely, the map $\mathcal{RS}^{J}$ gives,
in each pipe, the strengths of the waves of the second family generated by
4.4; see [13, Theorem 2].

####  4.1.3. The Riemann problem at the external boundary

Assuming, without loss of generality, that a pipe is modeled by the semiline
line $(-\infty,1)$, we consider the Riemann problem

(4.6)
$\left\\{\begin{array}[]{ll}\partial_{t}\rho+\partial_{x}q=0,&t>0,\,x<1,\\\
\partial_{t}q+\partial_{x}P(\rho,q)=0,&t>0,\,x<1,\\\
\left(\rho(0,x),q(0,x)\right)=\left(\rho_{l},q_{l}\right),&x<1.\end{array}\right.$

For $k>0$ and $\bar{u}\in\mathbb{R}^{+}\times\mathbb{R}$, we denote with

(4.7)
$\begin{array}[]{rccc}\mathcal{RS}^{e}:&\left(\mathbb{R}^{+}\times\mathbb{R}\right)^{2}\times(0,1)&\longrightarrow&\mathbb{R}\\\
&\left(\left(\rho_{l},q_{l}\right),\bar{u},k\right)&\longmapsto&\sigma_{1}\end{array}$

the Riemann solver for 4.6. More precisely, the map $\mathcal{RS}^{e}$ gives
the strength of the wave of the first family generated by 4.6, such that
$\left(\rho_{r},q_{r}\right)=\mathcal{L}_{1}\left(\mathcal{RS}^{e}\left(\left(\rho_{l},q_{l}\right),\bar{u},k\right)\right)(\rho_{l},q_{l})$
is the trace of the solution to 4.6 at $x=1$ and

$v_{2}\left(\rho_{r},q_{r}\right)=kv_{1}\left(\rho_{r},q_{r}\right)-kv_{1}\left(\bar{u}\right)+v_{2}\left(\bar{u}\right).$

Note that the Riemann solver $\mathcal{RS}^{e}$ is well defined, since the
following result holds.

######  Lemma 4.2.

Fix $k>0$ and $\bar{u}\in\mathbb{R}^{+}\times\mathbb{R}$. There exists a
neighborhood $\mathcal{U}\subseteq\mathbb{R}^{+}\times\mathbb{R}$ of $\bar{u}$
with the following property. For every $u_{l}\in\mathcal{U}$, there is a
unique $u_{r}\in\mathcal{U}$ such that the relations

$u_{r}=\mathcal{L}_{1}\left(\mathcal{RS}^{e}\left(u_{l},\bar{u},k\right)\right)(u_{l})\qquad\textrm{
and }\qquad
v_{2}\left(u_{r}\right)=v_{2}\left(\bar{u}\right)+kv_{1}\left(u_{r}\right)-kv_{1}\left(\bar{u}\right)$

hold.

###### Proof.

Consider the function

$\Psi(u,\sigma)\coloneqq
v_{2}\left(\mathcal{L}_{1}\left(\sigma\right)(u)\right)-kv_{1}\left(\mathcal{L}_{1}\left(\sigma\right)(u)\right)-v_{2}\left(\bar{u}\right)+kv_{1}\left(\bar{u}\right).$

We clearly have that $\Psi\left(\bar{u},0\right)=0$. Moreover,

$\partial_{\sigma}\Psi\left(u,\sigma\right)=\left(\nabla
v_{2}\left(\mathcal{L}_{1}\left(\sigma\right)(u)\right)-k\nabla
v_{1}\left(\mathcal{L}_{1}\left(\sigma\right)(u)\right)\right)\cdot\frac{d}{d\sigma}\mathcal{L}_{1}\left(\sigma\right)(u)$

and so, by 2.4,

$\partial_{\sigma}\Psi\left(\bar{u},0\right)=\left(\nabla
v_{2}\left(\bar{u}\right)-k\nabla v_{1}\left(\bar{u}\right)\right)\cdot
r_{1}(\bar{u})=\nabla v_{2}\left(\bar{u}\right)\cdot r_{1}(\bar{u})\neq 0.$

The Implicit Function Theorem permits to conclude. The $C^{1}$-regularity of
$\Psi$ follows from the regularity result established in [9, Chapter 5,
Section 5.2, Eq. (5.38)]. ∎

###  4.2. Glimm functionals

First, let us introduce the definition of approaching waves in the case of the
$p$-system. For the general definition see [9, Chapter 7.3] or [12, Definition
7].

######  Definition 4.3.

The waves $\sigma_{\ell,i,\alpha}$ and $\sigma_{\ell,j,\beta}$ are said to be
_approaching_ if $x_{\alpha}<x_{\beta}$ and $i>j$ or if $i=j$ and
$\min\\{\sigma_{\ell,i,\alpha},\sigma_{\ell,j,\beta}\\}<0$.

For a given $\gamma>0$, we introduce the following weighted functionals

(4.8)
$V_{\gamma}(t)=\sum_{\ell=1}^{n}\sum_{\alpha\in\mathcal{J}_{\ell}}\left(2K_{J}\left|\sigma_{\ell,1,\alpha}\right|e^{\gamma
x_{\alpha}}+\left|\sigma_{\ell,2,\alpha}\right|e^{-\gamma x_{\alpha}}\right),$

where $x_{\alpha}$ denotes the position of the discontinuity. Note that, for
every $\ell\in\\{1,\ldots,n\\}$ and $\alpha\in\mathcal{J}_{\ell}$, either
$\sigma_{\ell,1,\alpha}=0$ or $\sigma_{\ell,2,\alpha}=0$, since every
discontinuity corresponds to a wave of the first family or to a wave of the
second family. Moreover, we define the quadratic weighted functionals

(4.9) $Q_{\gamma}=Q_{\gamma}^{1,1}+Q_{\gamma}^{2,2}+Q_{\gamma}^{1,2},$

where

$\displaystyle Q_{\gamma}^{1,1}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum\left\\{\left|\sigma_{\ell,1,\alpha}e^{\gamma
x_{\alpha}}\sigma_{\ell,1,\beta}e^{\gamma
x_{\beta}}\right|:\left(\sigma_{\ell,1,\alpha},\sigma_{\ell,1,\beta}\right)\in\mathcal{A}_{\ell}\right\\},$
$\displaystyle Q_{\gamma}^{2,2}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum\left\\{\left|\sigma_{\ell,2,\alpha}e^{-\gamma
x_{\alpha}}\sigma_{\ell,2,\beta}e^{-\gamma
x_{\beta}}\right|:\left(\sigma_{\ell,2,\alpha},\sigma_{\ell,2,\beta}\right)\in\mathcal{A}_{\ell}\right\\},$
$\displaystyle Q_{\gamma}^{1,2}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum\left\\{\left|\sigma_{\ell,1,\alpha}e^{\gamma
x_{\alpha}}\sigma_{\ell,2,\beta}e^{-\gamma
x_{\beta}}\right|:\left(\sigma_{\ell,1,\alpha},\sigma_{\ell,2,\beta}\right)\in\mathcal{A}_{\ell}\right\\},$

where, as in [11, Section 7.3], $\mathcal{A}_{\ell}$ denotes the set of
approaching waves in the $\ell$-th pipe, and $\mathcal{J}_{\ell}$ is the set
of jumps of the solution in the in the $\ell$-th pipe. Finally, for
$\kappa>0$, we define

(4.10) $J_{\gamma}(t)=V_{\gamma}(t)+\kappa\,Q_{\gamma}(t).$

######  Remark 4.4.

The functional $J_{\gamma}$ is a Glimm-type functional with exponential
weights inspired by the one proposed in [17] for the stabilization of a
$2\times 2$ system with strictly positive velocities. It is essentially
composed by a linear part, namely $V_{\gamma}$, and by the quadratic part
$Q_{\gamma}$.

The term $V_{\gamma}$ measures the total strength of the waves of a wave-front
tracking approximate solution. The strengths of the waves of the first family
are weighted by the constant $2K_{J}$ for controlling the total variation
increment due to waves’ interactions with the junction. Moreover, the
exponential terms produce an exponential decay of $V_{\gamma}$ along a wave-
front tracking approximate solution. Note that $V_{\gamma}$ is equivalent to
the total variation of the approximate solution; see for example [11,
Proposition 4.3].

Finally, $Q_{\gamma}$ is a quadratic interaction potential and is composed by
three parts, namely $Q_{\gamma}^{h_{1},h_{2}}$ for $h_{1},h_{2}\in\\{1,2\\}$.
The term $Q_{\gamma}^{h_{1},h_{2}}$ considers all the possible interactions
between waves of the $h_{1}$ and $h_{2}$ families.

###  4.3. Interaction estimates

First, we recall the interaction estimates inside a duct; see [9, Lemma 7.2]
and [15, Lemma 4.1 & Proposition 4.2].

######  Lemma 4.5.

There exists a constant $K$ with the following property.

  1. (1)

If there is an interaction in a duct between two waves $\sigma_{1}^{-}$and
$\sigma_{2}^{-}$, respectively of the first and the second family, producing
the waves $\sigma_{1}^{+}$and $\sigma_{2}^{+}$, then

$\displaystyle\left|\sigma_{1}^{+}-\sigma_{1}^{-}\right|+\left|\sigma_{2}^{+}-\sigma_{2}^{-}\right|\leq
K\cdot\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|;$

see Figure 2 (left).

  2. (2)

If there is an interaction in a duct between two waves $\sigma_{i}^{\prime}$
and $\sigma_{i}^{\prime\prime}$ of the same $i$-th family producing waves of
total size $\sigma_{1}^{+}$and $\sigma_{2}^{+}$, then

$\displaystyle\begin{array}[]{ll}\left|\sigma_{1}^{+}-\left(\sigma_{1}^{\prime\prime}+\sigma_{1}^{\prime}\right)\right|+\left|\sigma_{2}^{+}\right|\leq
K\cdot\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|&\text{ if
}i=1,\\\
\left|\sigma_{1}^{+}\right|+\left|\sigma_{2}^{+}-\left(\sigma_{2}^{\prime\prime}+\sigma_{2}^{\prime}\right)\right|\leq
K\cdot\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|&\text{ if
}i=2;\end{array}$

see Figure 2 (middle and right).

As a consequence we deduce the monotonicity of the functional $J_{\gamma}$,
defined in 4.10.

$\sigma^{+}_{2}$ 2$\sigma^{-}_{1}$ 1$\sigma_{1}^{+}$ 1$\sigma_{2}^{-}$
2$\bar{t}$$\bar{x}$$\sigma_{2}^{+}$ 2$\sigma_{2}^{\prime\prime}$
2$\sigma_{1}^{+}$ 1$\sigma_{2}^{\prime}$ 2$\bar{t}$$\bar{x}$$\sigma_{2}^{+}$
2$\sigma_{1}^{\prime\prime}$ 1$\sigma_{1}^{+}$ 1$\sigma_{1}^{\prime}$
1$\bar{t}$$\bar{x}$ Figure 2. Wave-front interactions at $\bar{x}\in(0,1)$ at
$t=\bar{t}$.

######  Corollary 4.6.

Consider $\kappa>4KK_{J}(e^{\gamma}+e^{3\gamma})$. Assume that at time
$\bar{t}$ there is an interaction in a tube between two waves. Then

(4.11) $\Delta J_{\gamma}(\bar{t})<0.$

###### Proof.

Assume first that the interaction happens at the location $\bar{x}$ between a
wave of the first family with strength $\sigma_{1}^{-}$ with a wave of the
second family with strength $\sigma_{2}^{-}$. By Lemma 4.5 the emerging waves
$\sigma_{1}^{+}$ and $\sigma_{2}^{+}$ satisfy

$\left|\sigma_{1}^{+}-\sigma_{1}^{-}\right|+\left|\sigma_{2}^{+}-\sigma_{2}^{-}\right|\leq
K\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|.$

Then, since $V_{\gamma}$ is sufficiently small,

$\displaystyle\Delta J_{\gamma}(\bar{t})$ $\displaystyle=\Delta
V_{\gamma}(\bar{t})+\kappa\Delta Q_{\gamma}(\bar{t})$ $\displaystyle\leq
2K_{J}\left|\sigma_{1}^{+}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}\right|e^{-\gamma\bar{x}}-2K_{J}\left|\sigma_{1}^{-}\right|e^{\gamma\bar{x}}-\left|\sigma_{2}^{-}\right|e^{-\gamma\bar{x}}$
$\displaystyle\quad-\kappa\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|+\kappa
K\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|e^{2\gamma}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq
2K_{J}(\left|\sigma_{1}^{+}-\sigma_{1}^{-}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}-\sigma_{2}^{-}\right|e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|$
$\displaystyle\leq
2K_{J}K\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|$
$\displaystyle\leq\left(2K_{J}K(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}\right)\left|\sigma_{1}^{-}\sigma_{2}^{-}\right|<0.$

Assume now that the interacting waves, $\sigma_{1}^{\prime}$ and
$\sigma_{1}^{\prime\prime}$, are both of the first family and that the
interaction happens at the location $\bar{x}$. By Lemma 4.5 the emerging waves
$\sigma_{1}^{+}$ and $\sigma_{2}^{+}$ satisfy

$\left|\sigma_{1}^{+}-(\sigma_{1}^{\prime}+\sigma_{1}^{\prime\prime})\right|+\left|\sigma_{2}^{+}\right|\leq
K\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|.$

Then, since $V_{\gamma}$ is sufficiently small,

$\displaystyle\Delta J_{\gamma}(\bar{t})$ $\displaystyle=\Delta
V_{\gamma}(\bar{t})+\kappa\Delta Q_{\gamma}(\bar{t})$ $\displaystyle\leq
2K_{J}\left|\sigma_{1}^{+}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}\right|e^{-\gamma\bar{x}}-2K_{J}\left|\sigma_{1}^{\prime}\right|e^{\gamma\bar{x}}-2K_{J}\left|\sigma_{1}^{\prime\prime}\right|e^{\gamma\bar{x}}$
$\displaystyle\quad-\kappa\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|e^{2\gamma\bar{x}}+\kappa
K\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|e^{2\gamma\bar{x}}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq
2K_{J}(\left|\sigma_{1}^{+}-\sigma_{1}^{\prime}-\sigma_{1}^{\prime\prime}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}\right|e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|e^{2\gamma\bar{x}}$
$\displaystyle\leq
2K_{J}K\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|e^{2\gamma\bar{x}}$
$\displaystyle\leq\left(2K_{J}K(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}e^{2\gamma\bar{x}}\right)\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|<0.$

Assume now that the interacting waves, $\sigma_{2}^{\prime}$ and
$\sigma_{2}^{\prime\prime}$, are both of the second family and that the
interaction happens at the location $\bar{x}$. By Lemma 4.5, the emerging
waves $\sigma_{1}^{+}$ and $\sigma_{2}^{+}$ satisfy

$\left|\sigma_{1}^{+}\right|+\left|\sigma_{2}^{+}-(\sigma_{2}^{\prime}+\sigma_{2}^{\prime\prime})\right|\leq
K\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|.$

Then, since $V_{\gamma}$ is sufficiently small,

$\displaystyle\Delta J_{\gamma}(\bar{t})$ $\displaystyle=\Delta
V_{\gamma}(\bar{t})+\kappa\Delta Q_{\gamma}(\bar{t})$ $\displaystyle\leq
2K_{J}\left|\sigma_{1}^{+}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}\right|e^{-\gamma\bar{x}}-2K_{J}\left|\sigma_{2}^{\prime}\right|e^{-\gamma\bar{x}}-2K_{J}\left|\sigma_{2}^{\prime\prime}\right|e^{-\gamma\bar{x}}$
$\displaystyle\quad-\kappa\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|e^{-2\gamma\bar{x}}+\kappa
K\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|e^{-2\gamma\bar{x}}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq
2K_{J}(\left|\sigma_{1}^{+}\right|e^{\gamma\bar{x}}+\left|\sigma_{2}^{+}-\sigma_{2}^{\prime}-\sigma_{2}^{\prime\prime}\right|e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|e^{-2\gamma\bar{x}}$
$\displaystyle\leq
2K_{J}K\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|e^{-2\gamma\bar{x}}$
$\displaystyle\leq\left(2K_{J}K(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}e^{-2\gamma\bar{x}}\right)\left|\sigma_{2}^{\prime}\sigma_{2}^{\prime\prime}\right|<0,$

concluding the proof. ∎

The following result deals with the interactions of waves with the junction.

######  Lemma 4.7.

There exist $\delta_{J}>0$ and $K_{J}\geq 1$ with the following property. For
any $\bar{u}\in(\mathring{A}_{0})^{N}$ that yields a equilibrium solution to
the Riemann problem, for any 1-waves
$\sigma_{l}^{-}\in(-\delta_{J},\delta_{J})$ hitting the junction and producing
the 2-waves $\sigma_{l}^{+}$, it holds that

$\displaystyle\sum_{\ell=1}^{N}\left|\sigma_{\ell}^{+}\right|\leq
K_{J}\cdot\sum_{\ell=1}^{n}\left|\sigma_{\ell}^{-}\right|;$

see Figure 3.

For the proof, see [15, Proposition 4.2].

$\sigma_{2}^{+}$ 2$\sigma_{1}^{-}$ 1$x=0$$\bar{t}$ Figure 3. Wave-front
interactions at the junction $x=0$ at $t=\bar{t}$.

######  Corollary 4.8.

Assume that at time $\bar{t}$ there is an interaction between a wave and the
junction $J$. Then

(4.12) $\Delta J_{\gamma}(\bar{t})<0.$

###### Proof.

Assume, without loss of generality, that the interaction is due to a wave of
the first family of strength $\sigma_{1}^{-}$ from the pipe $\ell=1$. Denote
with $\sigma_{\ell}^{+}$ the strength of the emerging waves of the second
family in the pipe $\ell=1,\ldots,n$. By Lemma 4.7, we obtain that

$\displaystyle\sum_{l=1}^{n}\left|\sigma_{l}^{+}\right|\leq
K_{J}\cdot\left|\sigma_{1}^{-}\right|.$

Then, since $V_{\gamma}$ is sufficiently small,

$\displaystyle\Delta J_{\gamma}(\bar{t})$ $\displaystyle=\Delta
V_{\gamma}(\bar{t})+\kappa\Delta Q_{\gamma}(\bar{t})$
$\displaystyle\leq\sum_{\ell=1}^{n}\left|\sigma_{\ell}^{+}\right|-2K_{J}\left|\sigma_{1}^{-}\right|+\kappa\sum_{l=1}^{n}\left|\sigma_{\ell}^{+}\right|V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq
K_{J}\left|\sigma_{1}^{-}\right|-2K_{J}\left|\sigma_{1}^{-}\right|+\kappa
K_{J}\left|\sigma_{1}^{-}\right|V_{\gamma}(\bar{t}^{-})$ $\displaystyle\leq-
K_{J}\left|\sigma_{1}^{-}\right|+\kappa
K_{J}\left|\sigma_{1}^{-}\right|V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq-\frac{K_{J}}{2}\left|\sigma_{1}^{-}\right|<0,$

concluding the proof. ∎

Finally, we deal with the interaction of a wave with the external boundary.

######  Lemma 4.9.

There exists a constant $C>0$ with the following property. Assume that a wave
of the second family with strength $\sigma_{2}^{-}$ interacts with the
external boundary $x=1$ at a time $\bar{t}$ from the $\ell$-th pipe,
$\ell\in\left\\{1,\ldots,N\right\\}$; see Figure 4. Then, the emerging wave of
the first family has a strength $\sigma_{1}^{+}$ satisfying the estimate

(4.13) $\left|\sigma_{1}^{+}\right|\leq Ck_{\ell}\left|\sigma_{2}^{-}\right|.$

$\sigma_{1}^{+}$ 1$\sigma_{2}^{-}$ 2$x=1$$\bar{t}$ Figure 4. Interaction of a
wave of the second family at time $\bar{t}$ with the external boundary located
at position $x=1$. A wave of the first family is generated.

###### Proof.

Let us denote by $u_{l}$ and $u_{m}$ the states on the left and on the right
of the interacting wave, respectively, so that
$u_{m}=\mathcal{L}_{2}\left(\sigma_{2}^{-}\right)\left(u_{l}\right)$.

Since $u_{m}$ is an equilibrium at the external boundary, then

(4.14)
$v_{1}\left(u_{m}\right)=k_{\ell}v_{2}\left(u_{m}\right)-k_{\ell}v_{2}\left(\bar{u}_{\ell}\right)+v_{1}\left(\bar{u}_{\ell}\right).$

The emerging wave $(u_{l},u_{r})$ satisfies
$u_{r}=\mathcal{L}_{1}\left(\sigma_{1}^{+}\right)\left(u_{l}\right)$ and

(4.15)
$v_{1}\left(u_{r}\right)=k_{\ell}v_{2}\left(u_{r}\right)-k_{\ell}v_{2}\left(\bar{u}_{\ell}\right)+v_{1}\left(\bar{u}_{\ell}\right).$

Consider the function

$\psi\left(\sigma_{1},\sigma_{2}\right):=v_{2}\left(\mathcal{L}_{1}\left(\sigma_{1}\right)\left(u_{l}\right)\right)-k_{\ell}v_{1}\left(\mathcal{L}_{1}\left(\sigma_{1}\right)\left(u_{l}\right)\right)-v_{2}\left(\mathcal{L}_{2}\left(\sigma_{2}\right)\left(u_{l}\right)\right)+k_{\ell}v_{1}\left(\mathcal{L}_{2}\left(\sigma_{2}\right)\left(u_{l}\right)\right),$

whose zeros represent the possible strengths of the interacting and generated
waves at the right boundary. The functions $v_{2}$ and $v_{1}$, defined in
(2.3), are of class $C^{2}$ thanks to (P). Also the Lax curves have the same
regularity; see [9, Chapter 5]. Therefore, $\psi$ is of class $C^{2}$.
Clearly,

$\psi\left(0,0\right)=v_{2}\left(u_{l}\right)-k_{\ell}v_{1}\left(u_{l}\right)-v_{2}\left(u_{l}\right)+k_{\ell}v_{1}\left(u_{\ell}\right)=0.$

Moreover,

$\displaystyle\partial_{\sigma_{1}}\psi\left(\sigma_{1},\sigma_{2}\right)$
$\displaystyle=\nabla
v_{2}\left(\mathcal{L}_{1}\left(\sigma_{1}\right)\left(u_{l}\right)\right)\cdot\mathcal{L}_{1}^{\prime}\left(\sigma_{1}\right)\left(u_{l}\right)-k_{\ell}\nabla
v_{1}\left(\mathcal{L}_{1}\left(\sigma_{1}\right)\left(u_{l}\right)\right)\cdot\mathcal{L}_{1}^{\prime}\left(\sigma_{1}\right)\left(u_{l}\right),$
$\displaystyle\partial_{\sigma_{2}}\psi\left(\sigma_{1},\sigma_{2}\right)$
$\displaystyle=-\nabla
v_{2}\left(\mathcal{L}_{2}\left(\sigma_{2}\right)\left(u_{l}\right)\right)\cdot\mathcal{L}_{2}^{\prime}\left(\sigma_{2}\right)\left(u_{l}\right)+k_{\ell}\nabla
v_{1}\left(\mathcal{L}_{2}\left(\sigma_{2}\right)\left(u_{l}\right)\right)\cdot\mathcal{L}_{2}^{\prime}\left(\sigma_{2}\right)\left(u_{l}\right),$

and so, by (2.4),

$\displaystyle\partial_{\sigma_{1}}\psi\left(0,0\right)$ $\displaystyle=\nabla
v_{2}\left(u_{l}\right)\cdot r_{1}\left(u_{l}\right)-k_{\ell}\nabla
v_{1}\left(u_{l}\right)\cdot r_{1}\left(u_{l}\right)=\nabla
v_{2}\left(u_{l}\right)\cdot r_{1}\left(u_{l}\right)>0,$
$\displaystyle\partial_{\sigma_{2}}\psi\left(0,0\right)$
$\displaystyle=-\nabla v_{2}\left(u_{l}\right)\cdot
r_{2}\left(u_{l}\right)+k_{\ell}\nabla v_{1}\left(u_{l}\right)\cdot
r_{2}\left(u_{l}\right)=k_{\ell}\nabla v_{1}\left(u_{l}\right)\cdot
r_{2}\left(u_{l}\right)>0.$

Since $\partial_{\sigma_{1}}\psi(0,0)\neq 0$, by the implicit function
theorem, there exists a $C^{2}$ function $\sigma_{1}=\sigma_{1}(\sigma_{2})$
with bounded derivative such that $\sigma_{1}(0)=0$ and
$\psi(\sigma_{1}(\sigma_{2}),\sigma_{2})=0$ for every $\sigma_{2}$ in a
suitable neighborhood of $0$. Moreover

$\sigma_{1}^{\prime}(0)=-\frac{k_{\ell}\nabla v_{1}\left(u_{l}\right)\cdot
r_{2}\left(u_{l}\right)}{\nabla v_{2}\left(u_{l}\right)\cdot
r_{1}\left(u_{l}\right)}<0,$

which implies the existence of a constant $C>0$, depending only on $\bar{u}$
and $\delta$, such that

$\left|\sigma_{1}(\sigma_{2})\right|\leq Ck_{\ell}\left|\sigma_{2}\right|$

for every $\sigma_{2}$ in a suitable neighborhood of $0$. This permits to
prove (4.13). ∎

######  Remark 4.10.

Note that in the previous proof we deduce that the implicit function
$\sigma_{2}\mapsto\sigma_{1}(\sigma_{2})$ is strictly decreasing since its
derivative is strictly negative. By the choice of the parametrization of the
Lax curves, see (2.2), we deduce that if the interacting wave is a shock
(resp., a rarefaction), then the emerging wave is a rarefaction (resp., a
shock).

######  Corollary 4.11.

Let $K_{J}$ be as in Lemma 4.7 and $C>0$ as in Lemma 4.9. Fix
$k_{\ell}\leq\frac{e^{-2\gamma}}{4CK_{J}}$ for every $\ell=1,\ldots,n$. Assume
that at time $\bar{t}$ there is an interaction between a wave and the external
boundary. Then

(4.16) $\Delta J_{\gamma}(\bar{t})<0.$

###### Proof.

Assume, without loss of generality, that the interaction is due to a wave of
the second family of strength $\sigma_{2}^{-}$ from the $\ell$-th pipe. Denote
with $\sigma_{1}^{+}$ the strength of the emerging waves of the first family
in that pipe. By Lemma 4.9, we obtain that

$\displaystyle\left|\sigma_{1}^{+}\right|\leq
Ck_{\ell}\left|\sigma_{2}^{-}\right|.$

Then, since $V_{\gamma}$ is sufficiently small,

$\displaystyle\Delta J_{\gamma}(\bar{t})$ $\displaystyle=\Delta
V_{\gamma}(\bar{t})+\kappa\Delta Q_{\gamma}(\bar{t})$ $\displaystyle\leq
2K_{J}\left|\sigma_{1}^{+}\right|e^{\gamma}-\left|\sigma_{2}^{-}\right|e^{-\gamma}+\kappa\left|\sigma_{1}^{+}\right|e^{\gamma}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq
2CK_{J}k_{\ell}e^{\gamma}\left|\sigma_{2}^{-}\right|-\left|\sigma_{2}^{-}\right|e^{-\gamma}+\kappa\left|\sigma_{1}^{+}\right|e^{\gamma}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq(2CK_{J}k_{\ell}e^{\gamma}-e^{-\gamma})\left|\sigma_{2}^{-}\right|+C\kappa
k_{\ell}\left|\sigma_{2}^{-}\right|e^{\gamma}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq-\frac{e^{-\gamma}}{2}\left|\sigma_{2}^{-}\right|+C\kappa
k_{\ell}\left|\sigma_{2}^{-}\right|e^{\gamma}V_{\gamma}(\bar{t}^{-})$
$\displaystyle\leq-\frac{e^{-\gamma}}{4}\left|\sigma_{2}^{-}\right|<0,$

concluding the proof. ∎

###  4.4. Existence of an approximate wave-front tracking solution

In this subsection, we deal with the existence of a wave-front tracking
approximate solution, in the sense of Definition 4.1. In the wave-front
tracking approximate solution, we do not consider the so-.called non-physical
waves, which is used to control the total number of waves and interactions in
the case of systems (see [9, Chapter 7]). Indeed for a $2\times 2$ system with
suitable assumptions, it is possible to avoid such technicality (see [2, Lemma
2.3] and [6]).

######  Lemma 4.12.

Fix $\varepsilon>0$ and consider a wave-front tracking approximate solution
$u_{\varepsilon}$ as in Section 4.1.

Then, inside every pipe and except for a finite number of interactions, there
is at most one outgoing wave of each family for every interaction.

###### Proof.

Inside a pipe, by construction, only the interactions of waves of the same
family can produce a rarefaction fan at positive times. More precisely, if two
waves of the $k_{1}$ family ($k_{1}\in\\{1,2\\}$) interacts at some positive
time $\bar{t}$ inside a pipe, then the emerging wave of the $k_{2}$ family
($k_{2}\in\\{1,2\\}\setminus\\{k_{1}\\}$) can be a rarefaction of strength
bigger than $\varepsilon$ and so it is split in a rarefaction fan. Without
loss of generality, let us assume that at time $\bar{t}>0$ two waves of the
first family of strength $\sigma_{1}^{\prime}$ and $\sigma_{1}^{\prime\prime}$
interact together producing a rarefaction of the second family of strength
$\sigma_{2}$ with $\left|\sigma_{2}\right|>\varepsilon$. Then, using Lemma 4.5
and Corollary 4.6, we deduce that

$\varepsilon<\left|\sigma_{2}\right|\leq
K\left|\sigma_{1}^{\prime}\sigma_{1}^{\prime\prime}\right|\leq\frac{K}{\left(2K_{J}K(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}e^{2\gamma\bar{x}}\right)}\Delta
J_{\gamma}(\bar{t}).$

This implies that

$\Delta
J_{\gamma}(\bar{t})\leq\frac{\varepsilon\left(2K_{J}K(e^{\gamma\bar{x}}+e^{-\gamma\bar{x}})-\frac{\kappa}{2}e^{2\gamma\bar{x}}\right)}{K}<0.$

Since the functional $J_{\gamma}$ is non increasing, then we deduce that such
interactions can happen at most a finite number of times. ∎

######  Proposition 4.13.

For every $\varepsilon>0$, the construction illustrated in Section 4.1
produces a wave-front tracking approximate solution, defined for every time
$t\geq 0$.

###### Proof.

Fix $\varepsilon>0$ and consider a wave-front tracking approximate solution
$u_{\varepsilon}$. We need to prove that the total number of interactions (and
consequently of waves) remain finite.

First, assume, by contradiction, that there exists an infinite number of
interactions. Define $T$ such that, for $t<T$ there are a finite number of
waves and interactions, but at time $t=T$ there are infinitely many,
accumulating at a point $(T,\bar{x})$.

Case 1: $\bar{x}\in(0,1)$. This case is completely similar to the one of [2,
Lemma 2.5]. We report here for completeness.

Fix $\alpha>0$ such that $\bar{x}-2\alpha>0$ and $\bar{x}+2\alpha<1$. Fix
$\Delta t>0$ such that $\Delta t<\frac{\alpha}{\Lambda_{\max}}$. Consider the
rectangle

$R=[T-\Delta t,T]\times[\bar{x}-\alpha,\bar{x}+\alpha].$

There exists a sequence $(t_{i},x_{i})$ of interaction points belonging to $R$
such that

$(t_{i},x_{i})\to(T,\bar{x})$

and $t_{1}<t_{2}<\cdots.$ Define $\mathcal{I}$ as the set of all the points
$(t_{i},x_{i})$.

Define $\mathcal{F}$ the sets of all the waves which can be joined, forward in
time, to some points on $\mathcal{I}$ and which intersects the set $R$. We
split $\mathcal{F}=\mathcal{F}_{1}\cup\mathcal{F}_{2}$, where
$\mathcal{F}_{1}$, resp. $\mathcal{F}_{2}$, is the subset of $\mathcal{F}$ of
waves of the first family, resp. of the second family.

Consider the following sets.

  1. (1)

$\mathcal{I}_{1}$: the set of all interaction points of $R$ in which there are
exatcly two outgoing waves, one of the first family and one of the second
family.

  2. (2)

$\mathcal{I}_{2}$: the set of all interaction points of $R$ in which the two
interacting waves belongs to $\mathcal{F}$, but there is at most one ougoing
waves belonging to $\mathcal{F}$.

  3. (3)

$\mathcal{I}_{3}$: the set of all interaction points of $R$ in which the two
interacting waves do not belong to $\mathcal{F}$.

  4. (4)

$\mathcal{I}_{4}$: the set of all interaction points of $R$ in which the two
interacting waves belong to $\mathcal{F}$ and there are at least two outgoing
waves of the same family belonging to $\mathcal{F}$.

By Lemma 4.12, the set $\mathcal{I}_{4}$ is finite.

Define, for $t\in[T-\Delta t,T]$, the functional $\mathcal{V}(t)$ as the
number of waves, that at time $t$ belongs to $\mathcal{F}$. Note that, for
interactions in $\mathcal{I}_{1}$ and $\mathcal{I}_{3}$ the functional
$\mathcal{V}$ does not change, while $\mathcal{V}$ strictly decreases by $1$
or $2$ for interactions in $\mathcal{I}_{2}$. Finally, $\mathcal{V}$ can
increase for interactions in $\mathcal{I}_{4}$.

Moreover, $\mathcal{V}(T-\Delta t)$ is finite, $\mathcal{V}(t)\geq 0$ for all
$t$, therefore, since $\mathcal{I}_{4}$ is finite, also $\mathcal{I}_{2}$ is
finite.

Note that all the points in $\mathcal{I}_{3}$ do not belong to $\mathcal{I}$.

Starting from $\left(t_{1},x_{1}\right)$, we go forward in time with two
continuous lines: the first one made of waves of $\mathcal{F}_{1}$ (first
family) and the second one using waves of $\mathcal{F}_{2}$ (second family).
When we reach an interaction point $(\tilde{t},\tilde{x})$ belonging to
$\mathcal{I}_{2}$ or $\mathcal{I}_{4}$, we stop and start over from a point
$\left(t_{j},x_{j}\right)$, with $t_{j}>\tilde{t}$.

Since $\mathcal{I}_{2}$ and $\mathcal{I}_{4}$ are finite sets, there exists a
point $\left(t^{*},x^{*}\right)\in\mathcal{I}$ from which we can construct two
lines $\gamma_{1}(t),\gamma_{2}(t)$ until the time $T$.

The bounds on the velocities imply

$\gamma_{1}(t)\leqslant
x^{*}-c\left(t-t^{*}\right),\quad\gamma_{2}(t)\geqslant
x^{*}+c\left(t-t^{*}\right).$

Choose $\bar{T}\in(t^{*},T)$ such that

(4.17) $(T-\bar{T})\Lambda_{\max}<c(\bar{T}-t^{*})$

and fix $\bar{n}$ such that $t_{\bar{n}}>\bar{T}$.

Since $\gamma_{1}$ is composed of segments of $\mathcal{F}$, the point
$\left(t_{\bar{n}},\gamma_{1}\left(t_{\bar{n}}\right)\right)$ can be joined to
some point $(t_{h},x_{h})$ of $\mathcal{I}$. Moreover, the bounds on the
velocities imply that

$x_{h}\leqslant\gamma_{1}\left(t_{\bar{n}}\right)+\Lambda_{\max}\left(t_{h}-t_{\bar{n}}\right),\quad
h\geqslant\bar{n}.$

Analogously, since $\gamma_{2}$ is composed of segments of $\mathcal{F}$, the
point $\left(t_{\bar{n}},\gamma_{2}\left(t_{\bar{n}}\right)\right)$ can be
joined to some point $(t_{k},x_{k})$ of $\mathcal{I}$. The bounds on the
velocities imply that

$x_{k}\geqslant\gamma_{2}\left(t_{\bar{n}}\right)-\Lambda_{\max}\left(t_{h}-t_{\bar{n}}\right),\quad
k\geqslant\bar{n}.$

Putting these estimates together, we conclude

$\displaystyle x_{k}-x_{h}$
$\displaystyle\geqslant\gamma_{2}\left(t_{\bar{n}}\right)-\Lambda_{\max}\left(t_{k}-t_{\bar{n}}\right)-[\gamma_{1}\left(t_{\bar{n}}\right)+\Lambda_{\max}\left(t_{h}-t_{\bar{n}}\right)]$
$\displaystyle\geq
2c(t_{n}-t^{*})-\Lambda_{\max}(t_{k}-t_{\bar{n}}+t_{h}-t_{\bar{n}})$
$\displaystyle\geq 2c(\bar{T}-t^{*})-2\Lambda_{\max}(T-\bar{T})$
$\displaystyle>0.$

This contradicts the fact that the sequence $x_{i}$ tends to $\bar{x}$.

Case 2: $\bar{x}=1$. First, we suppose $T<\frac{1}{\Lambda_{\max}}$. Once we
obtain the result under this restriction, we can reproduce the same argument
in the time-intervals $[T,T+\frac{1}{\Lambda_{\max}}]$,
$[T+\frac{1}{\Lambda_{\max}},T+\frac{2}{\Lambda_{\max}}]$, and so on.

First, note that the number of interactions at $x=1$ is finite. The assumption
$T<\frac{1}{\Lambda_{\max}}$ implies that a wave of the second family
generated at the junction does not reach the boundary $x=1$ within time $T$.
Moreover, at positive times, new waves of the second family can be generated
only when two waves of the first family interact together, but this, by Lemma
4.12, can happen at most a finite number of times.

Therefore, there exists a sequence $(t_{i},x_{i})$ of interaction points such
that

$(t_{i},x_{i})\to(T,1)$

and $t_{1}<t_{2}<\cdots$ and $x_{1}<x_{2}<\cdots$. Define $\mathcal{I}$ as the
set of all the points $(t_{i},x_{i})$.

Define $\mathcal{F}$ the sets of all the waves which can be joined, forward in
time, to some points on $\mathcal{I}$. We split
$\mathcal{F}=\mathcal{F}_{1}\cup\mathcal{F}_{2}$, where $\mathcal{F}_{1}$,
resp. $\mathcal{F}_{2}$, is the subset of $\mathcal{F}$ of waves of the first
family, resp. of the second family.

Consider the following sets.

  1. (1)

$\mathcal{I}_{1}$: the set of all interaction points in which there are
exactly two outgoing waves, one of the first family and one of the second
family.

  2. (2)

$\mathcal{I}_{2}$: the set of all interaction points in which the two
interacting waves belongs to $\mathcal{F}$, but there is at most one ougoing
waves belonging to $\mathcal{F}$.

  3. (3)

$\mathcal{I}_{3}$: the set of all interaction points in which the two
interacting waves do not belong to $\mathcal{F}$.

  4. (4)

$\mathcal{I}_{4}$: the set of all interaction points in which the two
interacting waves belong to $\mathcal{F}$ and there are at least two outgoing
waves of the same family belonging to $\mathcal{F}$.

  5. (5)

$\mathcal{I}_{5}$: the set of all interaction points at $x=1$.

By Lemma 4.12, the set $\mathcal{I}_{4}$ is finite. Also $\mathcal{I}_{5}$ is
finite as already noted.

Define, for $t\in[0,T]$, the functional $\mathcal{V}(t)$ as the number of
waves, that at time $t$ belongs to $\mathcal{F}$. Note that, for interactions
in $\mathcal{I}_{1}$ and $\mathcal{I}_{3}$ the functional $\mathcal{V}$ does
not change, while $\mathcal{V}$ strictly decreases by $1$ or $2$ for
interactions in $\mathcal{I}_{2}$. Finally, $\mathcal{V}$ can increase for
interactions in $\mathcal{I}_{4}$ and $\mathcal{I}_{5}$.

Moreover, $\mathcal{V}(0)$ is finite, $\mathcal{V}(t)\geq 0$ for all $t$,
therefore, since $\mathcal{I}_{4}$ and $\mathcal{I}_{5}$ are finite, also
$\mathcal{I}_{2}$ is finite.

Note that all the points in $\mathcal{I}_{3}$ do not belong to $\mathcal{I}$.

Starting from $\left(t_{1},x_{1}\right)$, we go forward in time with two
continuous lines: the first one made of waves of $\mathcal{F}_{1}$ (first
family) and the second one using waves of $\mathcal{F}_{2}$ (second family).
When we reach an interaction point $(\tilde{t},\tilde{x})$ belonging to
$\mathcal{I}_{2}$, $\mathcal{I}_{4}$, or $\mathcal{I}_{5}$, we stop and start
over from a point $\left(t_{j},x_{j}\right)$, with $t_{j}>\tilde{t}$.

Since $\mathcal{I}_{2}$, $\mathcal{I}_{4}$, and $\mathcal{I}_{5}$ are finite
sets, there exists a point $\left(t^{*},x^{*}\right)\in\mathcal{I}$ from which
we can construct two lines $\gamma_{1}(t),\gamma_{2}(t)$ until the time $T$.
We conclude now exactly as in Case 1.

Case 3: $\bar{x}=0$. We proceed as in Case 2, by noticing that there are at
most a finite number of wave interactions with the junction at $x=0$, provided
$\Lambda_{\max}T<1$. ∎

###  4.5. Existence of a solution

Let us consider a $\varepsilon$-wave-front tracking solution
$u_{\varepsilon}$. By the interaction estimates, using the functional $J_{0}$
on $u_{\varepsilon}$ (i.e., $J_{\gamma}$ with $\gamma=0$), we deduce, owing to
Corollary 4.6, Corollary 4.8, and Corollary 4.11

(4.18) $J_{0}(t)\leq J_{0}(0^{+})$

since $J_{0}$ varies only at times of interaction. Therefore, we have

(4.19) $\mathrm{TV}(u_{\varepsilon}(t))\leq J_{0}(t)\leq J_{0}(0^{+}).$

Since it is standard to prove that there exists $L>0$ such that

$\sum_{\ell=1}^{N}\left\|(\rho_{\ell,\varepsilon}(t),q_{\ell,\varepsilon}(t))-(\rho_{\ell,\varepsilon}(s),q_{\ell,\varepsilon}(s))\right\|_{L^{1}(I_{\ell})}\leq
L\left|t-s\right|$

for every $t,s\geq 0$, then we conclude, by Helly’s compactness theorem (see
[9, Theorem 2.4]), that $u_{\varepsilon}$ converges strongly in $L^{1}$ to a
limit point $u$. Moreover, it is standard to deduce that $u$ satisfies Theorem
3.1-(i) as well as an entropy condition, i.e., Theorem 3.1-(iv).

###  4.6. Existence of a semigroup

Let us consider two $\varepsilon$-wave-front tracking solutions $u$ and
$\tilde{u}$, and define the functional

(4.20)
$\Phi(u,\tilde{u})\coloneqq\sum_{\ell=1}^{N}\sum_{i=1}^{2}\int_{I_{\ell}}\left|s_{\ell,i}(x)\right|W_{\ell,i}(x)\,\mathrm{d}x.$

Here, $s_{\ell,i}(x)$ measures the strengths of the $i$-th shock wave in the
$\ell$ pipe at the point $x$; the weights $W_{\ell,i}(x)$ are defined by

$W_{\ell,i}(x)\coloneqq
1+\kappa_{1}A_{\ell,i}(x)+\kappa_{1}\kappa_{2}(J_{0}(u)+J_{0}(v))$

for suitable $\kappa_{1}>0$ and $\kappa_{2}>0$;
$A_{\ell,i}(x)=A_{\ell,i}^{1}(x)+A^{2}_{\ell,i}(x)$, with

(4.21) $\displaystyle A_{\ell,i}^{1}(x)$
$\displaystyle=\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(u)\cup\mathcal{J}(\tilde{u})\\\
x_{\alpha}<x\\\ k_{\alpha}\neq i\\\
k_{\alpha}=2\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(u)\cup\mathcal{J}(\tilde{u})\\\
x_{\alpha}>x\\\ k_{\alpha}\neq i\\\
k_{\alpha}=1\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|,$
(4.24) $\displaystyle A_{\ell,i}^{2}(x)$
$\displaystyle=\left\\{\begin{array}[]{ll}\displaystyle\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(u)\\\
x_{\alpha}<x\\\
k_{\alpha}=i\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(\tilde{u})\\\
x_{\alpha}>x\\\
k_{\alpha}=i\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|,&\textrm{
if }s_{\ell,i}(x)<0,\vspace{0.2cm}\\\
\displaystyle\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(\tilde{u})\\\
x_{\alpha}<x\\\
k_{\alpha}=i\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|+\sum_{\begin{subarray}{c}\alpha\in\mathcal{J}(u)\\\
x_{\alpha}>x\\\
k_{\alpha}=i\end{subarray}}\left|\sigma_{\ell,k_{\alpha},\alpha}\right|,&\textrm{
if }s_{\ell,i}(x)\geq 0,\end{array}\right.$

(see [9, Eq. 8.9] and [15]). We set $\kappa_{1},\kappa_{2}$ to satisfy $1\leq
W_{\ell,i}(x)\leq 2$ for every $\ell\in\\{1,\ldots,N\\}$ and $i=1,2$. In this
way, the functional $\Phi$ is equivalent to the $L^{1}$-distance.

Exactly the same calculations as in [9, Chapter 8] imply that, for every $t>0$
when no interaction happens,

(4.25) $\frac{\mathrm{d}}{\mathrm{d}t}\Phi(u(t),\tilde{u}(t))\leq
C\varepsilon.$

If $t>0$ is an interaction time for $u$ or $\tilde{u}$, then Corollary 4.6,
Corollary 4.8, and Corollary 4.11 imply that

(4.26) $\Delta[J_{0}(u(t))+J_{0}(\tilde{u}(t))]<0,$

and so, choosing $\kappa_{2}$ large enough, we obtain

(4.27) $\Delta\Phi(u(t),\tilde{u}(t))<0.$

Thus, for every $0\leq s\leq t$, we obtain that

(4.28) $\Phi(u(t),\tilde{u}(t))-\Phi(u(s),\tilde{u}(s))\leq
C\varepsilon(t-s),$

proving Theorem 3.1-(iii).

In particular, from this Lipschitz dependence result, it is standard to deduce
the semigroup property claimed in Theorem 3.1-(ii).

###  4.7. Decay of the Glimm-type functional

In this part we prove that the Glimm-type functional $J_{\gamma}$, for
$\gamma>0$, has an exponential-in-time decay, similarly as in [17, Lemma 3.2].
Let a time $\bar{t}$ be fixed; we consider the variation of the functional
$J_{\gamma}$ locally around $\bar{t}$ according to four cases:

  1. (1)

there are no waves’ interactions at $t=\bar{t}$ inside the pipes nor an
interaction of a wave with the junction or the external boundary;

  2. (2)

there is an interaction between two waves inside a pipe at $t=\bar{t}$;

  3. (3)

there is a wave hitting the external boundary at time $t=\bar{t}$;

  4. (4)

there is a wave hitting the junction at time $t=\bar{t}$.

In the latter three cases, the estimates in Section 4.3 imply that $\Delta
J_{\gamma}(\bar{t})\leq 0$ provided $V_{\gamma}$ sufficiently small. Hence we
focus only on the first case, where no interaction happens.

######  Lemma 4.14.

Assume that at the time $\bar{t}$ no interactions of any type happen. Then the
functional $J_{\gamma}$, defined in (4.10), is differentiable and it holds

(4.29) $\frac{\mathrm{d}}{\mathrm{d}t}J_{\gamma}(\bar{t})\leq-c\gamma
J_{\gamma}(\bar{t}).$

###### Proof.

Assume that there is no any interactions at time $t=\bar{t}$. Since there is a
finite number of fronts at time $\bar{t}$, then there exists a neighborhood
$\mathcal{U}$ of $\bar{t}$ in which no interaction happens. Hence, in
$\mathcal{U}$, $V_{\gamma}(t)$ and $Q_{\gamma}$ are smooth functional.

First compute

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}V_{\gamma}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum_{\alpha\in\mathcal{J}_{\ell}}\Big{(}2K_{J}|\sigma_{\ell,1,\alpha}|e^{\gamma
x_{\alpha}(t)}\gamma\dot{x}_{\alpha}(t)-|\sigma_{\ell,2,\alpha}|e^{-\gamma
x_{\alpha}(t)}\gamma\dot{x}_{\alpha}(t)\Big{)}$
$\displaystyle\leq\sum_{\ell=1}^{n}\sum_{\alpha\in\mathcal{J}_{\ell}}\Big{(}-2c\gamma
K_{J}|\sigma_{\ell,1,\alpha}|e^{\gamma
x_{\alpha}(t)}-c\gamma|\sigma_{\ell,2,\alpha}|e^{-\gamma
x_{\alpha}(t)}\Big{)}$
$\displaystyle=-c\gamma\sum_{\ell=1}^{n}\sum_{\alpha\in\mathcal{J}_{\ell}}\Big{(}2K_{J}|\sigma_{\ell,1,\alpha}|e^{\gamma
x_{\alpha}(t)}+|\sigma_{\ell,2,\alpha}|e^{-\gamma
x_{\alpha}(t)}\Big{)}=-c\gamma V_{\gamma}(t).$

Moreover, we have

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}Q_{\gamma}^{1,1}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,1,\alpha},\sigma_{\ell,1,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,1,\alpha}\sigma_{\ell,1,\beta}\right|e^{\gamma(x_{\alpha}(t)+x_{\beta}(t))}\gamma(\dot{x}_{\alpha}(t)+\dot{x}_{\beta}(t))$
$\displaystyle\leq-c\gamma\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,1,\alpha},\sigma_{\ell,1,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,1,\alpha}\sigma_{\ell,1,\beta}\right|e^{\gamma(x_{\alpha}(t)+x_{\beta}(t))}=-c\gamma
Q_{\gamma}^{1,1}(t),$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}Q_{\gamma}^{2,2}(t)$
$\displaystyle=-\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,2,\alpha},\sigma_{\ell,2,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,2,\alpha}\sigma_{\ell,2,\beta}\right|e^{-\gamma(x_{\alpha}(t)+x_{\beta}(t))}\gamma(\dot{x}_{\alpha}(t)+\dot{x}_{\beta}(t))$
$\displaystyle\leq-c\gamma\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,2,\alpha},\sigma_{\ell,2,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,2,\alpha}\sigma_{\ell,2,\beta}\right|e^{-\gamma(x_{\alpha}(t)+x_{\beta}(t))}=-c\gamma
Q_{\gamma}^{2,2}(t),$
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}t}Q_{\gamma}^{1,2}(t)$
$\displaystyle=\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,1,\alpha},\sigma_{\ell,2,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,1,\alpha}\sigma_{\ell,2,\beta}\right|e^{\gamma(x_{\alpha}(t)-x_{\beta}(t))}\gamma(\dot{x}_{\alpha}(t)-\dot{x}_{\beta}(t))$
$\displaystyle\leq-c\gamma\sum_{\ell=1}^{n}\sum_{(\sigma_{\ell,1,\alpha},\sigma_{\ell,2,\beta})\in\mathcal{A}_{\ell}}\left|\sigma_{\ell,1,\alpha}\sigma_{\ell,2,\beta}\right|e^{\gamma(x_{\alpha}(t)-x_{\beta}(t))}=-c\gamma
Q_{\gamma}^{1,2}(t),$

and so

$\frac{\mathrm{d}}{\mathrm{d}t}Q_{\gamma}(t)\leq-c\gamma Q_{\gamma}(t).$

Therefore, we deduce that

$\frac{\mathrm{d}}{\mathrm{d}t}J_{\gamma}(t)\leq-c\gamma J_{\gamma}(t),$

concluding the proof. ∎

######  Corollary 4.15.

Fix $\gamma>0$. Then the functional $J_{\gamma}$, defined in (4.10), satisfies
the inequality

(4.30) $J_{\gamma}(t)\leq e^{-c\gamma}J_{\gamma}(0^{+}).$

The proof is immediate and so we omit it. This completes the proof of Theorem
3.1-(v).

Putting together all the results obtained in the previous sections, we
conclude the proof of Theorem 3.1.

## Acknowledgments

We thank J. M. Coron, M. Gugat, and E. Zuazua for their helpful comments.

GMC and NDN are members of the Gruppo Nazionale per l’Analisi Matematica, la
Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta
Matematica (INdAM).

GMC has been partially supported by the Project funded under the National
Recovery and Resilience Plan (NRRP), Mission 4, Component 2, Investment 1.4
(Call for tender No. 3138 of 16/12/2021), of Italian Ministry of University
and Research funded by the European Union (NextGenerationEU Award, No.
CN000023, Concession Decree No. 1033 of 17/06/2022) adopted by the Italian
Ministry of University and Research (CUP D93C22000410001), Centro Nazionale
per la Mobilità Sostenibile. He has also been supported by the Italian
Ministry of Education, University and Research under the Programme “Department
of Excellence” Legge 232/2016 (CUP D93C23000100001), and by the Research
Project of National Relevance “Evolution problems involving interacting
scales” granted by the Italian Ministry of University and Research (MUR PRIN
2022, project code 2022M9BKBC, CUP D53D23005880006).

NDN has been funded by the Swiss State Secretariat for Education, Research and
Innovation (SERI) under contract number MB22.00034 through the project TENSE.

MG and FM were partially supported by the GNAMPA-INdAM project
_Modellizzazione ed Analisi attraverso Leggi di Conservazione_ , (CUP
E53C23001670001).

MG was partially supported by the Project funded under the National Recovery
and Resilience Plan (NRRP) of Italian Ministry of University and Research
funded by the European Union (NextGenerationEU Award, No. CN000023, Concession
Decree No. 1033 of 17/06/2022 adopted by the Italian Ministry of University
and Research, CUP: H43C22000510001, Centro Nazionale per la Mobilità
Sostenibile).

MG was also partially supported by the Project funded under the National
Recovery and Resilience Plan (NRRP) of Italian Ministry of University and
Research funded by the European Union (NextGenerationEU Award, No.
ECS_00000037, CUP H43C22000510001, MUSA – Multilayered Urban Sustainability
Action).

MG and FM were partially supported by the PRIN 2022 project _Modeling, Control
and Games through Partial Differential Equations_ (D53D23005620006), funded by
the European Union (NextGenerationEU).

## References

  * [1] Fatiha Alabau-Boussouira, Vincent Perrollaz and Lionel Rosier  “Finite-time stabilization of a network of strings”  In _Math. Control Relat. Fields_ 5.4, 2015, pp. 721–742  DOI: 10.3934/mcrf.2015.5.721
  * [2] Debora Amadori and Graziano Guerra  “Global BV solutions and relaxation limit for a system of conservation laws”  In _Proc. Roy. Soc. Edinburgh Sect. A_ 131.1, 2001, pp. 1–26  DOI: 10.1017/S0308210500000767
  * [3] Fabio Ancona and Giuseppe Maria Coclite  “On the attainable set for Temple class systems with boundary controls”  In _SIAM J. Control Optim._ 43.6, 2005, pp. 2166–2190  DOI: 10.1137/S0363012902407776
  * [4] Fabio Ancona and Giuseppe Maria Coclite  “On the boundary controllability of first-order hyperbolic systems”  In _Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods_ 63.5-7, 2005, pp. e1955–e1966  DOI: 10.1016/j.na.2005.02.096
  * [5] Fabio Ancona and Andrea Marson  “Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point”  In _Control methods in PDE-dynamical systems_ 426, Contemp. Math.  Amer. Math. Soc., Providence, RI, 2007, pp. 1–43  DOI: 10.1090/conm/426/08182
  * [6] Paolo Baiti and Edda Dal Santo  “Front tracking for a $2\times 2$ system of conservation laws” Id/No 220  In _Electron. J. Differ. Equ._ 2012, 2012, pp. 14
  * [7] G. Bastin, B. Haut, J.-M. Coron and B. d’Andréa-Novel  “Lyapunov stability analysis of networks of scalar conservation laws”  In _Netw. Heterog. Media_ 2.4, 2007, pp. 751–759  DOI: 10.3934/nhm.2007.2.751
  * [8] Georges Bastin and Jean-Michel Coron  “Stability and boundary stabilization of 1-D hyperbolic systems” 88, Prog. Nonlinear Differ. Equ. Appl.  Basel: Birkhäuser/Springer, 2016  DOI: 10.1007/978-3-319-32062-5
  * [9] Alberto Bressan  “Hyperbolic systems of conservation laws. The one-dimensional Cauchy problem” 20, Oxford Lecture Series in Mathematics and its Applications  Oxford University Press, Oxford, 2000, pp. xii+250
  * [10] Alberto Bressan and Giuseppe Maria Coclite  “On the boundary control of systems of conservation laws”  In _SIAM J. Control Optim._ 41.2, 2002, pp. 607–622  DOI: 10.1137/S0363012901392529
  * [11] G.. Coclite, M. Garavello and B. Piccoli  “Traffic flow on a road network”  In _SIAM J. Math. Anal._ 36.6, 2005, pp. 1862–1886  DOI: 10.1137/S0036141004402683
  * [12] Rinaldo M. Colombo  “BV solutions to hyperbolic conservation laws”  In _One-dimensional hyperbolic conservation laws and their applications_ 21, Ser. Contemp. Appl. Math. CAM  World Sci. Publ., Hackensack, NJ, 2019, pp. 87–159
  * [13] Rinaldo M. Colombo and Mauro Garavello  “A well posed Riemann problem for the $p$-system at a junction”  In _Netw. Heterog. Media_ 1.3, 2006, pp. 495–511  DOI: 10.3934/nhm.2006.1.495
  * [14] Rinaldo M. Colombo and Mauro Garavello  “Comparison among different notions of solution for the $p$-system at a junction”  In _Discrete Contin. Dyn. Syst._ , 2009, pp. 181–190
  * [15] Rinaldo M. Colombo and Mauro Garavello  “On the Cauchy problem for the $p$-system at a junction”  In _SIAM J. Math. Anal._ 39.5, 2008, pp. 1456–1471  DOI: 10.1137/060665841
  * [16] Jean-Michel Coron and Georges Bastin  “Dissipative boundary conditions for one-dimensional quasi-linear hyperbolic systems: Lyapunov stability for the $C^{1}$-norm”  In _SIAM J. Control Optim._ 53.3, 2015, pp. 1464–1483  DOI: 10.1137/14097080X
  * [17] Jean-Michel Coron et al.  “Dissipative boundary conditions for $2\times 2$ hyperbolic systems of conservation laws for entropy solutions in BV”  In _J. Differential Equations_ 262.1, 2017, pp. 1–30  DOI: 10.1016/j.jde.2016.09.016
  * [18] Constantine M. Dafermos  “Hyperbolic conservation laws in continuum physics” 325, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]  Springer-Verlag, Berlin, 2016, pp. xxxviii+826  DOI: 10.1007/978-3-662-49451-6
  * [19] René Dáger and Enrique Zuazua  “Controllability of star-shaped networks of strings”  In _Mathematical and numerical aspects of wave propagation. 5th international conference, Santiago de Compostela, Spain, July 10–14, 2000_ Philadelphia, PA: SIAM, Society for Industrial and Applied Mathematics; Rocquencourt: INRIA, Institut National de Recherche en Informatique et en Automatique, 2000, pp. 1006–1010
  * [20] Markus Dick, Martin Gugat and Günter Leugering  “A strict $H^{1}$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction”  In _Numer. Algebra Control Optim._ 1.2, 2011, pp. 225–244  DOI: 10.3934/naco.2011.1.225
  * [21] Mathias Dus  “BV exponential stability for systems of scalar conservation laws using saturated controls”  In _SIAM J. Control Optim._ 59.2, 2021, pp. 1656–1679  DOI: 10.1137/20M1323837
  * [22] Mathias Dus  “Exponential stability of a general slope limiter scheme for scalar conservation laws subject to a dissipative boundary condition”  In _Math. Control Signals Syst._ 34.1, 2022, pp. 37–65  DOI: 10.1007/s00498-021-00301-2
  * [23] Qilong Gu and Tatsien Li  “Exact boundary controllability for quasilinear hyperbolic systems on a tree-like network and its applications”  In _SIAM J. Control Optim._ 49.4, 2011, pp. 1404–1421  DOI: 10.1137/080739902
  * [24] Martin Gugat, Markus Dick and Günter Leugering  “Gas flow in fan-shaped networks: classical solutions and feedback stabilization”  In _SIAM J. Control Optim._ 49.5, 2011, pp. 2101–2117  DOI: 10.1137/100799824
  * [25] Martin Gugat, Günter Leugering, Simona Tamasoiu and Ke Wang  “$H^{2}$-stabilization of the isothermal Euler equations: a Lyapunov function approach”  In _Chinese Ann. Math. Ser. B_ 33.4, 2012, pp. 479–500  DOI: 10.1007/s11401-012-0727-y
  * [26] Martin Gugat and Mario Sigalotti  “Stars of vibrating strings: switching boundary feedback stabilization”  In _Netw. Heterog. Media_ 5.2, 2010, pp. 299–314  DOI: 10.3934/nhm.2010.5.299
  * [27] Helge Holden and Nils Henrik Risebro  “Front tracking for hyperbolic conservation laws” 152, Applied Mathematical Sciences  Springer, Heidelberg, 2015, pp. xiv+515  DOI: 10.1007/978-3-662-47507-2
  * [28] Tatsien Li and Bopeng Rao  “Local exact boundary controllability for a class of quasilinear hyperbolic systems.”  In _Chin. Ann. Math., Ser. B_ 23.2, 2002, pp. 209–218  DOI: 10.1142/S0252959902000201
  * [29] Tatsien Li and Lei Yu  “Local exact boundary controllability of entropy solutions to linearly degenerate quasilinear hyperbolic systems of conservation laws”  In _ESAIM, Control Optim. Calc. Var._ 24.2, 2018, pp. 793–810  DOI: 10.1051/cocv/2017072
  * [30] Tatsien Li and Lei Yu  “The exact boundary controllability of nodal profile for entropy solutions to 1-D quasilinear hyperbolic systems of conservation laws with linearly degenerate negative (resp., positive) characteristic fields”  In _SIAM J. Control Optim._ 61.5, 2023, pp. 2761–2776  DOI: 10.1137/21M1395946
  * [31] Ta-Tsien Li and Bo-Peng Rao  “Exact boundary controllability for quasilinear hyperbolic systems”  In _SIAM J. Control Optim._ 41.6, 2003, pp. 1748–1755  DOI: 10.1137/S0363012901390099
  * [32] Yunguang Lu  “Hyperbolic conservation laws and the compensated compactness method” 128, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics  Chapman & Hall/CRC, Boca Raton, FL, 2003, pp. xii+241
  * [33] Vincent Perrollaz  “Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law”  In _Ann. Inst. H. Poincaré C Anal. Non Linéaire_ 30.5, 2013, pp. 879–915  DOI: 10.1016/j.anihpc.2012.12.003
  * [34] Julie Valein and Enrique Zuazua  “Stabilization of the wave equation on 1-d networks”  In _SIAM J. Control Optim._ 48.4, 2009, pp. 2771–2797  DOI: 10.1137/080733590