\subsection*{Keywords.}
Vehicular traffic; multi-lane; macroscopic limit; balance laws; stability.


\subsection*{AMS subject classifications.}
76A30; 35L60; 35B35.


\section{Introduction}

The interest in the modeling of dynamics of traffic flow dates back to the first half of the twentieth century and the related mathematical literature is today extensive, see e.g.~\cite{MR1600250,Helbing20011067}.
Three different scales of description of  vehicular traffic are typically considered, leading to microscopic, mesoscopic and macroscopic mathematical models, see e.g.~\cite{piccoli2009ENCYCLOPEDIA,albi2019vehicular} 

In this paper we focus on the macroscopic scale, inspired by fluid dynamics equations. Here, traffic flow is represented by the study of average quantities, such as the traffic density. First macroscopic model for traffic flow for single lane roads have been proposed in the 1950s by Lighthill and Whitham~\cite{lighthill1955PRSL}, and,  Richards~\cite{richards1956OR}. This model is derived by mass conservation principle and, it is described by a scalar conservation law of the form
\begin{equation} \label{eq:lwr}
\partial_t \rho + \partial_x f(\rho) = 0,
\end{equation}
where $\rho=\rho(x,t)$ represents the density of vehicles, with $\rho \in [0,\rho^{\max}]$, and $\rho^{\max}$ is the maximum density of vehicles on the road; $(x,t)\in \mathbb{R}\times \mathbb{R}^+$ represent the space and time variables, respectively. The nonlinear flux function $f(\rho)=\rho v(\rho)$ s a given function of the density and it is called fundamental diagram, where$v$ is the average velocity of the flow, function of the density. Equation~\eqref{lwr}, is endowed with an initial condition $\rho(x,0)=\rho_0(x)$. Improvements and further evolution of the basic macroscopic single lane description~\eqref{eq:lwr} have been proposed over the years by several authors, e.g.~Aw and Rascle~\cite{aw2000SIAP} and, independently, by Zhang~\cite{Zhang2002}. In second order models the scalar conservation law~\eqref{eq:lwr} is coupled to an additional equation for the evolution of the average speed $v$.

Single lane models are scarcely predictive for multi-lane roads, such as highways and freeways. Therefore, mathematical models have been extended in order to describe vehicular dynamics on multiple lanes. This allows to analyse  effects of lane changing maneuvers on the global behavior of traffic flow, such as the appearance of unstable phenomena (e.g.~stop and go waves). Typically, multi-lane models are based on systems of one-dimensional equations for the density of each lane and describe lane changes through source terms modeled directly at the macroscopic level, see, e.g., \cite{GoatinRossi2019,HoldenRisebro2019,SongKarni2019}, using empirical considerations.
In these models the source term is assumed to be proportional to local density on both the current and the target lane. Other fluid dynamics models, instead, describe the cumulative density on all lanes using a two-dimensional system of conservation
laws~\cite{SukhinovaTrapeznikovaChetverushkinChurbanova2009,HertyMoutariVisconti2018}. In
those studies, the modeling of traffic dynamics assume that vehicles move to lanes with faster speed or lower density, and that the evolution for the lateral velocity is proportional to the local density and the mean speed along the road.

In this paper we consider a different approach. We present a novel first order macroscopic model for multi-lane traffic flow whose derivation is motivated by a microscopic scale description of lane changing dynamics. The position of vehicles is modeled by a system of first order ordinary differential equations, namely by continuous dynamics, whereas lane changes are formulated as discrete events and occur instantaneously, contrary to other approaches based on the definition of a cool-down time~\cite{GongPiccoliVisconti,ChiriGongPiccoli}. The presence of both continuous and discrete dynamics leads us a hybrid system, e.g.~see~\cite{garavello2005hybrid,4806347,piccoli1998hybrid}. The macroscopic model is described by a system of hyperbolic balance laws for the evolution of the density in each lane, including source terms. Here, lane changes are obtained as macroscopic limit of microscopic lane changing maneuvers. The microscopic-to-macroscopic limit  does not need to postulate specific ad hoc scalings\cite{aw2002SIAP}. Moreover, in contrast to existing models, the derivation using the microscopic scale leads to source terms that depend on the current lane only, and not on adjacent lanes. The model is analysed by classifying all possible equilibrium solutions for the two-lane case and the stability with respect to small perturbations.

The paper is organized as follows. Section~\ref{sec:micro} is devoted to the definition of the microscopic framework, with the description of the lane changing conditions, based on safety and incentive criteria as, e.g., in~\cite{PiuPuppo2022,GongPiccoliVisconti}. Then, in Section~\ref{sec:derivation} we derive the first order continuum model as macroscopic limit of the microscopic continuous dynamics and of the discrete events describing the lane changing. Section~\ref{sec:analysis} is aimed to present the characterization of the equilibria of the macroscopic model in the case of a two-lane road and the analysis of the stability under some specific perturbations. Numerical simulations in Section~\ref{sec:numerics} show an evidence of the theoretical findings, as well as the consistency between the microscopic and the macroscopic models, and an application to lane closure problem. Finally, in Section~\ref{sec:conclusion} we summarize the results. 

\section{First order multi-lane microscopic model} \label{sec:micro}

In this section we present a first order microscopic multi-lane model characterized by continuous dynamics combined with discrete events for  lane changing.

\subsection{Microscopic dynamics with lane changing}

In the following we consider a homogeneous population of $N\in\mathbb{N}$ vehicles and we denote by $x_n=x_n(t)$ the time-dependent position of the $n$-th vehicle, with $n=1,\dots,N$. More precisely, we denote by $x_n$  the position of the rear bumper, therefore the position of the front bumper is given by $x_n+l$, where $l>0$ denotes the length of the vehicle which is assumed constant for all vehicles.

In the case of multi-lane traffic, vehicles travel along multiple lanes of length $L$ with the possibility to change lane. Consider $J$ lanes and  denote by $j \in {1,\dots,J}$ the lane index. Given a vehicle $n$ traveling in a lane $j$, we denote with $p_n^{j}$ and $s_n^{j}$ the following and leading vehicles, respectively, in lane $j$. Further, $p_n^{k}$ and $s_n^{k}$, $k=j-1,j+1$, denote the vehicles in the two adjacent lanes, see Fig.~\ref{strada}. In the following, we avoid the introduction of an ordering on the vehicle indices, so that  labels are  assigned among vehicles on the road and  remain unchanged even after a lane change occurs.

\begin{figure}[t]
\centering
\begin{tikzpicture}
[auto,
 block/.style ={rectangle, draw=blue, thick, fill=blue!20},
 block1/.style ={rectangle, draw=green, thick, fill=green!20},
 block2/.style ={rectangle, draw=red, thick, fill=red!20}
]
\draw [dashed](0,3)--(8,3);
\draw [dashed](0,2)--(8,2);
\draw [dashed](0,1)--(8,1);
\draw [dashed](0,0)--(8,0);
\draw (0,-1)--(8,-1);
\draw (0,4)--(8,4);



\path(-1,0.5)node{lane $j-1$};
\path (-1,1.5)node{lane $j$};
\path (-1,2.5)node{lane $j+1$};


\path(1.6,0.5)node[block]{$p_n^{j-1}$};
\path (1.2,1.5)node[block]{$p_n^{j}$};
\path(1.8,2.5)node[block]{$p_n^{j+1}$};

\path(5.5,0.5)node[block1]{$s_n^{j-1}$};
\path (6.5,1.5)node[block1]{$s_n^{j}$};
\path(7,2.5)node[block1]{$s_n^{j+1}$};

\path (3.5,1.5)node[block2]{$n$};
\end{tikzpicture}
\caption{Schematic representation of the multi-lane road. Here, the reference vehicle is $n$, traveling in lane $j$. Whereas, $p_n^{k}$ and $s_n^{k}$ represent the vehicles just behind and in front of vehicle $n$, respectively, in lane $k=j-1,j,j+1$.}\label{strada}
\end{figure}

Let $\Delta x^j_n = x_{s^j_n}-x_n$ be the headway between the $n$-th vehicle and its leading vehicle in lane $j$. The first order multi-lane microscopic model is written for $j\in \{1,\dots,J\}$ as a system of ordinary differential equations (ODEs) given by
\begin{equation}\label{bftl_2lane}
\dot x_n =V_j(\Delta x^j_n) \qquad n\in I_j(t), 
\end{equation}
where $I_j(t)$ denotes the set of vehicles in lane $j$ at time $t$. The function $V_j(h)$ prescribes the desired velocity  depending on the headway $h$. We assume $V_j$ to be a monotonic increasing function with $V_j(0)=0$ and  bounded by a lane-dependent maximum  value $V^{\max}_j$.



For multi-lane models, equation~\eqref{bftl_2lane} is coupled to a set of  discrete rules that describe the occurrence of a lane change. We assume to deal with instantaneous lane changing occurring from the current lane to one of the two adjacent lanes, the target lane, only. A vehicle performs a lane change based on two criteria~\cite{PiuPuppo2022,GongPiccoliVisconti}: an \emph{incentive} criterion, i.e.~a lane change may occur if local speed in the target lane is greater than the local speed in the current lane; and a \emph{safety} criterion, i.e.~a vehicle move to a target lane if it is possible to keep a safety distance $d_s>0$ in order to avoid collisions. In this way the lane changing conditions from lane $j$ to a lane $j'\in\{j-1,j+1\}$ are written as
\begin{equation}\label{microLCC}
\begin{cases}
V_{j'}(x_{s^{j'}_n}-x_n)>V_j(x_{s^{j}_n}-x_n) & \text{(incentive criterion)} \\[1ex]
x_{s^{j'}_n}-x_n>l+d_{s} \quad \text{and} \quad x_n-x_{p_n^{j'}}>l+d_{s} & \text{(safety criterion)} \end{cases}
\end{equation}
Note that, if the current lane is either $j=1$ or $j=J$ the target lane is one, namely lane $2$ or lane $J-1$, respectively. Instead, when the current lane is $j\in\{2,\dots,J-1\}$ there are two target lanes $j-1$ and $j+1$. If both satisfy the lane changing rules we assume that a vehicle moves to the most advantageous one.

For the purposes of this paper, of particular interest is the discussion of the steady states of~\eqref{bftl_2lane}.

\begin{definition} \label{def:ss:micro}
We say that system~\eqref{bftl_2lane}-\eqref{microLCC} is locally at equilibrium if in each lane vehicles are equally spaced and, thus, move with  constant velocity. Namely, for any fixed $j\in\{1,\dots,J\}$
$$
    \Delta x_n^j = C_j, \quad \forall n\in I_j,
$$
where $C_j > 0$ are lane-dependent constants. Moreover, we say that system~\eqref{bftl_2lane}-\eqref{microLCC} is at equilibrium if all lanes are locally at equilibrium and there are no lane changing, i.e.~\eqref{microLCC} is violated for any fixed $j\in\{1,\dots,J\}$ and $j'\in\{j-1,j+1\}$.
\end{definition}

\begin{remark}
    For the purposes of this paper, without loss of generality in the following we consider a multi-lane ring road of length $L > 0$. Thus, system~\eqref{bftl_2lane} is endowed with periodic boundary conditions on the domain $\left[-\frac{L}{2},\frac{L}{2}\right]$, $\forall j\in \{1,\dots,J\}$.
\end{remark}

\subsection{Evolution of the local density: a discrete description}

As first step to derive a macroscopic model including the microscopic rules~\eqref{bftl_2lane}-\eqref{microLCC} we provide a definition for the (local) density. We assume that on each lane the maximum value of the density, $\rho^{\max}$, is reached when the vehicles are equally spaced with distance equal to $d_s$. Imposing $\rho^{\max}=1$, we define the local density of the $n$-th vehicle in lane $j$ (at time $t$) as 
\begin{equation*}
	\rho^{(n)}_j(t):=\frac{l+d_s}{\Delta x^j_n}.
\end{equation*}
In the following, we analyze the behavior of the local density after a single lane change. The vehicle $n$ moves from the current lane $j$ to the target lane $j'\in\{j-1,j+1\}$ assuming that conditions~\eqref{microLCC} are satisfied at  time $t$. At time $t^-$, before the lane change occurs, the local density around the vehicle $p_n^{j'}$ is given by
\begin{equation*}
	\rho_{j'}^{(p^{j'}_n,\text{before})} (t^-)= \frac{l+d_s}{x_{s^{j'}_n}-x_{p^{j'}_n}}.
\end{equation*}
Whereas,  after the lane change has occurred at time $t^+$, the local density around the vehicle $p_n^{j'}$ becomes
\begin{equation*}
\rho_{j'}^{(p^{j'}_n,\text{after})} (t^+)= \frac{l+d_s}{\tilde{x}-x_{p^{j'}_n}}
\end{equation*}
where $\tilde{x}$ denotes the new position of vehicle $n$ in lane $j'$. We write $\tilde{x}$ as a convex combination of the positions of the follower and the leading vehicles in the new lane $j'$. Recalling the safety criterion in~\eqref{microLCC} yields, we have
\begin{equation*}
	\tilde{x}=\alpha (x_{s^{j'}_n}-d_s-l) + (1-\alpha) (x_{p^{j'}_n}+l+d_s) \qquad \alpha\in[0,1].
\end{equation*}
The parameter $\alpha$ can be modelled as a constant value or as a function of the local density of vehicle $n$ in its current lane $j$. This allows to describe the following scenario: if the density $\rho_{j}^{(n,\text{before})}$ is large, it is more probable that the position $\tilde{x}$ of vehicle $n$ in the new lane $j'$ is closer to the follower vehicle $p_n^{j'}$:
\begin{equation*}
\alpha=\alpha(\rho_{j}^{(n,\text{before})}) \quad \text{s.t.} \quad \alpha'(\cdot)<0  \quad \text{and}
\quad \alpha(0) =1,\,\,\alpha(1) =0.
\end{equation*}
As a linear example we use the function $ \alpha(r)=1-r$.


The density increment on lane $j'$ after the lane changing of vehicle $n$ from lane $j$ can be computed as the difference $\rho_{j'}^{(p^{j'}_n,\text{after})}-\rho_{j'}^{(p^{j'}_n,\text{before})}$:
\begin{align*}
\rho_{j'}^{(p^{j'}_n,\text{after})}-\rho_{j'}^{(p^{j'}_n,\text{before})}
&= \left(\frac{1}{\alpha + (1-2\alpha)\rho_{j'}^{(p^{j'}_n,\text{before})}}-1 \right)\rho_{j'}^{(p^{j'}_n,\text{before})} \\
&=:  A\left(\rho_j^{(n,\text{before})},\rho_{j'}^{(p^{j'}_n,\text{before})}\right)\rho_{j'}^{(p^{j'}_n,\text{before})}
\end{align*}
Summarizing, we obtain the new local density in lane $j'$ when a vehicle $n$ moves from lane $j$ to lane $j'$ as:

\begin{equation*}
\rho_{j'}^{(p^{j'}_n,\text{after})}=
\rho_{j'}^{(p^{j'}_n,\text{before})}+ A\left(\rho_{j}^{(n,\text{before})},\rho_{j'}^{(p^{j'}_n,\text{before})}\right)\rho_{j'}^{(p^{j'}_n,\text{before})}.
\end{equation*}




\section{First order multi-lane macroscopic model} \label{sec:derivation}
We present the derivation of the first order  multi-lane macroscopic model obtained as continuum limit of the first order microscopic model with lane changing conditions introduced in the previous section.

\subsection{Evolution of the local density: a continuum description} \label{sec:macro:evolution}

The previous description of the local discrete density allows us to derive a macroscopic scale formulation for the evolution of the density in each lane. Using a piecewise constant reconstruction of the local densities 

\begin{equation}\label{piece}
\rho_{j}(x,t):=\rho_{j}^{(m)}(t), \quad n \in I_j, \quad x \in [x_n,x_{n+1}) \subset \left[-\frac{L}{2},\frac{L}{2}\right],
\end{equation}

Since mass is conserved, the space-time evolution of $\rho_j=\rho_j(x,t)$ without lane changing must satisfy a scalar conservation law of the form 
\begin{equation}\label{lwr}
\partial_t \rho_j + \partial_x f_j(\rho_j) = 0, \quad \text{on } (x,t)\in\left[-\frac{L}{2},\frac{L}{2}\right]\times \mathbb{R}^+, \quad \forall j\in \{1,\dots,J\},
\end{equation}
For first order models
$
    f_j(\rho_j)=\rho_j v_j(\rho_j).
$
Here, $v_j$ is defined by the desired velocity function of the microscopic model~\eqref{bftl_2lane} as
\begin{equation} \label{eq:micro:macro:speed}
    v_j(\rho)= V_j \left(\frac{l+d_s}{\rho}\right).
\end{equation}

From~\eqref{lwr}, using  a Taylor expansion for the time derivative we obtain  
\begin{equation*}
    \rho_j(x,t+\Delta t) = \rho_j - \Delta t \partial_x f_j(\rho_j)+ o(\Delta t), \qquad \mbox{ for } j=1,\dots,J
\end{equation*}
In order to study the effects of the lane changes on the macroscopic evolution of the density, we consider, for instance, the lane changes from lane $j'$ to lane $j$. Observe that $j'$ can be either $j-1$ or $j+1$. Let $X\sim\text{Ber}(\gamma)$ be a random variable with Bernoulli distribution, parametrized by $\gamma$ denoting the probability of a lane changing ($X=1$) within a unit of time. Thus, $\gamma \Delta t$ is the probability that a vehicle changes lane in the interval $\Delta t$, namely $\mathbb{P}(X=1)=\gamma \Delta t$, whereas $\mathbb{P}(X=0)=1-\gamma \Delta t$. 
In this setting, the density changes stochastically depending on the outcome of $X$ as: 
\begin{equation}\label{casi}
\tilde{\rho}_j(x,t+\Delta t,X)= 
\rho_j - \Delta t \partial_x f_j(\rho^{n}_j) + X A(\rho_{j'},\rho_{j})\rho_j+ o(\Delta t)
\end{equation}
We  compute the time evolution of the density as  expected value of \eqref{casi}:
\begin{equation*}
\rho_j (x,t+\Delta t)
=\mathbb{E}[\tilde{\rho}_j(x,t+\Delta t,X)]  \\ 
= \rho_j - \Delta t \partial_x f_j(\rho_j) + \gamma \Delta t A(\rho_{j'},\rho_{j})\rho_j+ o(\Delta t),
\end{equation*}
and finally,
\begin{equation*}
\frac{\rho_j (x,t+\Delta t)- \rho_j(x,t)}{\Delta t}= - \partial_x f_j(\rho_j(x,t)) + \gamma A(\rho_{j'}(x,t),\rho_j(x,t))\rho_j(x,t)+o(1).
\end{equation*}
Renaming the probability $\gamma$ as $\gamma^{j' \to j}$ to emphasize the lane changing from lane $j'$ to lane $j$, we obtain for $\Delta t \to 0$ the continuous description
\begin{equation} \label{eq:macro:jpTOj}
\partial_t \rho_{j} + \partial_x f_j(\rho_{j}) = \gamma^{j' \to j} A(\rho_{j'},\rho_{j})\rho_{j}, \quad \text{on } (x,t)\in\left[-\frac{L}{2},\frac{L}{2}\right]\times \mathbb{R}^+, \quad \forall j\in \{1,\dots,J\},
\end{equation}
with
\begin{equation} \label{eq:A}
A(\rho_{j'},\rho_{j})= \left(\left(\alpha(\rho_{j'}) + (1-2\alpha(\rho_{j'}))\rho_{j}\right)^{-1}-1 \right),
\end{equation}
where the source term describes the increase of the density in lane $j$ due to lane changing maneuvers from lane $j'$ to lane $j$.

\begin{remark}
We remark that in our micro-macro derivation in the case of a constant $\alpha$ the source term of the evolution equation of the density in lane $j$ is directly proportional to $\rho_{j}$ itself. Whereas, in other models, e.g.~see~\cite{HoldenRisebro2019,SongKarni2019}, the source term is typically directly proportional to the density of the adjacent lanes. Furthermore, we point-out that in the microscopic framework we have  assumed the presence of vehicles on the road, namely  non zero densities on all lanes.
\end{remark}


\subsection{Lane changing conditions in the macroscopic scale} \label{sec:macro:lcc}

In order to complete the macroscopic scale description of the evolution of the density in lane $j$ due to maneuvers from lane $j'$ to lane $j$, we  relate the lane changing probability with the microscopic lane changing conditions~\eqref{microLCC}. We start recalling the incentive and safety criteria, rewriting them in terms of the previously introduced macroscopic notation.


After simple algebraic manipulations, the incentive condition for a lane changing from $j'$ to $j$, i.e.
\begin{equation*}
	V_{j}(x_{s_n^{j}}-x_n) > V_{j'}(x_{s_n^{j'}}-x_n),
\end{equation*}
is given by the macroscopic setting using~\eqref{eq:micro:macro:speed} as
\begin{equation*}
	v_{j}\left(\frac{l+d_s}{x_{s_n^{j}}-x_n}\right) >v_{j'}\left(\frac{l+d_s}{x_{s_n^{j'}}-x_n}\right),
\end{equation*}
which implies 
\begin{equation*}
	v_{j}(\rho^{(n)}_{j}) > v_{j'}(\rho^{(n)}_{j'}),
\end{equation*}
and hence to the following condition on the macroscopic speeds:
\begin{equation*}
	v_{j}(\rho_{j}) > v_{j'}(\rho_{j'}).
\end{equation*}
Therefore, no lane change occurs if the flow speed in the target lane $j$ is lower than the flow speed in the current lane $j'$.
Whereas, the safety criterion for a lane changing from $j'$ to $j$ can be rewritten as follows. Summing the two microscopic inequalities in~\eqref{microLCC}
\begin{equation*}
x_{s_n^{j}}-x_{p_n^{j}}>2(l+d_s),
\end{equation*}
computing the reciprocal and multiplying by $l+d_s$
\begin{equation} \label{eq:mu:def}
\frac{l+d_s}{x_{s_n^{j}}-x_{p_n^{j}}}<\frac{l+d_s}{2(l+d_s)}=\frac{1}{2}=:\mu 
\end{equation}
implies that
\begin{equation} 
\rho^{(j)}_{p_n^{j}} < \mu.
\end{equation}
Hence we obtain the following condition on the macroscopic density:
\begin{equation*}
\rho_{j} < \mu.
\end{equation*}

Therefore, no lane change occurs if $\rho_{j}\geqslant \mu$, and we expect that the probability of changing lanes increases as $\mu -\rho_{j}$ becomes larger.

Summarizing the previous discussion, we  write the probability $\gamma^{j' \to j}$ in  macroscopic setting, proportional to the density of the target lane $j$, as
\begin{equation*}
\begin{aligned}
\gamma^{j' \to j}(\rho_j',\rho_{j})= g(\rho_{j}) \max\left\{0, \min\left\{ \text{sign}\left(v_{j}(\rho_{j}) - v_{j'}(\rho_{j'})\right), \text{sign}\left(\mu -\rho_{j}\right) \right\}\right\}.
\end{aligned}
\end{equation*}
where the function $g:[0,\mu]\to[0,1]$ models the percentage of mass going from lane $j'$ to lane $j$.
Here, $g(\cdot)$ is a monotonically decreasing function of its argument, an example is provided by the linear function $ g(r)=1-2r$.

We point-out that so far we have focused on the evolution of the density in lane $j$ when a lane change from $j'\in\{j-1,j+1\}$ to $j$ occurs. This causes an increase of $\rho_j$. Similarly, there is a decrease of $\rho_{j'}$ which is modeled by the same source term in~\eqref{eq:macro:jpTOj} with a negative sign, hence we conserve the mass of all lanes. 


\subsection{Derivation of the first order macroscopic multi-lane model}

The derivation in Section~\ref{sec:macro:evolution} and Section~\ref{sec:macro:lcc} leads to the following final first order macroscopic model for a multi-lane road with $J$ lanes:
\begin{equation}\label{modello}
	\partial_t \rho_{j} + \partial_x f_j(\rho_{j}) = S(\rho_j,\rho_{j'}), \quad \text{on } (x,t)\in\left[-\frac{L}{2},\frac{L}{2}\right]\times \mathbb{R}^+, \quad \forall j\in \{1,\dots,J\},
\end{equation}
where the source term is given by
\begin{equation} \label{eq:source}
S(\rho_j,\rho_{j'}) = \sum\limits_{j' \in T_j}\gamma^{j' \to j}(\rho_j,\rho_{j'}) A(\rho_{j'},\rho_{j})\rho_{j} - \sum\limits_{j' \in T_j} \gamma^{j \to j'}(\rho_{j'},\rho_{j}) A(\rho_{j},\rho_{j'})\rho_{j'}
\end{equation}
and
\begin{equation} \label{eq:gamma:and:A}
\begin{aligned}
\gamma^{h \to k}(\rho_h,\rho_k)  
&= g(\rho_k) \max\left\{0, \min\left\{ \text{sign}\left(v_{k}(\rho_{k}) - v_{h}(\rho_{h})\right), \text{sign}\left(\mu -\rho_{k}\right) \right\}\right\} \\
A(\rho_{h},\rho_{k}) &= \left(\left(\alpha(\rho_{h}) + (1-2\alpha(\rho_{h}))\rho_{k}\right)^{-1}-1 \right)
\end{aligned}
\end{equation} 
for all $ k\in\{1,\dots,J\}$ and $h\in T_j$. The  the set $T_j$ of the target lanes of a given lane $j$ is
\begin{equation*}
T_j=
\begin{cases}
\{2\} & \mbox{ if }j=1\\
\{j-1,j+1\} & \mbox{ if } j=2,\dots,J-1\\
\{J-1\} & \mbox{ if }j=J.
\end{cases}
\end{equation*}

System~\eqref{modello} is endowed with given initial conditions $\rho_j^0(x):=\rho_j(x,t=0)$ and suitable boundary conditions.





\section{Steady state analysis} \label{sec:analysis}

This section is devoted to the discussion of the stability of the steady states of the macroscopic multi-lane model~\eqref{modello}. We are interested in a particular class of steady states defined as follows.

\begin{definition} \label{def:ss}
A steady state $\{\overline{\rho}_j(x)\}_{j=1}^J$ of the macroscopic multi-lane model \eqref{modello} is characterized by

$$
    \overline{\rho}_j(x) = \overline{\rho}_j\in (0,\rho^{\max}] \ \text{ and } \ S(\overline{\rho}_j) = 0, \quad \forall j=1,\dots,J,
$$
i.e.~in each lane the density is constant and no lane changes occurs.
\end{definition}

Moreover, the class of steady states described in Definition~\ref{def:ss} is the macroscopic counterpart of the equilibrium state of the microscopic model defined in Definition~\ref{def:ss:micro}.


\subsection{Explicit characterization of the equilibria} \label{ssec:charact:eq}


The explicit characterization of these states depends on the choice of the lane changing conditions. For the sake of presentation, we consider the case of a two-lane road, i.e.~$J=2$, described by the system of two balance laws:
\begin{equation}\label{duecorsie}
\begin{cases}
\partial_t \rho_{1} + \partial_x f_1(\rho_{1}) = \gamma^{2 \to 1} A(\rho_{2},\rho_{1})\rho_{1} - \gamma^{1 \to 2} A(\rho_{1},\rho_{2})\rho_{2}, \\

\partial_t \rho_{2} + \partial_x f_2(\rho_{2}) = \gamma^{1 \to 2} A(\rho_{1},\rho_{2})\rho_{2}- \gamma^{2 \to 1} A(\rho_{2},\rho_{1})\rho_{1}. \\
\end{cases}
\end{equation}

 Then, we denote $\overline{\rho}_1$ and $\overline{\rho}_2$ the steady state in lane 1 and lane 2, respectively. Furthermore, we assume that the two speed functions $v_1(\rho),v_2(\rho)$ are monotonically decreasing with respect to the density $\rho$, such that
\begin{equation} \label{eq:speed:functions}
v_1(\rho)<v_2(\rho),  \quad \rho\in[0,\rho^{\max}) \quad \text{ and } \quad \begin{aligned}
v_1(\rho^{\max})&=v_2(\rho^{\max})=0 \\ v_j(0)&=v_j^{\max}, \quad j=1,2.
\end{aligned}
\end{equation}
In the following, we denote  $v_j^\mu=v_j(\mu)$ the flow speed on the $j$-th lane corresponding to the critical density $\mu$ introduced in~\eqref{eq:mu:def}. In addition, we use the notation
$$
\rho_j^\mu:=(v_j)^{-1}(v_{j'}^\mu), \quad j,j'=1,2, \ j\neq j'
$$
to denote the unique value of the density in lane $j$ whose corresponding flow speed is equal to the speed of lane $j'$ at critical density.

We remember that lane changes are possible when both the incentive and safety criteria are satisfied.
Depending on the realization of the two criteria, the following four characterizations of the steady states given in Definition~\ref{def:ss} arise:


\begin{itemize}
\item[(A)] The density in both lanes is smaller than the critical density and the lane speeds are equal, namely
\begin{equation} \label{eq:eq:A}
v^{\text{eq}}:=v_1(\overline{\rho}_1)=v_2(\overline{\rho}_2) \qquad \text{ and }\qquad  0\leqslant \overline{\rho}_1 < \mu \,\,\land  \,\, 0\leqslant \overline{\rho}_2< \mu;
\end{equation}
\item[(B)] At least one of the two lanes has density greater than the critical value and the lane speeds are equal, namely
\begin{equation} \label{eq:eq:B}
v^{\text{eq}}:=v_1(\overline{\rho}_1)=v_2(\overline{\rho}_2) \qquad \text{ and }\qquad  \overline{\rho}_1\geqslant \mu \,\,\lor  \,\,  \overline{\rho}_2\geqslant \mu;
\end{equation}
\item[(C)] The density in both lanes is greater than the critical value and the lane speeds are different, namely
\begin{equation} \label{eq:eq:C}
v_1(\overline{\rho}_1)\neq v_2(\overline{\rho}_2) \qquad \text{ and }\qquad  \overline{\rho}_1 \geqslant \mu \,\,\land  \,\,  \overline{\rho}_2 \geqslant \mu;
\end{equation}
\item[(D)] The density in lane 1 is smaller than the critical value, whereas the density in lane 2 is greater, and the flow in lane 1 is slower than the flow in lane 2, namely
\begin{equation} \label{eq:eq:D}
v_1(\overline{\rho}_1)< v_2(\overline{\rho}_2) \qquad \text{ and }\qquad  \overline{\rho}_1< \mu \,\,\lor  \,\,  \overline{\rho}_2 \geqslant \mu.
\end{equation}
\end{itemize}

To these equilibria we also add the case in which all the mass is in the fastest lane and lane changes do not occur because the incentive criterion is not satisfied:
\begin{itemize}
\item[(E)] The density in lane 1 is zero, whereas the density in lane 2 is less or equal to $v_2^{-1}( v_1^{\text{max}})$, namely
\begin{equation} \label{eq:eq:E}
  \overline{\rho}_1 =0 \,\,\land  \,\,  \overline{\rho}_2\leqslant v_2^{-1}( v_1^{\text{max}}).
\end{equation}
\end{itemize}


\subsection{Linear stability analysis}

In the following, we assume that system~\eqref{duecorsie} is at equilibrium, according to Definition~\ref{def:ss}, and we consider a uniform perturbation in space on the two lanes such that the total mass is conserved. More precisely, the perturbed state $(\rho_1,\rho_2)$ of the equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$ is given by
\begin{equation}\label{perturbazione_modello}
\begin{cases}
\rho_1=\overline{\rho}_1+\varepsilon \rho\\
\rho_2=\overline{\rho}_2-\varepsilon \rho
\end{cases} \quad \text{with } \ \rho=\rho(t), \ \rho(0)=\rho_0>0
\end{equation}
with $|\varepsilon|\ll 1$. The aim is to investigate the stability of the five types of steady states previously described.


The left column of Table~\ref{tabella_convergenze} reports all the possible types of steady states of the two-lane model~\eqref{duecorsie}, which are characterized in the equations from~\eqref{eq:eq:A} to~\eqref{eq:eq:E}. When these are perturbed according to~\eqref{perturbazione_modello}, system~\eqref{duecorsie} converges, for large times, to the corresponding steady states listed in the right column of Table~\ref{tabella_convergenze}. In particular, we notice that in some cases the perturbed system is driven towards the initial steady state, and that in other cases the system converges to stationary states, different from the initial one.

\begin{table}[t]
\centering
\caption{Left: type of all equilibrium states of system~\eqref{duecorsie} defined in~\eqref{eq:eq:A}-\eqref{eq:eq:E}. Right: equilibrium states of system~\eqref{duecorsie} when the corresponding states on the left column are perturbed according to~\eqref{perturbazione_modello}.}\label{tabella_convergenze}
\begin{tabular}{ |c|c|  }
\hline
Initial equilibrium & Final equilibrium \\
\hline \hline
(A) & (A) \\
(B1) or (B2b) & (C) \\
(B2a) & (B2a) or (C) or (D) \\
(C) & (B1) or (B2a) or (C) or (D) \\
(D) & (B2a) or (C) or (D)\\
(E) & (E) \\
\hline
\end{tabular}
\end{table}

The following definition allows us to characterize the steady states of system~\eqref{duecorsie}.

\begin{definition} \label{def:stability}
    We say that an equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$ of system~\eqref{duecorsie} according to Definition~\ref{def:ss} is
    \begin{itemize}
    \item \emph{asymptotically stable} with respect to the space homogeneous perturbation~\eqref{perturbazione_modello} if there exists $\varepsilon\in\mathbb{R}$ such that $(\rho_1,\rho_2)\to(\overline{\rho}_1,\overline{\rho}_2)$ as $t\to\infty$;
    \item \emph{globally asymptotically stable} with respect to the space homogeneous perturbation~\eqref{perturbazione_modello} if $\forall\,\varepsilon\in\mathbb{R}$ one has $(\rho_1,\rho_2)\to(\overline{\rho}_1,\overline{\rho}_2)$ as $t\to\infty$;
    \item \emph{marginally stable} with respect to the space homogeneous perturbation~\eqref{perturbazione_modello} if for a given $\varepsilon\in\mathbb{R}$ one has $(\rho_1,\rho_2)\to(\tilde{\rho}_1,\tilde{\rho}_2)$ as $t\to\infty$, where $(\tilde{\rho}_1,\tilde{\rho}_2)$ is an equilibrium of different type from $(\overline{\rho}_1,\overline{\rho}_2)$.
    \end{itemize}
\end{definition}

The definition of a marginally stable equilibrium is inspired by the concept of marginal stability for dynamical systems, which is referred to as having trajectories around a perturbed steady state that neither converge to it, nor go away from where they started without limit. Furthermore, we note that the perturbed state of a marginal equilibrium corresponds, in this case, to a new equilibrium with different values of the equilibrium velocities. 

Using Definition~\ref{def:stability} we state the nature of all steady states of system~\eqref{duecorsie} in the following theorem.  We denote by (B2a) the equilibrium that belongs to the (B) equilibria set such that $v_2^\mu < v^{\text{eq}} \leqslant v_1^{\mu}$, whereas we call (B1) or (B2b) the equilibria in (B) with $0 \leqslant v^{\text{eq}} \leqslant v_2^{\mu}$. For the sake of clarity we include equilibria of type (B1) in the class of equilibria of type (C).  

\begin{theorem} \label{teorema}
Consider a space homogeneous perturbation as~\eqref{perturbazione_modello}. Then the steady states $(\overline{\rho}_1,\overline{\rho}_2)$ of the two-lane model~\eqref{duecorsie} can be characterized as follows according to Definition~\ref{def:stability}:
\begin{itemize}
    \item equilibrium solutions of type (A) and (E) are globally asymptotically stable;
    \item equilibrium solutions of type (B2a) and (C) with $\overline{\rho}_1=\mu$ are asymptotically stable if $\varepsilon<0$, and marginally stable otherwise;
    \item equilibrium solutions of type (C) and (D) with $\overline{\rho}_2=\mu$ are asymptotically stable if $\varepsilon>0$, and marginally stable otherwise;
    \item equilibrium solutions of type (C) and (D) with $\overline{\rho}_1\neq \mu$ and $\overline{\rho}_2\neq \mu$ are marginally stable.
\end{itemize}
\end{theorem}



\begin{figure}[t]
\centering
\begin{tikzpicture}
\begin{scope}[scale=0.85]
\draw[thick, ->](0,0)--(7,0);
\draw[thick, ->](0,0)--(0,7);
\path[below](7,0)node{$\rho_1$};
\path[left](0,7)node{$\rho_2$};
\path[below](6,0)node{$\rho^{\max}$};
\path[left](0,6)node{$\rho^{\max}$};
\path[below](3,0)node{$\mu$};
\path[left](0,3)node{$\mu$};
\path[anchor=north east](0,0)node{$0$};
\draw[name path= mu1, very thin] (0,3)--(6,3);
\draw[name path= mu2, very thin] (3,0)--(3,6);
\draw[very thin] (0,6)--(6,6);
\draw[very thin] (6,0)--(6,6);
\draw[white, name path=eq](0, 0.3*6)--(6,6);
\path [name intersections={of=eq and mu1,by=E}];
\path [name intersections={of=eq and mu2,by=F}];
\filldraw[pattern=crosshatch] (E)--(F)--(3,3)--cycle;
\filldraw[pattern=crosshatch dots] (3,3)--(3,6)--(6,6)--(6,3)--cycle;
\draw[very thick,violet,densely dotted] (0,0.3*6)--(E);
\draw[very thick,red] (E)--(F);
\draw[very thick,blue,densely dashed] (F)--(6,6);
\draw[thick,white] (0,0)--(0,0.3*6);
\draw[very thick,orange,densely dash dot] (0,0)--(0,0.3*6);
\end{scope}
\begin{scope}[xshift=7.5cm,scale=0.85]
\draw[thick, ->](0,0)--(7,0);
\draw[thick, ->](0,0)--(0,7);
\path[below](7,0)node{$\rho_1$};
\path[left](0,7)node{$\rho_2$};
\path[below](6,0)node{$\rho^{\max}$};
\path[left](0,6)node{$\rho^{\max}$};
\path[below](3,0)node{$\mu$};
\path[left](0,3)node{$\mu$};
\path[anchor=north east](0,0)node{$0$};
\draw[name path= mu1, very thin] (0,3)--(6,3);
\draw[name path= mu2, very thin] (3,0)--(3,6);
\draw[name path= a1, very thin] (0,6)--(6,6);
\draw[name path= a2, very thin] (6,0)--(6,6);
\draw[white, name path=eq](0, 0.3*6)--(6,6);
\path [name intersections={of=eq and mu1,by=E}];
\path [name intersections={of=eq and mu2,by=F}];


\draw[->](6*0.2,0)--(0,6*0.2);
\draw[->](6*0.1,0)--(0,6*0.1);

\draw[white, name path=eq1](6*0.3,0)--(0,6*0.3);
\path [name intersections={of=eq1 and eq,by=E0}];
\draw[->](6*0.3,0)--(E0);
\draw[->](0,6*0.3)--(E0);

\draw[white, name path=eq1](6*0.4,0)--(0,6*0.4);
\path [name intersections={of=eq1 and eq,by=E1}];
\draw[->](6*0.4,0)--(E1);
\draw[->](0,6*0.4)--(E1);

\draw[white, name path=eq1](6*0.5,0)--(0,6*0.5);
\path [name intersections={of=eq1 and eq,by=E2}];
\draw[->](6*0.5,0)--(E2);
\draw[->](0,6*0.5)--(E2);

\draw[white, name path=eq1](6*0.6,0)--(0,6*0.6);
\path [name intersections={of=eq1 and eq,by=E3}];
\draw[->](6*0.6,0)--(E3);
\draw[->](0,6*0.6)--(E3);

\draw[white, name path=eq1](6*0.7,0)--(0,6*0.7);
\path [name intersections={of=eq1 and eq,by=E4}];
\draw[->](6*0.7,0)--(E4);
\draw[->](0,6*0.7)--(E4);

\draw[white, name path=eq1](6*0.8,0)--(0,6*0.8);
\path [name intersections={of=eq1 and eq,by=E51}];
\path [name intersections={of=eq1 and mu1,by=E52}];
\draw[->](6*0.8,0)--(E52);
\draw[->](0,6*0.8)--(E51);

\draw[white, name path=eq1](6*0.9,0)--(0,6*0.9);
\path [name intersections={of=eq1 and eq,by=E61}];
\path [name intersections={of=eq1 and mu1,by=E62}];
\draw[->](6*0.9,0)--(E62);
\draw[->](0,6*0.9)--(E61);

\draw[white, name path=eq1](6*1,0)--(0,6*1);
\path [name intersections={of=eq1 and eq,by=E71}];
\path [name intersections={of=eq1 and mu1,by=E72}];
\draw[->](6*1,0)--(E72);
\draw[->](0,6*1)--(E71);

\draw[white, name path=eq1](6*1.1,0)--(0,6*1.1);
\path [name intersections={of=eq1 and a1,by=A1}];
\path [name intersections={of=eq1 and a2,by=A2}];
\path [name intersections={of=eq1 and eq,by=E81}];
\path [name intersections={of=eq1 and mu1,by=E82}];
\draw[->](A1)--(E81);
\draw[->](A2)--(E82);

\draw[white, name path=eq1](6*1.2,0)--(0,6*1.2);
\path [name intersections={of=eq1 and a1,by=A3}];
\path [name intersections={of=eq1 and a2,by=A4}];
\path [name intersections={of=eq1 and mu2,by=E91}];
\path [name intersections={of=eq1 and mu1,by=E92}];
\draw[->](A3)--(E91);
\draw[->](A4)--(E92);


\draw[white, name path=eq1](6*1.3,0)--(0,6*1.3);
\path [name intersections={of=eq1 and a1,by=A5}];
\path [name intersections={of=eq1 and a2,by=A6}];
\path [name intersections={of=eq1 and mu2,by=E101}];
\path [name intersections={of=eq1 and mu1,by=E102}];
\draw[->](A5)--(E101);
\draw[->](A6)--(E102);

\draw[white, name path=eq1](6*1.4,0)--(0,6*1.4);
\path [name intersections={of=eq1 and a1,by=A7}];
\path [name intersections={of=eq1 and a2,by=A8}];
\path [name intersections={of=eq1 and mu2,by=E111}];
\path [name intersections={of=eq1 and mu1,by=E112}];
\draw[->](A7)--(E111);
\draw[->](A8)--(E112);


\fill[red!40!white] (E)--(F)--(3,3)--cycle;
\fill[red!40!white] (3,3)--(3,6)--(6,6)--(6,3)--cycle;
\draw[very thick,red!40!white] (0,0.3*6)--(E);
\draw[very thick,red!40!white] (E)--(F);
\draw[very thick,red!40!white] (F)--(6,6);

\draw[very thick,red!40!white] (0,0)--(0,0.3*6);
\end{scope}
\end{tikzpicture}
\caption{Left: plane $(\rho_1,\rho_2)$ where we mark the steady states of model~\eqref{duecorsie}. The violet dotted line represents the equilibria of type (A), the red solid line the equilibria of type (B2a), the blue dashed line the equilibria of type (B1), the orange dash-dotted line the equilibria of type (E), the region shaded with criss-crossing lines represents the equilibria of type (D), and the region shaded with dots represents the equilibria of type (C). Right: phase portrait $(\rho_1,\rho_2)$ of model~\eqref{duecorsie} where the light red zones represent the equilibria of the system.}\label{fasi}
\end{figure} 



It is possible to represent the set of the equilibrium solutions of the two-lane model~\eqref{bftl_2lane} in the plane $(\rho_1,\rho_2)$. In particular, for linear speed functions given by
\begin{equation} \label{lineari}
\begin{aligned}
v_j(\rho) &= v_j^{\max}\left(1-\frac{\rho}{\rho^{\max}}\right), \quad j=1,2 \\
0&<v_1^{\max} < v_2^{\max}
\end{aligned}
\end{equation}
the steady states are shown in the left panel of Fig.~\ref{fasi}. The different hatchings and colors represent the different steady states characterized in Section~\ref{ssec:charact:eq}. It is easy to verify that the points on the line connecting the equilibria of type (A) with the equilibria of type (B), namely the line identified by the violet dotted, the red solid and the blue dashed segments, fulfill
$
    \rho_2=v_2^{-1}(v_1(\rho_1)).
$
In the case~\eqref{lineari}, is
\begin{equation*}
\rho_2=\rho^{\max}\left( 1-\frac{v_1^{\max}}{v_2^{\max}} \left(1-\frac{\rho}{\rho^{\max}}\right) \right).
\end{equation*}
Moreover, we report in the right panel of Fig.~\ref{fasi} the phase portrait of system~\eqref{duecorsie}. Some trajectories are depicted with black lines and they are obtained considering several initial data as perturbation of equilibrium solutions. Here, the steady states are identified by the light red zones. We observe that the trajectories are straight lines since the perturbation we are considering is homogeneous in space. 

\begin{remark}
    The plane $(\rho_1,\rho_2)$ showing the steady states, as well as the phase portrait, both depicted in Fig.~\ref{fasi}, can be also studied for more general speed functions. In that case, the equilibria of type (A) and the equilibria of type (B) are connected by a curve, instead of a line, whose formulation depends on the particular mathematical description of the density-speed relation.
\end{remark}

In the following we  provide the proof of Theorem \ref{teorema}.

\begin{proof} Let us illustrate the analysis of each equilibrium class.
 \textbf{Steady state of type (A)}

In this case, the monotonicity of the velocity functions implies that
\begin{equation}\label{condA}
v_1^\mu < v^{\text{eq}} \leqslant v_1^{\max}. 
\end{equation}
We consider two sub-cases depending on the sign of $\varepsilon$:
\begin{itemize}
\item[(A1)] if $\varepsilon>0$ then:
\begin{equation*}
\rho_1>\overline{\rho}_1 \quad \text{and} \quad v_1(\rho_1)<v^{\text{eq}}, \qquad
\rho_2<\mu \quad \text{and} \quad v_2(\rho_2)>v^{\text{eq}}.
\end{equation*}
Thus, lane changes from lane 2 to lane 1 do not occur, i.e.~$\gamma^{2 \to 1}(\rho_2,\rho_1)=0$, because the incentive condition is not satisfied;
\item[(A2)] similarly in the case $\varepsilon<0$, in which  lane changes from lane 1 to lane 2 do not occur.
\end{itemize}

Let us study in detail the case (A1).
We have that the evolution of the perturbed states $(\rho_1,\rho_2)$ follows the following system of balance laws:
\begin{equation*}
\begin{cases}
\partial_t \rho_1 + \partial_x f_1(\rho_1) =  - \gamma^{1 \to 2}(\rho_1,\rho_2) A(\rho_1,\rho_2)\rho_2 \\
\partial_t \rho_2 + \partial_x f_2(\rho_2) = \gamma^{1 \to 2}(\rho_1,\rho_2) A(\rho_1,\rho_2)\rho_2,
\end{cases}
\end{equation*}
and substituting~\eqref{perturbazione_modello} we obtain
\begin{equation*}
\begin{cases}
\partial_t (\overline{\rho}_1+\varepsilon \rho) + \partial_x f_1(\overline{\rho}_1+\varepsilon \rho) =  - \gamma^{1 \to 2}(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho) A(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho)(\overline{\rho}_2-\varepsilon \rho)\\
\partial_t (\overline{\rho}_2-\varepsilon \rho) + \partial_x f_2(\overline{\rho}_2-\varepsilon \rho) =+\gamma^{1 \to 2}(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho) A(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho)(\overline{\rho}_2-\varepsilon \rho).
\end{cases}
\end{equation*}
If we linearize the previous equations around the steady state $(\overline{\rho}_1,\overline{\rho}_2)$, namely
\begin{align*}
\gamma^{1 \to 2}(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho) &= \gamma^{1 \to 2}(\overline{\rho}_1,\overline{\rho}_2)+\varepsilon \nabla  \gamma^{1 \to 2}(\overline{\rho}_1,\overline{\rho}_2) \cdot (\rho,-\rho)^T +O(\varepsilon^2) \\ &= -\varepsilon g'(\overline{\rho}_2)\rho + O(\varepsilon^2) \\
A(\overline{\rho}_1+\varepsilon \rho,\overline{\rho}_2-\varepsilon \rho) &= A(\overline{\rho}_1,\overline{\rho}_2)+\varepsilon \nabla A(\overline{\rho}_1,\overline{\rho}_2)\cdot (\rho,-\rho)^T +O(\varepsilon^2) \\ &=A(\overline{\rho}_1,\overline{\rho}_2)+\varepsilon(\partial_{\rho_1}A(\overline{\rho}_1,\overline{\rho}_2)-\partial_{\rho_2}A(\overline{\rho}_1,\overline{\rho}_2))\rho+O(\varepsilon^2),
\end{align*}
we get the evolution of the perturbation $\rho=\rho(t)$ as
\begin{equation*}
\partial_t \rho =  \overline{\rho}_2 A(\overline{\rho}_1, \overline{\rho}_2)g'( \overline{\rho}_2)\rho + O(\varepsilon
\end{equation*}
with analytical solution
\begin{equation} \label{eq:exp:decay:rho}
\rho(t)=\rho_0 \exp\left(\overline{\rho}_2 A(\overline{\rho}_1, \overline{\rho}_2)g'( \overline{\rho}_2) t\right).
\end{equation}
Recalling that $g'(r)<0 \ \forall r$, we find that $\rho(t)\to 0$ as $t\to +\infty$, and thus $(\rho_1,\rho_2)\to(\overline{\rho}_1,\overline{\rho}_2)$ as $t\to +\infty$. In other words, the perturbed state converges to the steady state of type (A1) under the perturbation~\eqref{perturbazione_modello}.

With similar computations we can obtain the same result for the sub-case (A2), therefore we omit explicit presentation.

\textbf{Steady state of type (B)}

In this case it is possible to distinguish between various situations due to the monotonicity assumption of the velocity functions.
\begin{itemize}
\item[(B1)] If $\overline{\rho}_1\geqslant \mu$, \eqref{eq:speed:functions} implies that also $\overline{\rho}_2>\mu$ and $\overline{\rho}_2>\overline{\rho}_1$; then, in this case
\begin{equation*}
0 \leqslant v^{\text{eq}} \leqslant v_2^{\mu}.
\end{equation*}
\item[(B2)] If $\overline{\rho}_2\geqslant \mu$, instead, we distinguish the other following two sub-cases:
\begin{itemize}
\item[(B2a)] if $\mu \leqslant \overline{\rho}_2<\rho_2^\mu$, \eqref{eq:speed:functions} implies that $\overline{\rho}_1<\mu$ and thus
\begin{equation*}
v_2^\mu < v^{\text{eq}} \leqslant v_1^{\mu}; 
\end{equation*}
\item[(B2b)] if $\overline{\rho}_2\geqslant \rho_2^\mu$ then $\overline{\rho}_1\geqslant \mu$ and this case reduces to (B1).
\end{itemize}
\end{itemize}


Let us analyze first the case (B1). Given an equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$ of type (B1), we can distinguish the following sub-cases depending on the sign of its perturbation:
\begin{itemize}
\item In the case $\varepsilon>0$:
\begin{itemize}
\item[(I)] if the perturbed densities $(\rho_1,\rho_2)$ are both above the critical value, i.e.~$\mu < \rho_2 < \rho_1$, then $(\rho_1,\rho_2)$ is an equilibrium of type (C) for the system because lane changing is not possible due to the safety criterion;
\item[(II)] if, instead, the perturbed density in the second lane is less than the critical value, i.e.~$\rho_2<\mu$, then the system converges to a different equilibrium solution $(\tilde{\rho}_1,\tilde{\rho}_2)$, which is of type (C) with $\tilde{\rho}_2=\mu$;
\end{itemize}
\item similar results hold for the case $\varepsilon<0$.
\end{itemize}
We prove the previous statements in the following.

Case (I) implies that lane changes are not possible neither from lane 1 to lane 2, nor from lane 2 to lane 1, because the safety criterion is violated. Thus, $\gamma^{1 \to 2}(\rho_1,\rho_2)=\gamma^{2\to 1}(\rho_2,\rho_1)=0$ for all $(x,t)$, and it is easy to check that the evolution of the perturbation $\rho$ satisfies
\begin{equation*}
\varepsilon \partial_t \rho =  0 \ \Rightarrow \ \rho(t) = \rho_0
\end{equation*}
Then, $(\rho_1,\rho_2)=(\overline{\rho}_1+\varepsilon\rho_0,\overline{\rho}_2-\varepsilon\rho_0)$ is an equilibrium of type (C) because the densities are both above the critical value $\mu$ and the speeds in the two lanes are different.

Case (II) implies that lane changes are possible only from lane 1 to lane 2, because of the safety condition. Thus, $\gamma^{1 \to 2}(\rho_1,\rho_2)>0$ and $\gamma^{2\to 1}(\rho_2,\rho_1)=0$. Repeating the same computations as for the steady states of type (A), it is possible to show that we obtain again the exponential decay~\eqref{eq:exp:decay:rho} of the perturbation $\rho$. However, in this case the density $\rho_1$ decreases as long as $\gamma^{1 \to 2}(\rho_1,\rho_2)>0$.
Therefore, there exists a time $\overline{t}$ such that $\gamma^{1\to 2}\to0$ as $t\to\overline{t}$, and
\begin{equation*}
\rho(t) =
\begin{cases}
\rho_0 \exp\left(\overline{\rho}_2 A(\overline{\rho}_1, \overline{\rho}_2)g'( \overline{\rho}_2) t\right), & t<\overline{t} \\[1ex]
\frac{1}{\varepsilon}(\overline{\rho}_2-\mu), & t\geqslant \overline{t}
\end{cases}
\end{equation*}
where the behaviour for $t\geqslant \overline{t}$ is obtained by imposing mass conservation principle.


Now, we focus on the case (B2a). Given an equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$ of type (B2a), we can distinguish the following sub-cases depending on the sign of its perturbation:
\begin{itemize}
\item In the case $\varepsilon>0$ we have: 
\begin{itemize}
\item[(I)] if the perturbed state $(\rho_1,\rho_2)$ is such that $\rho_1<\mu$ and $\rho_2>\mu$, then the system converges to a different steady state $(\tilde{\rho}_1,\tilde{\rho}_2)$, which is of type (D);
\item[(II)] if the perturbed state $(\rho_1,\rho_2)$ is such that $\rho_1>\mu$ and $\rho_2>\mu$, then the system converges to a different steady state $(\tilde{\rho}_1,\tilde{\rho}_2)$, which is of type (C);
\item[(III)] if the perturbed state $(\rho_1,\rho_2)$ is such that $\rho_2 \leqslant \mu$, then the system converges to a different steady state $(\tilde{\rho}_1,\tilde{\rho}_2)$, which is of type (C) with $\tilde{\rho}_2=\mu$;
\end{itemize}
\item in the case $\varepsilon<0$: 
\begin{itemize}
\item[(IV)] independently of the initial perturbed state $(\rho_1,\rho_2)$, the system returns to the equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$ of type (B2a). 
\end{itemize}
\end{itemize}

We observe that in case (I) lane changes are not possible, neither from lane 1 to lane 2 because of the safety criterion, nor from lane 2 to lane 1 because of the incentive criterion. Therefore, the perturbation $\rho$ remains constant in time

Case (II) is similar, since there are no lane changes because of the safety criterion. 

In case (III), instead, only lane changes from lane 1 to lane 2 are possible and until the safety criterion is verified, i.e.~until $\rho_2$ remains smaller than the critical value $\mu$. Therefore, the analysis of this case is similar to the analysis of case (II) for a steady state of type (B1).

Finally, for case (IV) we note that, since $\rho_1<\mu$ and $\rho_2>\mu$, only lane changing from lane 2 to lane 1 can occur. The velocity in lane 2 increases until its value becomes equal to the velocity in lane 1 which is decreasing. In this case, the system converges to the original equilibrium $(\overline{\rho}_1, \overline{\rho}_2)$.

Case (B2b) is analogous to case (B1), and therefore we omit the analysis.

\textbf{Steady state of type (C)}

If the perturbation~\eqref{perturbazione_modello} of an equilibrium state $(\overline{\rho}_1, \overline{\rho}_2)$ of type (C) is characterized by a small $\varepsilon \rho$ such that $\rho_1<\mu$ and $\rho_2<\mu$, then $\partial_t \rho = 0$ and $(\rho_1,\rho_2)=(\overline{\rho}_1+\varepsilon\rho,\overline{\rho}_2-\varepsilon\rho)$ establishes an equilibrium of type (C) for the system. Otherwise depending on the size of the initial perturbation $\varepsilon \rho_0$ the system converges either to a different equilibrium of type (C) characterized by the density on one lane equal to $\mu$, or of type (D), or of type (B1), or of type (B2a).   

 \textbf{Steady state of type (D)}

We observe that an equilibrium state $(\overline{\rho}_1, \overline{\rho}_2)$ of type (D) is characterized by $\overline{\rho}_1<\mu$, so that the safety condition for a lane change from lane $2$ to lane $1$ is satisfied. However, the lane change cannot occur because at the same time  $v_1(\overline{\rho_1})<v_2(\overline{\rho_2})$ holds
This situation is only possible if
\begin{equation}\label{caseD}
\mu \leqslant \overline{\rho}_2 \leqslant \rho_2^\mu \qquad \text{and} \qquad (v_1)^{-1}(v_{2}(\overline{\rho}_2))\leqslant\overline{\rho}_1 \leqslant \mu.
\end{equation}
If one has that $\overline{\rho}_1=(v_1)^{-1}(v_{2}(\overline{\rho}_2))$, case (D) reduces to case (B2a).
Contrary, if $\overline{\rho}_1\neq (v_1)^{-1}(v_{2}(\overline{\rho}_2))$ and the perturbed state $(\rho_1,\rho_2)$, cf.~\eqref{perturbazione_modello}, satisfies~\eqref{caseD}, then $\partial_t \rho = 0$ because the lane changing conditions are not satisfied, and $(\rho_1,\rho_2)$ establishes a new equilibrium of type (D) for the system. Otherwise, if $(\rho_1,\rho_2)$ does not satisfy~\eqref{caseD}, depending on the size of the perturbation the system converges to an equilibrium state either of type (C), or of type (B2a).

 \textbf{Steady state of type (E)}

In this case the equilibrium state is characterized by a zero density in the slowest lane, whereas the density in the fastest lane takes values in $[0,v_2^{-1}(v_1^{\text{max}})]$. Perturbation~\eqref{perturbazione_modello} is admissible only for $\varepsilon>0$, thus defining a perturbed state $(\rho_1,\rho_2)$ of the system which makes possible only lane changes from lane 1 to lane 2. With the same arguments of case (A1), we can prove that the perturbation $\rho(t)$ decays exponentially and $(\rho_1,\rho_2)\to(\overline{\rho}_1,\overline{\rho}_2)$ as $t\to\infty$. This result explains mathematically the situation in which all vehicles migrate from the slowest lane to the fastest one.
$\qed$
\end{proof}





\section{Numerical simulations} \label{sec:numerics}

In this section we present four numerical tests in order to investigate the behavior of the multi-lane macroscopic model~\eqref{modello} using two lanes $J=2$ and three lanes $J=3$. In particular, the first test aims to show the consistency between the microscopic multi-lane model and its macroscopic limit performed in Section~\ref{sec:derivation}. The following two tests are devoted to study the effects of space homogeneous and local (i.e.~space non-homogeneous) perturbations of equilibrium states. Finally, in the last test we propose a practical application of the multi-lane macroscopic model which is used to study the flow of vehicles on a three-lane highway with a local lane closure, e.g.~due to a work zone.

In all simulations the macroscopic system~\eqref{modello} is numerically integrated with a first order finite volume scheme with Rusanov numerical flux, considering a uniform grid of size $\Delta x$ and adaptive time steps $\Delta t$ in order to satisfy the Courant-Friedrichs-Lewy stability condition. 

\subsection{Numerical evidence of the macroscopic limit}

In order to investigate the consistency between the microscopic and the macroscopic model, we focus on a two-lane circular road of length $L=1500$~meters. However, in the numerical simulations we consider normalized parameters, namely $L=1$, $l=d_s=\frac{5~\text{meters}}{\tilde{L}}$ and a final time $T=100$.

The initial condition of the microscopic model~\eqref{bftl_2lane} are set in such a way the two lanes are locally at the equilibrium, but lane changes are possible. We refer to Definition~\ref{def:ss:micro}. Therefore, we initially fix a suitable number of vehicles in the two lanes and impose that vehicles are uniformly distributed on each lane. Precisely, let $N_j(0)$ be the number of vehicles in lane $j=1,2$ at time $t=0$. Then \begin{equation}\label{ic_micro}
\forall n\in I_j(0): x_n(0)\,\,\text{equally spaced with headway}\,\,h_j=\frac{L}{N_j(0)}
\end{equation}
For the macroscopic model~\eqref{duecorsie}, the initial condition is computed using the corresponding initial local discrete densities defined in~\eqref{piece}. Thus
\begin{equation}\label{ic_macro}
\rho_j(x,0)=\sum_{n\in I_j(0)}\rho_j^{(n)}(t=0) \chi_{[x_n,x_{s_n^j})}(x) \,\, j=1,2
\end{equation}
We employ the linear density-speed relations given in~\eqref{lineari} as macroscopic speed functions. In particular, we choose $v_2^{\max}=1$ and $v_1^{\max}=0.7$. Furthermore $\Delta x =0.003$.


\begin{table}[t]
	\centering
	\caption{Numerical experiment 1 on the consistency of the macroscopic limit. Values of the initial conditions and final states of the microscopic and macroscopic model.\label{tab:exp1}}
	\begin{tabular}{l|cc|cc}
	
		& \multicolumn{2}{c|}{Lane 1} & \multicolumn{2}{c}{Lane 2} \\
	
		& $t=0$ & $t=100$ & $t=0$ & $t=100$ \\
		\hline\hline
		\multirow{2}{*}{Microscopic} & $N_1=150$ & $N_1=104$ & $N_2=30$ & $N_2=76$ \\
		& $\rho_1^{(n)}=1$ & $\rho_1^{(n)}=0.69$ & $\rho_2^{(n)}=0.2$ & $\rho_2^{(n)}=0.51$ \\
		\hlin
		Macroscopic & $\rho_{1}(x,t)=1$ & $\rho_{1}(x,t)=0.70$ & $\rho_{2}(x,t)=0.2$ & $\rho_{2}(x,t)=0.50$ \\
		\hlin
	\end{tabular}
\end{table}

\begin{table}[t]
	\centering
	\caption{Numerical experiment 2 on the consistency of the macroscopic limit. Values of the initial conditions and final states of the microscopic and macroscopic model.\label{tab:exp2}}
	\begin{tabular}{l|cc|cc}
	
		& \multicolumn{2}{c|}{Lane 1} & \multicolumn{2}{c}{Lane 2} \\
	
		& $t=0$ & $t=100$ & $t=0$ & $t=100$ \\
		\hline\hline
		\multirow{2}{*}{Microscopic} & $N_1=100$ & $N_1=72$ & $N_2=50$ & $N_2=78$ \\
		& $\rho_1^{(n)}=0.67$ & $\rho_1^{(n)}=0.48$ & $\rho_2^{(n)}=0.33$ & $\rho_2^{(n)}=0.52$ \\
		\hlin
		Macroscopic & $\rho_{1}(x,t)=0.66$ & $\rho_{1}(x,t)=0.49$ & $\rho_{2}(x,t)=0.33$ & $\rho_{2}(x,t)=0.50$ \\
		\hlin
	\end{tabular}
\end{table}


The microscopic system is solved with a fifth-order Runge-Kutta method. The results of the two tests are summarized in Table~\ref{tab:exp1} and Table~\ref{tab:exp2}. Here, we report the chosen initial conditions, namely the number of vehicles and the local densities for the microscopic model and the macroscopic density for the continuum model, as well as the final states obtained via numerical evolution of both models.
Both simulations show the agreement between the microscopic multi-lane model and its macroscopic limit.






\subsection{Global  perturbation in space  of equilibrium solutions}

In this test we show a numerical evidence of the theoretical study on the stability of the steady states. Since the analytical results provides a detailed description of the behaviour of perturbations of equilibrium solutions, we present here only two numerical examples. We consider an initial datum for the two-lane model~\eqref{duecorsie} given by $(\rho_1,\rho_2)=(\overline{\rho}_1+\varepsilon\rho_0,\overline{\rho}_2-\varepsilon\rho_0)$, where $(\overline{\rho}_1,\overline{\rho}_2)=(0.27,0.49)$. Considering again the macroscopic density-speed relation as in~\eqref{lineari}, we have that $(\overline{\rho}_1,\overline{\rho}_2)$ belongs to the set of steady states of type (A). Observe that $(\rho_1,\rho_2)$ is a uniform space perturbation of the steady state, as given in~\eqref{perturbazione_modello}. In particular, we choose $\varepsilon\rho_0=0.485$ in the first test and $\varepsilon\rho_0=-0.27$ in the second test.

\begin{figure}[t]
\centering
\includegraphics[width=0.49\textwidth]{test_1} \includegraphics[width=0.49\textwidth]{test_2}
\caption{Global in space perturbation of the equilibrium solution $(\overline{\rho}_1,\overline{\rho}_2)=(0.27,0.49)$ of type (A). Left: the perturbation is $(\rho_1,\rho_2)=(\overline{\rho}_1+\varepsilon\rho_0,\overline{\rho}_2-\varepsilon\rho_0)$, with $\varepsilon\rho_0=0.485$. Right: the perturbation is $(\rho_1,\rho_2)=(\overline{\rho}_1+\varepsilon\rho_0,\overline{\rho}_2-\varepsilon\rho_0)$, with $\varepsilon\rho_0=-0.27$.}\label{evoA}
\end{figure} 

Here, the space domain is $[-0.5,0.5]$ with periodic boundary conditions, the final time is $T=50$, and $\Delta x =0.003$. In Fig.~\ref{evoA} we show the evolution in time of the mean value of the densities $\rho_1$ and $\rho_2$ on the road length. Namely, at each time $t$ we compute the quantities
\begin{equation} \label{eq:mean:density}
	\langle \rho_j \rangle(t) \approx \int_{-\frac12}^{\frac12} \rho_j(x,t) \mathrm{d}x, \quad j=1,2.
\end{equation}
We observe that $\langle \rho_j \rangle(t) \to \overline{\rho}_j$, $j=1,2$, as $t\to\infty$ with zero standard deviation. Therefore, $\rho_j(t) \to \overline{\rho}_j$, $j=1,2$, as $t\to\infty$ showing the convergence of the system to the original steady state. This confirms the theoretical finding on the global asymptotic stability of equilibria of type (A). 

\subsection{Local  perturbation in space of equilibrium solutions}

In the following test, we study the effects of a local in space perturbation, i.e.~non-homogeneous in space. In particular, the perturbation of the equilibrium state $(\overline{\rho}_1,\overline{\rho}_2)$ is a bump in the density located around $x=0$ and it modeled by a Gaussian profile. More precisely, we consider the following initial datum:
\begin{equation} \label{eq:local:pert}
	(\rho_1(x,0),\rho_2(x,0))=\left(\overline{\rho}_1+G(x),\overline{\rho}_2-G(x) \right)
\end{equation}
where

\begin{equation} \label{eq:gaussian}
	G(x)=\exp(-100x^2)\cdot R, \quad \text{with } R=\begin{cases} \min(\overline{\rho}_1, 1-\overline{\rho}_2) & \mbox{if } \overline{\rho}_1\geqslant \overline{\rho}_2 \\ \min(\overline{\rho}_2, 1-\overline{\rho}_1)  & \mbox{otherwise.}  \end{cases}
\end{equation}

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{gauss_inv}
\caption{Global in space perturbation of the equilibrium solution $(\overline{\rho}_1,\overline{\rho}_2)=(0.14,0.40)$ of type (A). The perturbation is given by~\eqref{eq:local:pert}-\eqref{eq:gaussian}. Density profiles at time $t=0$ (top panels) and final time $T=10$ (bottom panels) in lane 1 (left panels) and lane 2 (right panels).}\label{gauss}
\end{figure} 

\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{gaussiana3d}
\caption{Global in space perturbation of the equilibrium solution $(\overline{\rho}_1,\overline{\rho}_2)=(0.14,0.40)$ of type (A). The perturbation is given by~\eqref{eq:local:pert}-\eqref{eq:gaussian}. Evolution in time of the density profiles in lane 2 (left panel) and lane 1 (right panel)}\label{gauss2}
\end{figure} 

The space domain is again $[-0.5,0.5]$ with periodic boundary conditions, the final time is $T=10$, and $\Delta x = 0.01$. For this test we refer again to an initial equilibrium of type (A), precisely $(\overline{\rho}_1,\overline{\rho}_2)=(0.14,0.40)$ obtained with the equilibrium velocity $v^{\text{eq}}=0.6$. The density profiles at initial and final time on both lanes are provided in Fig.~\ref{gauss}, whereas the evolution in time can be observed in Fig.~\ref{gauss2}. Computing the mean value of the densities $\rho_1$ and $\rho_2$ as in the previous numerical test, cf.~\eqref{eq:mean:density} we obtain that $\langle \rho_1 \rangle(T)=0.143$, with standard deviation
$3\cdot10^{-3}$, and $\langle \rho_2 \rangle(T)=0.399$ with standard deviation $2\cdot10^{-3}$. We conclude that the numerical results show the convergence of the system to the original equilibrium $(\overline{\rho}_1,\overline{\rho}_2)$. Therefore, also in the case of a local perturbation, the system behaves as studied in the case of a homogeneous perturbation around the steady states.

\subsection{Lane closure}

Here we present an example of a lane drop on a three-lane road. Traffic flow undergoes lane closure, where part of the fastest lane, i.e.~lane 3, is closed due to, e.g., an accident or the presence of a working in progress area. The road design for this simulation is illustrated in Fig.~\ref{chiusura}. On the three-lane road we prescribe different desired velocity profiles such that $v_3(\rho)<v_2(\rho)<v_1(\rho)$, $\rho\in[0,\rho^{\text{max}})$.

\begin{figure}[t]
\centering
\begin{tikzpicture}
\draw (0,3)--(8,3);
\draw [dashed](0,2)--(8,2);
\draw [dashed](0,1)--(8,1);
\draw (0,0)--(8,0);
\draw[fill=black] (4,2) rectangle ++(2,1);
\path(-1,0.5)node{lane $1$};
\path (-1,1.5)node{lane $2$};
\path (-1,2.5)node{lane $3$};
\end{tikzpicture}
\caption{Schematic for traffic flow through local closure of lane 3.}\label{chiusura}
\end{figure}

We consider a road of unit length modeled by the space domain $I=[-0.5,0.5]$ with closure in lane 3 located in $C=[0,0.25]$. Linear velocities as in the previous tests are prescribed, with $v_3^{\max}=1$, $v_2^{\max}=0.7$ and $v_1^{\max}=0.6$. We present two tests with uniform initial densities given by
\begin{equation}
\begin{array}{l}
\rho_1(x,0)=0.4,\,\, x \in I\,\, \mbox{(test 1 \& 2)} \\ 
\rho_2(x,0)=0.6, \,\,x\in I\,\, \mbox{(test 1)}\,\,-\,\, \rho_2(x,0)=0.1, \,\,x\in I\,\, \mbox{(test 2)} \\
\rho_3(x,0)=\begin{cases} 0.2 & x<0 \\ 0 & x>0.25. \end{cases} \,\, \mbox{(test 1 \& 2)}
\end{array}
\end{equation}
Moreover, we assume free flow boundary conditions on $x=0.5$ and Dirichlet boundary conditions on $x=-0.5$ given by
\begin{equation*}
    \rho_j(-0.5,t)=\rho_j(-0.5,0),\quad\forall t>0.
\end{equation*}
We evolve the model~\eqref{modello} with $J=3$ up to final time $T=1.2$.


\begin{figure}[t]
\centering
\includegraphics[width=\textwidth]{work3d}\\
\includegraphics[width=\textwidth]{work3d_test2}
\caption{Evolution in time of the density in the three lanes. Top: test 1, bottom: test 2.}\label{3d}
\end{figure}


Fig.~\ref{3d} shows the evolution of the densities in the three lanes. We observe that the local closure of lane 3 leads to an increase of the densities in the two right-most lanes that could also create bottlenecks depending on the initial congestion of the road. Both tests show the presence of a queue in lane 3 that evolves backward in space. In the test 1 vehicles migrate to lane 2 where the density increase up to the critical value $\mu=0.5$  while in test 2 the density in lane 2 tends to $\mu$ locally near to $C$ due to the different inflow condition. Furthermore, we note that the the density in lane 1, the slowest one, stabilizes close to the value $\mu$ because vehicles tend to stay in lane 2 because of the incentive criterion.



\section{Conclusions} \label{sec:conclusion}
In this paper, we have presented a new first-order multi-lane macroscopic model for traffic flow, which has been derived as continuum limit of a microscopic follow-the-leader framework where we have proposed specific lane changing conditions. Exploiting the evolution of the discrete densities, we have described the effects of lane changes in terms of macroscopic quantities and without postulating specific scalings in space and time for the derivation of the final system of partial differential equations. The system thus obtained is a system of balance laws with source terms that refer to the lane changing and that model the increase of the density in a given lane, which is proportional to the density of the lane itself. After this derivation a detailed stability analysis of the system equilibria has been performed. The considered equilibria refer to a situation where traffic flow is constant in all lanes and there are no lane changes. We have studied the stability of these equilibria under global, i.e.~space homogeneous, perturbations proving the existence of asymptotically stable, globally asymptotically stable and marginally stable equilibria. We have performed some numerical tests devoted to the comparison between the microscopic model and the corresponding multi-lane macroscopic limit. Furthermore, we have studied numerically the evolution of perturbed equilibria using global perturbation, confirming the results of the theoretical stability analysis. We have also shown that the steady states behave similarly under local perturbations. Finally, we have proposed a numerical test aimed to study the behavior of the model in a real situation such as the closure of part of a lane on a three-lane road. The evolution of the densities show realistic flow of vehicles with the lane changes dynamics introduced here. 


\section*{Acknowledgments}

M.H. thanks the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) for the financial support through 320021702/GRK2326, 333849990/IRTG-2379, B04, B05 and B06 of CRC1481, HE5386/18-
1,19-2,22-1,23-1, ERS SFDdM035 and under Germany’s Excellence Strategy EXC-2023 Internet of Production 390621612 and under the Excellence Strategy of the Federal Government and the Länder.

M.H. and G.P. acknowledge support through
the EU DATAHYKING project.

M.P. and G.P. acknowledge support through Ateneo Sapienza project 2019 ``Metodi numerici per problemi evolutivi, networks ed applicazioni'', 2020 ``Algoritmi e modelli per sistemi di natura iperbolica, networks e applicazioni'', and 2021 ``Evolutionary problems:
analysis techniques and construction of numerical solutions''. This work was also carried out within the MUR (Ministry of University and Research) PRIN-2017 project ``Innovative Numerical Methods for Evolutionary Partial Differential Equations and Applications'' (number 2017KKJP4X).

M.P. and G.P. are members of the INdAM Research Group GNCS.

M. P. wishes to thank: Giuseppe Visconti for the valuable help and fruitful discussions; and Elisa Iacomini for the for the useful suggestions during the stay at the RWTH Aachen University.


\bibliographystyle{siamplain}
