# A Continuation Framework for Traffic Flow Modeling and Control on Ring Roads

Sébastien Fueyo, Carlos Canudas de Wit

► To cite this version:

Sébastien Fueyo, Carlos Canudas de Wit. A Continuation Framework for Traffic Flow Modeling and Control on Ring Roads. 2026. hal-05518999

HAL Id: hal-05518999

https://hal.science/hal-05518999v1

Preprint submitted on 19 Feb 2026

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# A Continuation Framework for Traffic Flow Modeling and Control on Ring Roads

1st Sébastien Fueyo

GIPSA-lab

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP

Grenoble, France

sebastien.fueyo@gipsa-lab.fr

2nd Carlos Canudas de Wit

GIPSA-lab

Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP

Grenoble, France

carlos.canudas-de-wit@gipsa-lab.fr

***Abstract***—A continuation-based method is proposed to systematically derive macroscopic partial differential equation (PDE) representations from first- and second-order microscopic multi-agent traffic models on a ring. The framework establishes a consistent bridge between microscopic vehicle dynamics and macroscopic traffic flow descriptions, thereby enabling the design of feedback control laws at the macroscopic level that can be coherently implemented in the underlying microscopic system. When deployed, the feedback effectively renders all vehicles autonomous, allowing precise regulation of the traffic state. The methodology is illustrated using the Optimal Velocity (OV) and Optimal Velocity-Follow-the-Leader (OV-FTL) models, demonstrating its capability to steer the system toward prescribed density and/or velocity profiles.

***Index Terms***—Traffic modeling, car-following models, macroscopic PDEs, continuation method, Optimal Velocity model, feedback control, autonomous vehicles, ring-road traffic

## I. INTRODUCTION

Road traffic systems consist of a large number of interacting vehicles, making direct microscopic analysis increasingly challenging as the system size grows. Vehicle interactions are commonly modeled using car-following laws formulated as first- or second-order ordinary differential equations, such as the optimal velocity (OV) first order model [3] and the optimal velocity-follow the leader (OV-FTL) second order model [6]. To reduce the complexity of large-scale microscopic descriptions, traffic flow is often represented at a macroscopic level by hyperbolic partial differential equations, including classical models such as the Lighthill-Whitham-Richards (LWR) first order model [15] and the second order Aw-Rascle-Zhang (ARZ) model [2].

Macroscopic traffic dynamics emerge from the collective behavior of individual drivers and can exhibit complex phenomena such as phantom traffic jams and stop-and-go waves. These instabilities arise when uniform traffic flow loses stability and small perturbations amplify into nonlinear oscillations, commonly referred to as jamitons [7]. Capturing such phenomena requires a consistent link between microscopic vehicle dynamics and macroscopic traffic models.

While the discretization of partial differential equations into systems of ordinary differential equations is well established, the inverse problem of deriving macroscopic PDEs

<figure><img src="image_2.png" /><figcaption>Fig. 1: Continuation paradigm</figcaption></figure>

from microscopic dynamics has received comparatively less attention. This inverse procedure, commonly referred to as the continuation method [20], provides a systematic framework for connecting micro- and macroscopic descriptions. Previous works on the continuation procedure have mainly focused on first-order traffic models, using convolution operators [5], [14], [16] or Taylor expansions [1], [13], [17], [19], [23], whereas a unified continuation approach applicable to both first- and second-order microscopic models remains largely unexplored. In this work, we develop a general continuation methodology for first- and second-order traffic models posed on a ring road. Ring-road configurations are widely used in traffic research because they provide a simple and reproducible experimental setting, eliminate boundary effects, and facilitate the observation of traffic instabilities [6], [11], [21], [22], [24]. The closed-loop geometry naturally gives rise to stop-and-go waves and other nonlinear phenomena, making it an ideal testbed for both modeling and control studies.

Building upon the methodology introduced in [9], which addressed second-order traffic models on a ring road where each vehicle interacts solely with its immediate leader, we introduce two main extensions. First, we extend the framework to first-order traffic models, which were not considered in the original study. Second, we generalize the interaction topology by allowing each vehicle to respond to multiple neighboring vehicles, rather than being limited to its direct predecessor.

The practical relevance of the proposed framework is further highlighted by reformulating microscopic control problems at the macroscopic level, thereby enabling the tracking of
arbitrary reference profiles for autonomous vehicles governed by microscopic dynamics. This is accomplished by designing feedback laws based on the macroscopic model derived through the continuation procedure, which can then be consistently implemented in the underlying microscopic system, as illustrated in Fig. 1.

## II. TRAFFIC SETTING AND MICROSCOPIC MODELLING

We consider a traffic stream composed of $N \in \mathbb{N}$ vehicles circulating on a single-lane ring road of length $2\pi$, chosen for simplicity. The longitudinal position of vehicle $i$ at time $t$ is denoted by $x_i(t) \in [0, 2\pi)$ and is defined modulo $2\pi$, reflecting the periodic geometry commonly adopted in traffic flow experiments and theoretical studies. At the microscopic scale, traffic dynamics are described by interaction-based car-following models. Depending on the desired level of realism, vehicle motion may be represented either by first-order or second-order dynamical systems.

a) First-order models: In first-order models, the vehicle speed adjusts instantaneously to the surrounding traffic configuration, for $t \ge 0$,

$$ \dot{x}_i(t) = F(x_1(t), \dots, x_N(t)), \quad i = 1, \dots, N, \quad (1) $$

where $F$ encodes the interaction rules and driving behavior, assumed identical for all vehicles. This formulation includes local car-following laws as well as nonlocal interaction-based models.

b) Second-order models: To account for inertial effects, finite reaction times, and smoother acceleration profiles, second-order models introduce velocity as a dynamic variable, for $t \ge 0$ and $i = 1, \dots, N$,

$$ \left\{ \begin{array}{l} \dot{x}_i(t) = v_i(t), \\ \dot{v}_i(t) = A(x_1(t), \dots, x_N(t), v_1(t), \dots, v_N(t)), \end{array} \right. \quad (2) $$

where $A$ models the response of drivers to the relative positions and velocities of surrounding vehicles.

c) Representative examples:

**Example 1 (Nonlocal interaction traffic model).** Each vehicle adapts its speed based on the positions of all other vehicles, representing anticipation effects or automated driving strategies:

$$ \dot{x}_i(t) = \sum_{j=1}^{N} f(x_i(t) - x_j(t)), \quad i = 1, \dots, N, \quad (3) $$

where $f$ quantifies the influence of surrounding vehicles.

**Example 2 (Spacing-based car-following model).** Drivers react primarily to the distance to the immediate leader:

$$ s_i(t) = x_{i+1}(t) - x_i(t), \quad \dot{x}_i(t) = V(s_i(t)), \quad (4) $$

where $V(\cdot)$ is the optimal velocity function. A typical choice is

$$ V(s) = v_{\max} \left( 1 - \frac{s_{\text{stop}}}{s} \right), $$

with $v_{\max} > 0$ the free-flow speed and $s_{\text{stop}} > 0$ the minimal spacing at standstill, see for instance [10].

**Example 3 (Second-order optimal velocity follow-the-leader (OV-FTL) model).** This model extends the optimal velocity car-following law to include relative velocity effects and collision avoidance as in [6]:

$$ \ddot{x}_i(t) = a \frac{\dot{x}_{i+1}(t) - \dot{x}_i(t)}{(x_{i+1} - x_i)^2} + b(V(x_{i+1} - x_i) - \dot{x}_i(t)), \quad (5) $$

for $t \ge 0$, $i = 1, \dots, N$ and where $a, b > 0$ are positive parameters describing the drivers' behavior and $V(\cdot)$ is an increasing, bounded optimal velocity function, typically the one given in (4).

The first term models response to relative velocity, dominant at short distances to prevent collisions, while the second term relaxes each vehicle toward the desired speed given its spacing. This is a prototypical second-order microscopic traffic model exhibiting both uniform flow equilibria and stop-and-go instabilities.

## III. CONTINUUM APPROXIMATION OF FIRST- AND SECOND- ORDER TRAFFIC DYNAMICS

We now derive a continuum representation of the microscopic traffic models (1) and (2). Rather than presenting a detailed derivation, we adopt a continuation principle that is well established in traffic flow theory and multi-agent systems (see, e.g., [9], [18], [20]). Our contribution lies in extending this framework to include first- and second-order vehicle dynamics while allowing interactions with multiple neighboring vehicles, rather than restricting attention to the immediate predecessor as in [9]. Such generalized interaction structures naturally arise in intelligent transportation systems and capture more heterogeneous driving behaviors.

The key idea consists of replacing the discrete vehicle index $i \in \{1, \dots, N\}$ with a continuous label $M \in [0, N]$, interpreted as a vehicle count variable. This leads to a continuous mapping $x(t, M)$ interpolating the discrete trajectories,

$$ x(t, M = i) = x_i(t), $$

from which macroscopic traffic quantities can be defined.

### A. Macroscopic traffic variables

Following classical formulations of cumulative-flow, we introduce the Moskowitz function $M(t, x)$, defined as the number of vehicles located upstream of position $x \in [0, 2\pi]$ at time $t \ge 0$ along the ring road. By construction,

$$ M(t, x_i(t)) = i, \quad M(t, 0) = M(t, 2\pi). $$

The vehicle density $\rho$ and flow $\phi$ are defined as

$$ \rho = \partial_x M, \quad \phi = -\partial_t M, \quad (6) $$

which directly yields the conservation of vehicles,

$$ \partial_t \rho + \partial_x \phi = 0. \quad (7) $$

Introducing the macroscopic speed $v = \phi/\rho$, we recover the standard traffic conservation law

$$ \partial_t \rho + \partial_x (\rho v) = 0. \quad (8) $$
## B. Closure via continuation of microscopic interactions

To close (8), the velocity field $v$ must be related to the microscopic dynamics (1) and (2). We begin with the first-order dynamics (1), where the velocity $v$ will be expressed as a function of the density $\rho$.

Using the inverse relation between $x(t, M)$ and $M(t, x)$, one obtains the classical identities

$$v = \partial_t x, \quad \partial_M x = \frac{1}{\rho}. \qquad (9)$$

Under the assumption of slowly varying density, relative vehicle positions can be locally approximated by a first Taylor expansion as

$$x_j(t) - x_i(t) \approx (j-i) \frac{1}{\rho(t, x_i(t))}, \qquad (10)$$

which allows the microscopic interaction law to be expressed as a function of the local density only. By performing a spatial continuation of the function $F$ and using equations (9)–(10) in (1), we derive an effective macroscopic speed law:

$$v(t, x) = \mathcal{F}(x, \rho), \qquad (11)$$

where the function $\mathcal{F}$ satisfies

$$\mathcal{F}(x, \rho) := F\left(x + (j-1)\frac{1}{\rho}, \dots, x + (j-N)\frac{1}{\rho}\right). \qquad (12)$$

Substituting (11) into (8) yields a first-order microscopic traffic model on the ring. Due to the closed-loop geometry of the ring road, the density satisfies $\rho(t, 0) = \rho(t, 2\pi)$. We summarize this result in the following proposition.

**Proposition III.1.** Under a continuum approximation, the first-order traffic system (1) admits the following macroscopic representation:

$$\left\{ \begin{array}{l} \partial_t \rho + \partial_x (\rho \mathcal{F}(x, \rho)) = 0, \\ \rho(t, 0) = \rho(t, 2\pi), \end{array} \right. \qquad (13)$$

where the function $\mathcal{F}$ is obtained by applying the continuation procedure to the microscopic interaction laws.

Proposition III.1 extends classical continuum limits of car-following models to traffic systems. In particular, it recovers the Lighthill–Whitham–Richards (LWR) model from the Optimal Velocity one, as illustrated in the following example.

**Example 4.** Consider the optimal velocity model of Example 2. The associated macroscopic speed law reads

$$\mathcal{F}(x, \rho) = v_{\max}(1 - s_{\text{stop}}\rho),$$

leading from (13) to a LWR-type traffic model, see [10].

We now turn to the second-order model (2). First, we have:

$$\partial_t v = \partial_t x, \qquad (14)$$

since the time derivative of the speed corresponds to the acceleration of the vehicle. Second, to handle the differences

between vehicle speeds, we approximate $v_j - v_i$ using a spatial derivative and by a first Taylor expansion:

$$v_j - v_i \approx (j-i) \partial_x v, \qquad (15)$$

which assumes that the velocity field varies smoothly along the road. This approximation allows us to express interactions between neighboring vehicles in terms of continuous fields, making it possible to derive a macroscopic representation of the second-order dynamics that is consistent with the microscopic model.

By performing a spatial continuation of the function $A$ and using (14)–(10) together with (15) in (2), we obtain an effective macroscopic acceleration law

$$\partial_t v = \mathcal{G}(x, \rho, \partial_x v, v), \qquad (16)$$

where the function $\mathcal{G}$ satisfies

$$\mathcal{G}(x, \rho, \partial_x v, v) := A(\mathbf{X}, \mathbf{V}), \qquad (17)$$

where the auxiliary vectors $\mathbf{X}$ and $\mathbf{V}$ are defined by

$$\mathbf{X} := (x + (j-1)\frac{1}{\rho}, \dots, x + (j-N)\frac{1}{\rho}), \qquad (18)$$

$$\mathbf{V} := (v + (j-1)\partial_x v, \dots, v + (j-N)\partial_x v). \qquad (19)$$

Using both (16) and (8), we obtain a second-order macroscopic traffic model on the ring. As in the first-order case, the closed-loop geometry imposes the periodic condition $\rho(t, 0) = \rho(t, 2\pi)$ and $v(t, 0) = v(t, 2\pi)$. The corresponding macroscopic formulation is summarized in the following proposition.

**Proposition III.2.** Under a continuum approximation, the second-order microscopic traffic system (2) admits the following macroscopic representation:

$$\left\{ \begin{array}{l} \partial_t \rho + \partial_x (\rho v) = 0, \\ \partial_t v = \mathcal{G}(x, \rho, \partial_x v, v), \\ \rho(t, 0) = \rho(t, 2\pi) \quad \text{and} \quad v(t, 0) = v(t, 2\pi), \end{array} \right. \qquad (20)$$

where the function $\mathcal{G}$ is obtained by applying the continuation procedure to the second-order microscopic interaction laws.

**Example 5.** As an illustration, the OV-FTL model from Example 3 can be expressed in the macroscopic form given in Proposition III.2 with

$$\mathcal{G}(x, \rho, \partial_x \rho, v) := a \rho^2 + b \left( V\left(\frac{1}{\rho}\right) - v \right), \qquad (21)$$

where $a$ and $b$ are positive parameters characterizing driver behavior, where $a$ and $b$ are model parameters, $v$ denotes the vehicle velocity, and $V(\cdot)$ represents the optimal velocity function. The resulting formulation is closely related to a second-order Aw–Rascle–Zhang (ARZ) type model; see [10].

This continuum formulation provides a systematic framework for analyzing qualitative properties of traffic systems, such as stability, controllability, and robustness, which are typically difficult to address directly at the microscopic level. By describing the collective dynamics of many interacting vehicles through distributed state variables, the macroscopic
model enables the use of established analytical tools from partial differential equations and control theory. In particular, the study of traffic wave propagation, disturbance attenuation, and the design of stabilizing feedback laws becomes more tractable, while preserving a consistent connection with the underlying multi-agent dynamics.

## IV. APPLICATION TO THE CONTROL OF AUTONOMOUS VEHICLES

In this section, we demonstrate the relevance of the continuation method for control problems involving first- and second-order microscopic traffic models. In particular, by designing feedback laws that render the vehicles autonomous, we derive microscopic control strategies that enable the system to track arbitrary reference trajectories and prescribed spatial distributions.

### A. Systems modeled by first order equations

**1) Macroscopic control problem:** Consider the first-order model (1) with control inputs $u_i$. The dynamics of each vehicle are given by

$$ \dot{x}_i(t) = F(x_1(t), \dots, x_N(t)) + u_i(t), \quad i = 1, \dots, N, \qquad (22) $$

Applying a continuation of the control, that is, extending $u_i$ to a continuous control field $u(t, x)$ satisfying $u(t, x_i(t)) = u_i(t)$, yields the macroscopic representation, as in Section III:

$$ \left\{ \begin{array}{l} \partial_t \rho + \partial_x (\rho (\mathcal{F}(x, \rho) + u)) = 0, \\ \rho(t, 0) = \rho(t, 2\pi). \end{array} \right. \qquad (23) $$

We now fix a target density $\rho^r$, representing the desired collective configuration, and assume that it evolves according to

$$ \left\{ \begin{array}{l} \partial_t \rho^r + \partial_x (\rho^r \mathcal{H}(x, \rho^r)) = 0, \\ \rho^r(t, 0) = \rho^r(t, 2\pi), \end{array} \right. \qquad (24) $$

where $\mathcal{H}(x, \rho^r)$ denotes the velocity field associated with the target density. To measure the deviation between the actual and desired densities, we introduce the error variable $\rho^e := \rho - \rho^r$. Subtracting the two conservation laws yields the following evolution equation for the error:

$$ \left\{ \begin{array}{l} \partial_t \rho^e + \partial_x (\rho(\mathcal{F}(x, \rho) + u) - \rho^r \mathcal{H}(x, \rho^r)) = 0, \\ \rho^e(t, 0) = \rho^e(t, 2\pi). \end{array} \right. \qquad (25) $$

We propose the following feedback control:

$$ u(t, x) = \frac{1}{\rho(t, x)} \left( -\rho(t, x) \mathcal{F}(x, \rho(t, x)) + \rho^r(t, x) \mathcal{H}(x, \rho^r(t, x)) + \alpha \rho^e(t, x) + \beta \int_0^x \rho^e(t, s) ds \right). \qquad (26) $$

From (25) with feedback (26), the error dynamics reduce to

$$ \partial_t \rho^e + \alpha \partial_x \rho^e + \beta \rho^e = 0, \qquad (27) $$

with periodic boundary $\rho^e(t, 0) = \rho^e(t, 2\pi)$, whose exponential convergence to zero is guaranteed by Theorem A.1 whose statement and the proof are delayed in the Appendix. The first term in the control (26) compensates for the original microscopic dynamics $\mathcal{F}$, effectively rendering the vehicles fully autonomous under the feedback law. The second terms dissipate energy to guarantee exponential stability.

**2) Optimal velocity example:** We return to the optimal velocity (OV) model introduced in Example 2. The controlled OV model is given by the following microscopic dynamics:

$$ \dot{x}_i(t) = v_{\max} \left( 1 - \frac{s_{\text{stop}}}{x_{i+1}(t) - x_i(t)} \right) + u_i(t), \qquad (28) $$

for $i = 1, \dots, N$, where $u_i$ is a control applied to the vehicle $i$. Applying the continuation procedure described in Section III, the macroscopic density $\rho(t, x)$ satisfies, for $x \in [0, 2\pi]$,

$$ \partial_t \rho(t, x) + \partial_x(\rho(t, x)(\mathcal{F}(x, \rho(t, x)) + u(t, x))) = 0, \qquad (29) $$

with periodic boundary conditions, and $\mathcal{F}(x, \rho) = v_{\max}(1 - s_{\text{stop}}\rho)$.

We aim to steer the OV model (28) toward a spatially uniform density while ensuring that all vehicles travel at uniform speed 1, which corresponds to the optimal and smoothest traffic regime that can be expected. We assume that the road length is $2\pi$; therefore, the target uniform density is $N/(2\pi)$. In addition, we require that the vehicle speeds converge to 1. In macroscopic terms, this means that the reference density $\rho^r$ satisfies Equation (24) with $\mathcal{H} = 1$, since $\mathcal{H}$ denotes the velocity field in the LWR model and the initial data is $\rho_0^r = N/(2\pi)$.

**a) Microscopic feedback law:** At the macroscopic level, the feedback control $u$ is

$$ u(t, x) = -\mathcal{F}(x, \rho(t, x)) + \frac{1}{\rho(t, x)} \times \left[ \rho^r(t, x) + \alpha \rho^e(t, x) + \beta \int_0^x \rho^e(t, s) ds \right]. \qquad (30) $$

The microscopic control $u_i(t) = u(t, x_i(t))$ is obtained by evaluating the macroscopic feedback law at the vehicle position. Since the feedback (30) depends on the density $\rho(t, \cdot)$ and on the integral term $\int_0^x \rho^e(t, s) ds$, its computation involves the macroscopic density field over the road segment. Therefore, the control has a nonlocal structure and, at the modeling level, uses global information on the density distribution of all vehicles.

**b) Numerical implementation:** The feedback control is derived at the macroscopic level and depends explicitly on the density field $\rho(t, x)$ and on the nonlocal integral term $\int_0^x \rho^e(t, s) ds$. Directly discretizing this control at the microscopic level is not straightforward, since the discrete vehicle system does not provide a density field on a spatial grid, nor a natural representation of the nonlocal integral. Computing the control directly from inter-vehicle distances would lead to a highly nonlinear and nonlocal discrete feedback law that is difficult to stabilize numerically.
To overcome this difficulty, we adopt a hybrid micro–macro numerical strategy. The macroscopic conservation law (29) with the feedback control (30) is first discretized on a uniform spatial and temporal grid using a first-order Lax–Friedrichs finite difference scheme, allowing us to compute the density field $\rho(t, x)$ and the corresponding feedback control $u(t, x)$ over the entire time interval. The feedback control is then evaluated at the vehicle positions $x_i(t)$ to define the micro- scopic controls $u_i(t) = u(t, x_i(t))$. The microscopic vehicle dynamics are finally advanced in time using these interpolated controls. This approach ensures that the control is designed and stabilized at the PDE level, while consistently transferring it to the discrete vehicle system, combining the robustness of the Lax–Friedrichs scheme with an effective micro–macro coupling.

c) Illustration.: Figure 2 and Figure 3 illustrate the effect of the macroscopic feedback law applied to the microscopic OV system on a ring road. The simulation considers $N = 30$ vehicles circulating on a ring of length $L = 2\pi$, with feedback gains $\alpha = 1$ and $\beta = 10$, and a final time $T = 10$.

The top panel of Figure 2 shows the evolution of the inter-vehicle distances $x_{i+1} - x_i$ over time, highlighting how the feedback law gradually regularizes the spacing between vehicles, while the bottom panel depicts the evolution of the vehicle speeds $\dot{x}_i$, illustrating how the control steers the system toward a more uniform and stable velocity profile. Figure 3 presents the spatial distribution of the vehicles on the ring at times $t = 0$ and $t = 7$.

This representation emphasizes that the macroscopic feedback law effectively regulates the microscopic dynamics: inter-vehicle distances converge toward a uniform spacing, while vehicle velocities stabilize around the desired values, demonstrating that macroscopic control objectives can be successfully enforced at the level of individual vehicles.

<figure><img src="image_3.png" /><figcaption>Fig. 2: Top: evolution of inter-vehicle distances $x_{i+1} - x_i$. Bottom: evolution of vehicle speeds $\dot{x}_i$.</figcaption></figure>

## B. Second order model:

1) Macroscopic control problem: We consider an extension of the classical second-order optimal velocity follow-the-leader (OV–FTL) model (3) with multiple vehicles ahead, as

<figure><img src="image_4.png" /><figcaption>Fig. 3: Vehicle positions on the ring at different times.</figcaption></figure>

well as an individual control applied to each vehicle. In this framework, each vehicle adapts its acceleration by taking into account not only the immediately preceding vehicle but also a finite number of leaders $K > 0$. The dynamics are given by, for $i = 1, \dots, N$ and we add a control

$$ \ddot{x}_i(t) = \sum_{k=1}^{K} \left[ a_k \frac{\dot{x}_{i+k}(t) - \dot{x}_i(t)}{(x_{i+k}(t) - x_i(t))^2} + b_k (V(x_{i+k}(t) - x_i(t)) - \dot{x}_i(t)) \right] + u_i, \quad (31) $$

where $N \ge K \ge 1$ denotes the anticipation range, $a_k, b_k > 0$ are sensitivity parameters typically chosen as decreasing functions of $k$, and $V(\cdot)$ is an increasing and bounded optimal velocity function. This formulation naturally generalizes the standard OV–FTL model (recovered for $K = 1$) and provides a microscopic description with nonlocal interactions, which is particularly suitable for stability analysis and for the derivation of nonlocal macroscopic traffic models.

Under the continuum approximation described above, the relative distances and velocity differences are approximated as

$$ x_{i+k}(t) - x_i(t) \approx \frac{k}{\rho(t, x_i)}, \qquad \dot{x}_{i+k}(t) - \dot{x}_i(t) \approx k \partial_x v(t, x_i). $$

Substituting these expressions into the microscopic acceleration law yields the macroscopic closure

$$ \partial_t v = \sum_{k=1}^{K} \left[ a_k \frac{k \partial_x v}{\left(\frac{k}{\rho}\right)^2} + b_k \left(V\left(\frac{k}{\rho}\right) - v\right) \right] + u, $$

where $u$ is a continuation of the controls $u_i$. Equivalently, the function $G$ in Proposition III.2 is given by

$$ G(x, \rho, \partial_x v, v) = \sum_{k=1}^{K} \left[ a_k \frac{\rho^2}{k} \partial_x v + b_k \left( V\left(\frac{k}{\rho}\right) - v \right) \right] + u. \quad (32) $$

This macroscopic model generalizes the classical second-order OV–FTL continuum limit by introducing nonlocal density-dependent interactions that account for anticipation of multiple leading vehicles.
Let us define the target density and velocity profiles ($\rho^r, v^r$) as the solution of the second-order macroscopic traffic system

$$ \left\{ \begin{array}{l} \partial_t \rho^r + \partial_x (\rho^r v^r) = 0, \\ \partial_t v^r = \tilde{\mathcal{H}}(x, \rho^r, \partial_x v^r, v^r), \\ \rho^r(t, 0) = \rho^r(t, 2\pi) \quad \text{and} \quad v^r(t, 0) = v^r(t, 2\pi), \end{array} \right. \qquad (33) $$

where the function $\tilde{\mathcal{H}}(x, \rho^r, \partial_x v^r, v^r)$ represents the acceleration law defining the evolution of the target velocity field. The pair $(\rho^r, v^r)$ is assumed to be sufficiently smooth and to satisfy $\rho^r(t, x) > 0$ for all $(t, x) \in [0, T] \times [0, 2\pi]$.

Let $\alpha, \beta > 0$ be two design parameters. The distributed feedback control law is defined as

$$ u = -G(x, \rho, \partial_x v) + \frac{1}{\rho} \left( v \partial_x (\rho v) + \rho^r \tilde{\mathcal{H}}(x, \rho^r, \partial_x v^r) - v^r \partial_x (\rho^r v^r) + \alpha(\rho^r v^r - \rho v) + \beta \partial_x (\rho^r - \rho) \right). \qquad (34) $$

As in the first-order case, this feedback law renders all vehicles autonomous by relying solely on macroscopic state information. It was shown in [8] that, under suitable regularity and boundedness assumptions, the closed-loop system is stable when tracking a time-invariant target corresponding to a constant density and a uniform velocity profile. In the following, we illustrate this result by considering a steady-state target defined by a uniform vehicle density along the ring, leading to a spatially uniform target velocity.

2) Convergence from a random initial configuration toward uniform density and speed: As in the first-order model, we consider an initial configuration in which vehicles are randomly distributed along the ring, with random perturbations in their velocities. The objective is to steer the system toward the steady state corresponding to the uniform density $N/(2\pi)$, assuming a ring of length $2\pi$, for which the associated equilibrium speed is $v = 1$. Applying the feedback law (34) with $\tilde{\mathcal{H}} = 1$ and $\rho^r = N/(2\pi)$, the vehicles progressively adjust both their positions and velocities, converging toward a configuration with uniform spacing and constant speed equal to 1.

Numerically, the feedback law (34) for the second-order model does not contain an integral term. As a consequence, the associated microscopic feedback is obtained from (34) through a first-order approximation, in which each vehicle uses only local information, namely the position and velocity of the vehicle directly ahead. In contrast with the first-order case, the control law therefore does not require a full connectivity pattern over all vehicles.

Figure 4, for $N = 40$, shows the time evolution of the vehicle speeds and inter-vehicle distances. Starting from an approximately uniform initial configuration, the system converges toward the uniform equilibrium corresponding to the prescribed target speed $v^r = 1$, with final time $T = 3.5$, $\alpha = 10$ and $\beta = 1$.

<figure><img src="image_5.png" /><figcaption>Fig. 4: Microscopic evolution under the second-order feedback law. Speeds and inter-vehicle distances converge from an almost uniform initial configuration toward the uniform equilibrium.</figcaption></figure>

## V. CONCLUSION

We have presented a continuation method that systematically derives PDE representations for first- and second-order microscopic multi-agent systems modeling traffic dynamics on a ring, allowing each vehicle to interact with multiple neighbors rather than just its immediate leader. This framework enables microscopic control problems to be reformulated at the macroscopic level, facilitating the design of feedback laws that render the vehicles effectively autonomous while ensuring consistent behavior at the microscopic scale. Illustrated on the first-order Optimal Velocity model and the second-order Optimal Velocity Follow-the-Leader model, our approach successfully steers the system toward a prescribed density profile in the first-order case, and toward prescribed density and speed profiles in the second-order case. Future work will explore extensions to higher-order vehicle dynamics, more complex interaction networks, and advanced control objectives.

## REFERENCES

[1] F. Ancona, M. Bentaibi, and F. Rossi. On the continuum limit of the follow-the-leader model and its stability. *Discrete and Continuous Dynamical Systems*, 2024.

[2] A. Aw and M. Rascle. Resurrection of "second order" models of traffic flow. *SIAM journal on applied mathematics*, 60(3):916–938, 2000.

[3] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama. Dynamical model of traffic congestion and numerical simulation. *Phys. Rev. E*, 51:1035–1042, Feb 1995.

[4] L. Baratchart, S. Fueyo, G. Lebeau, and J.-B. Pomet. Sufficient stability conditions for time-varying networks of telegrapher’s equations or difference-delay equations. *SIAM J. Math. Anal.*, 53(2):1831–1856, 2021.

[5] M. Bodnar and J. J. L. Velazquez. Derivation of macroscopic equations for individual cell-based models: a formal approach. *Mathematical Methods in the Applied Sciences*, 28(15):1757–1779, 2005.

[6] S. Cui, B. Seibold, R. Stern, and D. B. Work. Stabilizing traffic flow via a single autonomous vehicle: Possibilities and limitations. In *2017 IEEE Intelligent Vehicles Symposium (IV)*, pages 1336–1341, 2017.
[7] M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales, and B. Seibold. Self-sustained nonlinear waves in traffic flow. *Physical Review E—Statistical, Nonlinear, and Soft Matter Physics*, 79(5):056113, 2009.

[8] S. Fueyo and C. Canudas-de Wit. A Continuation-Based Control Strategy for Stabilizing Second-Order Macroscopic Traffic Flow on Circular Roads. *CDC (Conference on Decision and Control)*, 2025. To appear.

[9] S. Fueyo and C. Canudas de Wit. From microscopic driver models to macroscopic pdes in ring road traffic dynamics. *European Journal of Control*, 86:101364, 2025. Special Issue on the European Control Conference 2025.

[10] M. Garavello, K. Han, B. Piccoli, et al. Models for vehicular traffic on networks, volume 9. American Institute of Mathematical Sciences (AIMS), Springfield, MO, 2016.

[11] P. Goatin. Dissipation of stop-and-go waves in traffic flows using controlled vehicles: A macroscopic approach. *IEEE Control Systems Letters*, 8:628–633, 2024.

[12] J. K. Hale and S. M. Verduyn Lunel. Introduction to Functional Differential Equations, volume 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.

[13] H. Holden and N. H. Risebro. Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. *Networks and Heterogeneous Media*, 13(3):409–421, 2018.

[14] A. J. Leverentz, C. M. Topaz, and A. J. Bernoff. Asymptotic dynamics of attractive-repulsive swarms. *SIAM Journal on Applied Dynamical Systems*, 8(3):880–908, 2009.

[15] M. J. Lighthill and G. B. Whitham. On kinematic waves II. a theory of traffic flow on long crowded roads. *Proceedings of the royal society of London. series a. mathematical and physical sciences*, 229(1178):317–345, 1955.

[16] G. C. Maffettone, A. Boldini, M. Di Bernardo, and M. Porfiri. Continuation control of large-scale multiagent systems in a ring. *IEEE Control Systems Letters*, 7:841–846, 2023.

[17] T. G. Molnár, D. Upadhyay, M. Hopka, M. Van Nieuwstadt, and G. Orosz. Lagrangian models for controlling large-scale heterogeneous traffic. In *2019 IEEE 58th Conference on Decision and Control (CDC)*, pages 3152–3157, 2019.

[18] G. Newell. A simplified theory of kinematic waves in highway traffic, part I: General theory. *Transportation Research Part B: Methodological*, 27(4):281–287, 1993.

[19] D. Nikitin. Scalable Large-Scale Control of Network Aggregates. PhD dissertation, Université Grenoble Alpes, Sep 2021.

[20] D. Nikitin, C. Canudas-de Wit, and P. Frasca. A continuation method for large-scale modeling and control: From ODEs to PDE, a round trip. *IEEE Transactions on Automatic Control*, 67(10):5118–5133, 2022.

[21] R. E. Stern, S. Cui, M. L. Delle Monache, R. Bhadani, M. Bunting, M. Churchill, N. Hamilton, R. Haulcy, H. Pohlmann, F. Wu, B. Piccoli, B. Seibold, J. Sprinkle, and D. B. Work. Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments. *Transportation Research Part C: Emerging Technologies*, 89:205–221, 2018.

[22] S.-i. Tadaki, M. Kikuchi, M. Fukui, A. Nakayama, K. Nishinari, A. Shibata, Y. Sugiyama, T. Yosida, and S. Yukawa. Phase transition in traffic jam experiment on a circuit. *New Journal of Physics*, 15(10):103034, 2013.

[23] A. Tordeux, G. Costeseque, M. Herty, and A. Seyfried. From traffic and pedestrian follow-the-leader models with reaction time to first order convection-diffusion flow models. *SIAM Journal on Applied Mathematics*, 78(1):63–79, 2018.

[24] H. Yu and M. Krstic. Varying speed limit control of Aw-Rascle-Zhang traffic model. In *2018 21st International Conference on Intelligent Transportation Systems (ITSC)*, pages 1846–1851, 2018.

*Proof.* To remove the zeroth-order term, we introduce the change of variables

$$ \rho^e(t, x) = e^{-\frac{\beta}{\alpha}x} z(t, x). \qquad (35) $$

Substituting this into (27) yields the pure transport equation

$$ \partial_t z + \alpha \partial_x z = 0, \qquad (36) $$

with modified periodic condition

$$ z(t, 0) = \gamma z(t, 2\pi), \quad \gamma = e^{-\frac{\beta}{\alpha} \cdot 2\pi}. $$

Using the method of characteristics, $z$ is transported along straight lines with speed $\alpha$, so that after one spatial period, a particle initially at $x = 2\pi$ reaches $x = 0$ after a time $\tau = 2\pi/\alpha$. Consequently, the boundary condition can be written in equivalent delay form

$$ z(t, 0) = \gamma z(t - \tau, 0). \qquad (37) $$

This is a scalar delay equation, and its continuous solutions are in bijection with continuous solutions of Equation (27), see for instance [4]. The characteristic equation of (37) is

$$ e^{\lambda\tau} = \gamma, $$

and the equilibrium is exponentially asymptotically stable in $L^\infty([0, 2\pi])$ if $|\gamma| < 1$ as shown in [12], which holds here because $\alpha, \beta > 0$. $\square$

# APPENDIX

**Theorem A.1.** The equilibrium $\rho^e \equiv 0$ of the linear transport equation (27) with periodic boundary condition $\rho^e(t, 0) = \rho^e(t, 2\pi)$, is exponentially asymptotically stable in $L^\infty([0, 2\pi])$.