# Continuum Control Design for Traffic Flow Stabilization on Circular Roads: A Microscopic-Macroscopic Approach*

Sebastien Fueyo<sup>a</sup>, Carlos Canudas de Wit<sup>a</sup>

<sup>a</sup>Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, F-38000 Grenoble, France

## Abstract

This paper addresses the stabilization of traffic flow using a control strategy for autonomous vehicles. While previous approaches have stabilized traffic near equilibrium states, the proposed method extends this to time-varying density and velocity profiles. The strategy modulates vehicle accelerations based on a continuation-based framework, which is applied to macroscopic traffic models on ring-roads. A key feature of the method is its ability to adapt to various intervehicle connectivity structures, which depend on the communication capabilities between vehicles, allowing it to handle complex traffic scenarios with varying connectivity patterns. The main contributions include (i) deriving a macroscopic control model from a second-order microscopic framework, and (ii) proposing a general procedure to synthesize control inputs that guide traffic toward time-varying target profiles, offering a flexible approach for real-world traffic conditions.

**Key words:** Traffic flow stabilization; Autonomous vehicles; Time-varying profiles; Continuation-based control; Macroscopic traffic modeling.

## 1 Introduction

Analyzing road traffic systems presents a significant challenge due to the complex interactions among a large number of vehicles. To address this complexity, microscopic models are often employed, typically formulated as systems of first- or second-order ordinary differential equations (ODEs). These models capture the behavior of individual vehicles and their interactions, with notable examples including the Optimal Velocity-Follow the Leader (OV-FTL) model [2] and models incorporating driver delays [3].

However, in practice, continuous approaches are frequently used to study traffic flow on a larger scale. These methods replace discrete vehicle-based models with hyperbolic partial differential equations (PDEs), providing a macroscopic perspective on traffic behavior. Among the prominent macroscopic models are the Lighthill-Whitham-Richards (LWR) model [13] and the Aw-Rascle-Zhang (ARZ) model [1], both of which have been extensively studied in the literature [8, 23].

Emergent patterns in traffic, often driven by collective driver behavior, give rise to fascinating dynamics, such as phantom traffic jams and stop-and-go oscillations. These oscillations can destabilize traffic flow, as first experimentally observed by [21]. A major focus of contemporary research is improving traffic flow by influencing driver behavior through strategies like speed limit modulation, ramp metering, and direct driver interventions [5, 12, 20].

Ring-road configurations have been widely studied as fundamental models for traffic flow and stabilization strategies. These systems, commonly used as base models, naturally exhibit stop-and-go waves, a major cause of congestion, making them particularly suited for testing control strategies aimed at maintaining steady and uniform traffic flow [4, 11, 19, 22]. Compared to open roads, ring-road models differ fundamentally due to their closed geometry, which imposes periodic boundary conditions: the upstream and downstream boundaries coincide, leading to a global coupling effect. This configuration can amplify oscillations and thus poses greater challenges for stabilization. Yet, precisely because of this structural simplicity and the persistence of traffic patterns it induces, ring roads serve as a useful benchmark for analyzing and validating control strategies before extending them to more complex or open-road scenarios.

<footnote>* This paper was not presented at any IFAC meeting. Corresponding author S. Fueyo.</footnote>

<footnote>Email addresses: sebastien.fueyo@gipsa-lab.fr
(Sebastien Fueyo), carlos.canudas-de-wit@gipsa-lab.fr
(Carlos Canudas de Wit).</footnote>
In this context, recent studies have demonstrated that traffic flow can be stabilized through the control of a single autonomous vehicle. In particular, [4, 10] analyzed the local exponential stability of the microscopic OV-FTL model by introducing a feedback law acting on one autonomous car. At the macroscopic level, [9] proposed an open-loop control strategy for the ARZ model, where the velocity of an autonomous car is externally prescribed over a sufficiently long time interval. However, such approaches are typically effective only when the traffic conditions are already near a desired equilibrium. Achieving stabilization from arbitrary initial conditions in a closed-loop framework generally requires the coordination of multiple controlled vehicles.

This paper proposes a control strategy that modulates the accelerations, i.e. a second-order model, of autonomous vehicles to drive the system toward prescribed time-varying density and velocity profiles, which are not necessarily constant equilibrium states of the system. The control strategy builds on the continuation-based method pioneered in [17, 18], where microscopic systems are modeled as PDEs to facilitate the design of control strategies, which are then translated back into ODE-based control laws. This approach has been effectively applied to first-order models on ring roads [14, 15] and second-order models on linear roads [17].

A key advantage of the continuation-based method is its flexibility in accommodating different intervehicle connectivity structures, which can vary depending on the communication capabilities between vehicles. This adaptability allows the method to handle a wide range of traffic conditions, from scenarios with dense vehicle interaction to more sparsely connected systems. Building on these ideas, the present work extends the method to second-order traffic models on circular roads, where the closed-loop system introduces additional challenges due to its non-linear and periodic nature.

This work begins by extending the control methodology introduced in [7], providing a detailed derivation of the macroscopic control model obtained from a second-order microscopic control framework via a continuation method. Although this result had previously been stated without rigorous justification, a partial proof was given in [6]; this paper closes that gap. Building on this foundation, we introduce a more general control law capable of tracking arbitrary density and velocity profiles —extending beyond earlier approaches limited to equilibrium configurations; see [7].

The main contributions of this paper are as follows. First, we develop a macroscopic control strategy for traffic systems on a ring road, based on a continuation framework that enables the construction of control inputs capable of steering the system toward time-varying target profiles. This approach explicitly accounts for the propagation of perturbations inherent to the closed-loop

topology of the ring, allowing for fine control over the spatial structure of traffic patterns. The proposed control law turns each vehicle into an autonomous agent, using damping mechanisms to ensure convergence toward the desired profile.

Second, we introduce a general procedure based on continuation methods for designing acceleration-based control laws from prescribed macroscopic profiles. This strategy systematically derives microscopic inputs that drive the system toward the desired traffic state. Unlike earlier continuation-based approaches limited to equilibrium settings, our method accommodates time-dependent and spatially varying targets, and adapts to various intervehicle connectivity structures. Its effectiveness is demonstrated through numerical examples.

## 2 Microscopic control design problem

Let $N \in \mathbb{N}$ and consider the variables $\{x_i(t)\}_{i=1}^N$, representing the positions of $N$ controlled vehicles moving along a circular road. Each position $x_i(t)$ is defined modulo $2\pi$, and the vehicle dynamics are given by:

$$ \ddot{x}_i(t) = f(x_{i+1}(t) - x_i(t), \dot{x}_{i+1}(t) - \dot{x}_i(t), \dot{x}_i(t)) + u_i(t), \quad (1) $$

where $t \ge 0$, $x_{i+1}(t) \ge x_i(t)$, and the periodicity condition $x_{N+1}(t) = x_1(t)$ holds. The function $f: \mathbb{R}^3 \to \mathbb{R}$ captures the driver behavior of each vehicle, while $u_i(t) \in \mathbb{R}$ denotes an external control applied to the acceleration of vehicle $i$.

<figure><img src="image_1.png" /><figcaption>Fig. 1. Schematic representation of the controlled vehicle system on a circular road with $N = 8$ vehicles. Each vehicle $x_i(t)$ is influenced by the relative position and velocity of its successor $x_{i+1}(t)$, and is subject to a control input $u_i(t)$. The arrows illustrate the interaction structure and control inputs as described in Equation (1).</figcaption></figure>

Equation (1) models the acceleration of vehicle $i$.
as a function of the distance to the vehicle ahead $x_{i+1}(t) - x_i(t)$, the relative velocity $\dot{x}_{i+1}(t) - \dot{x}_i(t)$, and its own speed $\dot{x}_i(t)$. This framework captures typical car-following dynamics on a ring road, with control inputs introduced to influence or stabilize the overall behavior of the system.

A representative instance of the function $f$ takes the form

$$f(x, y, z) = a(V(x) - z) + b \frac{y}{x^2}, \quad (2)$$

which leads to a dynamical model consistent with the Optimal Velocity-Follow-the-Leader (OV-FTL) framework. For the original model, refer to [2], while [4] modified it by adding a term to prevent vehicle collisions. The model presented in equation (2) incorporates this collision-avoidance term. The constants $a$ and $b$, both strictly positive, quantify the drivers' behavioral traits, distinguishing between more cautious and more reactive responses.

We now present how, by applying the continuation method developed in [18], we can transform the microscopic controlled problem defined in Equation (1) into a macroscopic controlled problem, which is described by a system of two scalar PDEs. While this connection was mentioned in [7], it was not formally proven, although the result was implicitly suggested in [6]. The following sections provide the detailed mathematical derivation of this transformation and it is the first contribution of the paper.

## 3 Macroscopic Traffic Control via Continuation

The continuation method for transforming the microscopic controlled problem into a macroscopic one is based on the following idea. To account for the geometry of the ring road, we adopt a convenient convention for indexing the vehicles: the indices are understood modulo $N$, so that $N+1$ is identified with 1 and $N$ with 0. By definition, the index variable $i$ is discrete.

The core idea of the continuation method is to introduce a *virtual continuum* of vehicles between the discrete indices $i$ and $i+1$, thereby obtaining a continuously varying index. We denote this continuously varying index by $M \in [0, N]$.

This continuation of the indices allows us to reinterpret the discrete set of vehicle positions $\{x_i\}_{i=1,...,N}$ as a continuum of vehicles, described by a continuous function

$$x(t, M): \mathbb{R}_+ \times [0, N] \to [0, 2\pi],$$

where $x(t, M)$ represents the position of the vehicle labeled by the continuous index $M$ at time $t$. In particular,

at integer values of $M$, this function coincides with the discrete vehicle positions:

$$x(t, M) = x_i(t) \quad \text{for } M = i \in \{1, \dots, N\}.$$

Moreover, this continuously varying index $M$ can be interpreted as the *Moskowitz function*. This reinterpretation enables us to derive a macroscopic PDE model that describes the system in terms of the vehicle density $\rho$ and speed $v$ of this continuum. We now proceed with the mathematical details.

### 3.1 Moskowitz function and conservation law property

Following [16], the classical Moskowitz function $M(t, x)$, defined for $(t, x) \in \mathbb{R}_+ \times [0, 2\pi]$, represents the cumulative number of vehicles that have passed location $x$ by time $t$, starting from the reference vehicle position $x(t, 0)$. For computational convenience, the convention adopted in this paper defines the Moskowitz function as the number of vehicles located upstream of position $x$ at time $t$, counted from the reference vehicle position $x(t, 0)$. Both conventions are mathematically equivalent, differing only by a change of sign.

The function $M: \mathbb{R}_+ \times [0, 2\pi] \to \mathbb{R}_+$ is assumed to be continuous and smooth, reflecting the regular motion of vehicles. By definition, if $x_i(t)$ denotes the position of vehicle $i$ at time $t$, then the Moskowitz function satisfies $M(t, x_i(t)) = i$. In other words, the Moskowitz function corresponds to the continuous index variable introduced earlier. Since the road is circular, the configuration is periodic in space, and we have the identity $M(t, 0) = M(t, 2\pi)$ for all $t \ge 0$.

The *density* $\rho$ and the *flow rate* $\phi$ can be directly obtained from the *Moskowitz function* $M(t, x)$, which, in this work, is defined as the count of vehicles positioned *upstream* of location $x$ at time $t$, relative to the reference vehicle at $x(t, 0)$. These two quantities, defined as functions $\rho, \phi: \mathbb{R}_+ \times [0, 2\pi] \to \mathbb{R}_+$, are obtained through differentiation:

$$\rho := \partial_x M, \quad \phi := -\partial_t M. \qquad (3)$$

The interpretation is straightforward: taking the *spatial derivative* of $M$ at fixed time reveals how the cumulative vehicle count changes along the road—this gives the number of vehicles per unit length, which corresponds to the *density* $\rho$. On the other hand, the *time derivative* at fixed position describes how quickly the cumulative count increases, and thus directly gives the *flow rate* $\phi$. The minus sign in the flow rate expression accounts for the fact that $M$ decreases in the direction of increasing position when vehicles move forward, ensuring that the flow rate is positive when vehicles pass position $x$.
Since $M$ is assumed to be smooth, we can interchange the order of differentiation:

$$ \partial_t(\partial_x M) = \partial_x(\partial_t M). \quad (4) $$

Combining this identity with the definitions of $\rho$ and $\phi$ in (3), we obtain the conservation of vehicles:

$$ \partial_t \rho + \partial_x \phi = 0. \quad (5) $$

This equation expresses the local conservation of mass: the change in density at a given point results from the net imbalance of incoming and outgoing flow.

We denote by $v(t, x)$ the velocity field, which describes the vehicle speed at time $t$ and position $x$. It is a function from $\mathbb{R}_+ \times [0, 2\pi]$ into $\mathbb{R}_+$. Since the vehicle flux is given by the product of the density and velocity, we have

$$ v(t, x) := \frac{\rho(t, x)}{\phi(t, x)}. \quad (6) $$

Substituting this expression into the conservation law (5), we obtain:

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

This formulation highlights the transport nature of the density evolution: changes in density arise from spatial variations in the flux, itself modulated by both speed and density. This subsection ends by introducing important properties of the Moskowitz function, which will be crucial for understanding the relationship between vehicle density, flow, and velocity in the model. These properties will play a key role in deriving subsequent equations and control mechanisms discussed in the next sections.

By definition, the functions $x(t, \cdot)$ and $M(t, \cdot)$ are inverse to each other, in the sense that, for all $z \in [0, 2\pi]$ and $z_0 \in [0, N]$, we have

$$ x(t, M(t, z)) = z \quad \text{and} \quad M(t, x(t, z_0)) = z_0. $$

Differentiating the first identity with respect to $z$, and the second with respect to $t$, we obtain:

$$ \partial_M x \cdot \partial_x M = 1 \quad \text{and} \quad \partial_t M + \partial_t x \cdot \partial_x M = 0. \quad (8) $$

The first relation confirms the inverse function rule, while the second describes how the trajectory $x(t, z)$ evolves in time through its dependence on the Moskowitz function.

From (6), (3) and the second equation in (8), we recover the fact that the velocity $v$ is the partial derivative of $x$ with respect to time, i.e., $v = \partial_t x$. Moreover, from the first identity in (8), we obtain:

$$ \partial_M x = \frac{1}{\rho}, \quad (9) $$

which expresses the inverse relationship between the vehicle position $x$ and the Moskowitz function $M$. Geometrically, this implies that the spatial density $\rho$ can be interpreted as the inverse of the rate at which positions $x$ change with respect to $M$ along the ring road. In other words, the higher the density, the smaller the spatial distance between successive vehicles.

Finally, differentiating Equation (9) with respect to time, exchanging the order of partial derivatives, and substituting the definition of $v$, we obtain the following expression:

$$ \partial_M(\partial_t x) = \partial_x v. \quad (10) $$

This equation reveals a connection between the time evolution of the position $x$ and the spatial gradient of the velocity field $v$. Geometrically, this suggests that the rate of change of vehicle positions with respect to $M$ is influenced by the gradient of the velocity along the road, reflecting the way that variations in the speed of vehicles can affect their spatial distribution over time.

In summary, these relations highlight the interplay between vehicle positions, their speed, and their spatial distribution, providing a framework for understanding the flow dynamics on a ring road through the Moskowitz function. In particular, transforming Equation (1) into a PDE proves useful.

## 3.2 Taylor expansion

Up to this point, we have interpreted vehicle conservation on the ring road through the lens of the Moskowitz function. The remaining challenge is to provide a macroscopic interpretation of the dynamical term (1). To that end, we recall that the position variable $x_i(t)$ can be viewed as a function $x(t, i)$, and similarly, the control input $u_i(t)$ is interpreted as $u(t, i) = u_i(t)$, where $u: \mathbb{R}_+ \times [0, 2\pi] \to \mathbb{R}$ is a continuous extension of the discrete control inputs.

We begin with the following expression:

$$ \begin{aligned} x_{i+1} - x_i &= \frac{x(t, i+1) - x(t, i)}{(i+1) - i} \\ &= \frac{x(t, M(t, x_{i+1}(t))) - x(t, M(t, x_i(t)))}{M(t, x_{i+1}(t)) - M(t, x_i(t))}. \end{aligned} \quad (11) $$

In the second line of Equation (11), we reparametrized the positions using the Moskowitz function $M$. Then, applying a first order Taylor expansion, we approximate the finite difference $x_{i+1} - x_i$ by the partial derivative of $x$ with respect to $M$, yielding:

$$ x_{i+1}(t) - x_i(t) \approx \partial_M x(t, i). \quad (12) $$
Taking the time derivative of Equation (12) gives:

$$ \dot{x}_{i+1}(t) - \dot{x}_i(t) \approx \partial_t (\partial_M x)(t, i). \quad (13) $$

Under appropriate regularity assumptions, this time derivative can be related to spatial variations in the velocity field. Hence, the discrete variables can be approximated by their macroscopic counterparts as follows:

$$ \begin{aligned} u_i &\approx u, \quad \dot{x}_i \approx v, \\ x_{i+1} - x_i &\approx \partial_M x, \quad \dot{x}_{i+1} - \dot{x}_i \approx \partial_t (\partial_M x). \end{aligned} \quad (14) $$

Using the macroscopic relations established in (9) and (10), i.e.,

$$ \partial_M x \approx \frac{1}{\rho}, \quad \partial_t (\partial_M x) \approx \partial_x v, $$

we obtain the final approximations from (14):

$$ \begin{aligned} u_i &\approx u, \quad \dot{x}_i \approx v, \\ x_{i+1} - x_i &\approx \frac{1}{\rho}, \quad \dot{x}_{i+1} - \dot{x}_i \approx \partial_x v. \end{aligned} \quad (15) $$

Therefore, Equation (1) can be interpreted in the macroscopic limit as the following PDE:

$$ \partial_t v = f \left( \frac{1}{\rho}, \partial_x v, v \right) + u. \quad (16) $$

We now summarize the results obtained in the previous two sections.

### 3.3 Macroscopic controlled problem

We now consider the macroscopic formulation of the control problem, obtained as the limit of the microscopic model (1) when the number of vehicles tends to infinity. This limit yields a system of partial differential equations describing the evolution of traffic density and velocity along the ring road. The control input, originally acting at the microscopic level, appears in the macroscopic dynamics as a distributed source term in the velocity equation. The following lemma summarizes the resulting system and its boundary conditions.

**Lemma 1** In the macroscopic limit, the microscopic model (1) can be interpreted as the following system of one-dimensional partial differential equations defined on $\mathbb{R}_+ \times (0, 2\pi)$:

$$ \left\{ \begin{aligned} \partial_t \rho + \partial_x (v\rho) &= 0, \\ \partial_t v &= f \left( \frac{1}{\rho}, \partial_x v, v \right) + u, \end{aligned} \right. \quad (17a) \quad (17b) $$

subject to the periodic boundary conditions

$$ \rho(t, 0) = \rho(t, 2\pi), \quad v(t, 0) = v(t, 2\pi), \quad \text{for all } t \ge 0, $$

and the initial conditions

$$ \rho(0, x) = \rho_0(x), \quad v(0, x) = v_0(x), \quad \forall x \in [0, 2\pi], $$

for some scalar functions $\rho_0$ and $v_0$ defined on $[0, 2\pi]$.

**PROOF.** The macroscopic equations (17a)-(17b) follow directly by combining Equations (7) and (16). The periodic boundary conditions arise naturally from the topology of the system, since the vehicles are assumed to move on a ring. This implies that $\rho(t, 0) = \rho(t, 2\pi)$ and $v(t, 0) = v(t, 2\pi)$, for all $t \ge 0$.

We now exploit the link between the microscopic and macroscopic models, established through the continuation method, to design a control strategy that ensures tracking of prescribed density and velocity profiles.

## 4 Macroscopic control design

This section presents the problem formulation and the proposed control design approach. Subsection 4.1 presents the problem, focusing on the challenge of tracking a prescribed density and velocity profile in the traffic system. Subsection 4.2 then introduces the control design strategy developed to tackle these challenges, based on the models defined earlier in (17a)-(17b).

### 4.1 Formulation of the problem

We consider a macroscopic model of traffic flow on a ring road and aim to design a feedback control strategy that ensures asymptotic tracking of a desired evolution. Specifically, the target behavior is prescribed in terms of reference velocity and density profiles governed by a given dynamic model.

**Problem 1** Let $\bar{v}(t, x)$ and $\bar{\rho}(t, x)$ denote reference velocity and density profiles, respectively, satisfying the system

$$ \left\{ \begin{aligned} \partial_t \bar{\rho} + \partial_x (\bar{v} \bar{\rho}) &= 0, \\ \partial_t \bar{v} &= F \left( \frac{1}{\bar{\rho}}, \partial_x \bar{v}, \bar{v} \right), \end{aligned} \right. \quad (18a) \quad (18b) $$

where $F: \mathbb{R}^3 \to \mathbb{R}$ is a smooth function. The objective is to design a control input $u$ such that the solution $(\rho, v)$ of the controlled system (17a)-(17b) asymptotically tracks the reference trajectory $(\bar{\rho}, \bar{v})$.

To ensure physical relevance and well-posedness, we impose the following assumptions. First, mass conservation on the ring road requires

$$ \int_0^{2\pi} \rho_0(x) \, dx = \int_0^{2\pi} \bar{\rho}(0, x) \, dx. \quad (19) $$
Second, we assume that the reference profiles $\bar{\rho}$ and $\bar{v}$ are sufficiently regular and uniformly bounded: they belong to $C^3([0, \infty) \times [0, 2\pi])$, and there exist constants $\gamma_1, \gamma_2 > 0$ such that

$$ \bar{\rho}(t, x), \bar{v}(t, x) \in [\gamma_1, \gamma_2] \quad \text{for all } t \ge 0, x \in [0, 2\pi]. \qquad (20) $$

A natural subclass of reference trajectories is given by equilibrium (stationary) profiles. In that case, Problem 1 reduces to the following:

**Problem 2** Let $\bar{v}(x)$ and $\bar{\rho}(x)$ be time-independent profiles satisfying

$$ \left\{ \begin{array}{l} \partial_x(\bar{v}\bar{\rho}) = 0, \qquad (21a) \\ F\left(\frac{1}{\bar{\rho}}, \partial_x\bar{v}, \bar{v}\right) = 0. \qquad (21b) \end{array} \right. $$

The goal is to design a control input $u$ that steers the solution $(\rho, v)$ toward the steady-state profile $(\bar{\rho}, \bar{v})$.

A particularly relevant situation is that of uniform equilibria, where $\bar{\rho}(x) \equiv \bar{\rho}$ and $\bar{v}(x) \equiv \bar{v}$ are constant. The stabilization of such configurations was addressed in [6], where a solution to Problem 2 was proposed.

In the present work, we extend this approach to the more general setting of time-varying reference profiles $(\bar{\rho}(t, x), \bar{v}(t, x))$, thereby enabling the design of feedback laws that stabilize the system around dynamic, possibly non-stationary trajectories. This broader formulation accommodates a wider range of control objectives and provides a solution to the full Problem 1.

## 4.2 Control design

The proposed control strategy relies on a feedback law aimed at eliminating driver behavior, ensuring that all vehicles operate autonomously while incorporating damping. This is crucial because wave equations naturally allow information to propagate in both directions, and without sufficient damping, this can result in the onset of stop-and-go instabilities.

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

$$ u = -f + \frac{1}{\rho} \left( v \partial_x(\rho v) + \bar{\rho} \bar{F} - \bar{v} \partial_x(\bar{v} \bar{\rho}) + \alpha(\bar{\rho} \bar{v} - \rho v) + \beta \partial_x(\bar{\rho} - \rho) \right), \qquad (22) $$

where $f = f(\frac{1}{\rho}, \partial_x v, v)$ and $\bar{F} = F(\frac{1}{\bar{\rho}}, \partial_x \bar{v}, \bar{v})$ for notational convenience. We assume that, under the feedback law (22), the solution pair $(\rho, v)$ to the system (17a)-(17b) is at least three times continuously differentiable.

### 4.2.1 Error equations

Let $\tilde{\rho} := \rho - \bar{\rho}$ and $\tilde{v} := v - \bar{v}$ represent the error terms, i.e., the deviations of $(\rho, v)$—the solution of (17a)-(17b) with periodic boundary conditions under the feedback law defined in (22)—from the target state $(\bar{\rho}, \bar{v})$. The objective is to establish that both error terms decay to zero. We begin by deriving the error equations for the density and speed.

**Lemma 2** With the feedback law defined in (22), the open-loop system (17a)-(17b) leads to the following equations for the density and speed error dynamic:

$$ \left\{ \begin{aligned} \partial_{tt}\tilde{\rho} &= -\alpha\partial_t\tilde{\rho} + \beta\partial_{xx}\tilde{\rho}, & (23a) \\ \partial_t\tilde{v} &= \left(-\alpha - \frac{\partial_t\rho}{\rho}\right)\tilde{v} + h, & (23b) \end{aligned} \right. $$

with

$$ h = \frac{1}{\rho} \left( -\bar{v}\partial_t\tilde{\rho} - \tilde{\rho}\bar{F} - \alpha\bar{v}\tilde{\rho} - \beta\partial_x\tilde{\rho} \right), \qquad (24) $$

and periodic boundary conditions:

$$ \left\{ \begin{array}{ll} \tilde{\rho}(t, 0) = \tilde{\rho}(t, 2\pi), & \forall t \ge 0, \qquad (25a) \\ \tilde{v}(t, 0) = \tilde{v}(t, 2\pi), & \forall t \ge 0. \qquad (25b) \end{array} \right. $$

## PROOF.

We begin by subtracting the equations of the actual system (17a) from those of the target system (18a) for the densities, yielding the following expression:

$$ \partial_t \rho - \partial_t \bar{\rho} + \partial_x(\rho v) - \partial_x(\bar{\rho} \bar{v}) = 0. \qquad (26) $$

Next, we take the time derivative of both sides of (26), and use the equations from (17a)-(17b)-(18a)-(18b) to obtain:

$$ \partial_{tt}\tilde{\rho} + \partial_x(\rho(f+u) - v\partial_x(\rho v) - \bar{\rho}\bar{F} + \bar{v}\partial_x(\bar{\rho}\bar{v})) = 0. \qquad (27) $$

Substituting the feedback law from (22) into (27) yields the error equation (23a). Finally, the periodic boundary condition for the density follows directly from the definitions of $\rho$ and $\bar{\rho}$.

Subtracting the actual system equations (17b) from the target ones (18b) for the vehicle speeds leads to the following relation:

$$ \begin{aligned} \partial_t \tilde{v} &= -\bar{F} + \frac{1}{\rho} \left( v \partial_x(\rho v) + \bar{\rho} \bar{F} - \bar{v} \partial_x(\bar{\rho} \bar{v}) \right. \\ & \qquad \left. + \alpha(\bar{\rho} \bar{v} - \rho v) + \beta \partial_x(\bar{\rho} - \rho) \right) \\ &= \frac{1}{\rho} \left( -v \partial_t \rho + \bar{v} \partial_t \bar{\rho} - (\bar{\rho} - \rho) \bar{F} \right. \\ & \qquad \left. + \alpha(\bar{\rho} \bar{v} - \rho v) - \beta \partial_x \tilde{\rho} \right). \end{aligned} \qquad (28) $$
Thus

$$ \partial_t \tilde{v} = \frac{1}{\rho} \left( -\tilde{v} \partial_t \rho - \bar{v} \partial_t \tilde{\rho} - \alpha \bar{v} \tilde{\rho} - \alpha \rho \tilde{v} - \tilde{\rho} \bar{F} - \beta \partial_x \tilde{\rho} \right). \quad (29) $$

This leads to the conclusion of the lemma, along with the spatial periodicity of $v$ and $\bar{v}$.

To clarify the structure of the argument, we present a schematic representation of the coupled error dynamics. As shown in Fig. 2, the evolution of the density error $\tilde{\rho}$ is governed by an autonomous damped wave equation. In contrast, the velocity error $\tilde{v}$ depends on both $\tilde{\rho}$ and its spatial and temporal derivatives. This hierarchical structure suggests a two-step approach: we first analyze the convergence of the density error, and then deduce the convergence of the velocity error.

<figure><img src="image_2.png" /><figcaption>Fig. 2. Hierarchical structure of the error dynamics. The density error evolves autonomously, while the velocity error depends on the full spatio-temporal behavior of the density error.</figcaption></figure>

We now show that the error equations for the density and speed exhibit exponential convergence to zero.

### 4.2.2 Convergence toward the desired density $\bar{\rho}$

Since the problem involves an infinite-dimensional system, it is essential to define the notion of convergence expected for the PDE system. For a function $g: [0, 2\pi] \to \mathbb{R}$, we define the infinite norm as:

$$ \|g\|_{\infty} = \sup_{x \in [0, 2\pi]} |g(x)|. \quad (30) $$

We will demonstrate that the damping equation converges exponentially to zero. Although this result has been established in [6], we present it again here for the sake of completeness and to ensure the self-contained nature of the paper.

**Theorem 3** Under the feedback control $u$ defined in (22), the error dynamics for the density $\tilde{\rho} = \rho - \bar{\rho}$, as well as its partial derivatives in time and space, converge uniformly and exponentially to zero. Specifically, there exist positive constants $a, b > 0$ such that:

$$ \sup (\|\tilde{\rho}(t, x)\|_{\infty}, \|\partial_x \tilde{\rho}(t, x)\|_{\infty}, \|\partial_t \tilde{\rho}(t, x)\|_{\infty}) \le ae^{-bt}, \quad (31) \\ \text{for all } t \ge 0. $$

**PROOF.** To prove the exponential convergence of $\tilde{\rho}(t, x)$ to zero as $t \to \infty$, we begin by considering its Fourier series expansion in space. We assume periodic boundary conditions for simplicity, which allow us to express $\tilde{\rho}(t, x)$ as a Fourier series:

$$ \tilde{\rho}(t, x) = \sum_{n=-\infty}^{\infty} c_n(t) e^{inx}, $$

where $c_n(t)$ are the Fourier coefficients, defined by

$$ c_n(t) = \frac{1}{2\pi} \int_0^{2\pi} \tilde{\rho}(t, x) e^{-inx} dx. $$

At the initial time $t = 0$, the Fourier coefficients $c_n(0)$ are determined by the initial condition $\tilde{\rho}_0(x)$:

$$ c_n(0) = \frac{1}{2\pi} \int_0^{2\pi} \tilde{\rho}_0(x) e^{-inx} dx. $$

We substitute this Fourier expansion into the governing equation for $\tilde{\rho}(t, x)$, which results in a set of decoupled differential equations for the Fourier coefficients $c_n(t)$. The evolution of each coefficient $c_n(t)$ is governed by the equation

$$ \ddot{c}_n(t) + \alpha \dot{c}_n(t) + \beta n^2 c_n(t) = 0, $$

where $\alpha$ and $\beta$ are positive constants related to the system's damping and the frequency of the spatial modes.

**Step 1: Solving the characteristic equation** The characteristic equation associated with the above second-order differential equation is

$$ r^2 + \alpha r + \beta n^2 = 0. $$

Solving this quadratic equation for $r$, we obtain the roots

$$ r_{1,n}, r_{2,n} = \frac{-\alpha \pm \sqrt{\alpha^2 - 4\beta n^2}}{2}. $$

**Step 2: Analyzing the roots** We now analyze the behavior of these roots in different cases, depending on the discriminant $\alpha^2 - 4\beta n^2$.

**Case 1:** $\alpha^2 - 4\beta n^2 > 0$ When the discriminant is positive, the roots are real and distinct:

$$ r_{1,n} = \frac{-\alpha + \sqrt{\alpha^2 - 4\beta n^2}}{2}, \quad r_{2,n} = \frac{-\alpha - \sqrt{\alpha^2 - 4\beta n^2}}{2}. $$

Both roots are negative because $\alpha > 0$ and $\alpha^2 - 4\beta n^2 > 0$. Therefore, the general solution for $c_n(t)$ is

$$ c_n(t) = a_n e^{r_{1,n}t} + b_n e^{r_{2,n}t}, $$
where $a_n$ and $b_n$ are constants determined by the initial conditions. Since both roots are negative, it follows that $c_n(t) \to 0$ exponentially as $t \to \infty$.

**Case 2:** $\alpha^2 - 4\beta n^2 < 0$ When the discriminant is negative, the roots are complex conjugates:

$$r_{1,n} = -\frac{\alpha}{2} + i\omega_n, \quad r_{2,n} = -\frac{\alpha}{2} - i\omega_n,$$

where $\omega_n = \frac{\sqrt{4\beta n^2 - \alpha^2}}{2}$. The solution in this case is

$$c_n(t) = e^{-\frac{\alpha}{2}t} (a_n \cos(\omega_n t) + b_n \sin(\omega_n t)).$$

Since $\frac{\alpha}{2} > 0$, the solution decays exponentially, and hence $c_n(t) \to 0$ as $t \to \infty$.

**Case 3:** $\alpha^2 - 4\beta n^2 = 0$ In this case, the roots are real and equal:

$$r_{1,n} = r_{2,n} = -\frac{\alpha}{2}.$$

The solution is

$$c_n(t) = (a_n + b_n t)e^{-\frac{\alpha}{2}t}.$$

Since $\frac{\alpha}{2} > 0$, the solution still decays exponentially as $t \to \infty$.

**Step 3: Convergence of $\tilde{\rho}(t, x)$** Since $c_0 \equiv 0$ and in view of (19), the above analysis leads to the conclusion that for all $n \neq 0$, the Fourier coefficients $c_n(t)$ decay exponentially over time. Therefore, the solution $\tilde{\rho}(t, x)$, expressed as a Fourier series, also decays exponentially:

$$\tilde{\rho}(t, x) = \sum_{n=-\infty}^{\infty} c_n(t) e^{inx}.$$

As each $c_n(t) \to 0$ exponentially as $t \to \infty$, it follows that $\tilde{\rho}(t, x) \to 0$ uniformly in $x$. This convergence is exponential in $t$. Furthermore, the same reasoning show that the time derivatives $\partial_t \tilde{\rho}(t, x)$ and $\partial_x \tilde{\rho}(t, x)$ also decay exponentially, ensuring smooth convergence of $\tilde{\rho}(t, x)$.

### 4.2.3 Convergence toward the desired speed

At this stage, there is no guarantee that the speed error converges exponentially to zero. Nevertheless, the following theorem provides a sufficient condition under which such convergence holds.

**Theorem 4** Assume that $\partial_x \bar{v}$ is uniformly bounded above and that there exists a time $t_0 > 0$ and a constant

$\alpha > c > 0$ such that, for all $t \ge t_0$,

$$\left| \frac{\partial_t \bar{\rho}(t, x)}{\bar{\rho}(t, x)} \right| \le c. \qquad (32)$$

Then the velocity perturbation $\tilde{v} = v - \bar{v}$ converges uniformly and exponentially to zero as $t \to \infty$; that is, there exist constants $c_1, d_1 > 0$ such that

$$\|\tilde{v}(t, \cdot)\|_{\infty} \le c_1 e^{-d_1 t}, \quad \forall t \ge 0. \qquad (33)$$

**PROOF.** Since, by Theorem 3, both $\tilde{\rho}$ and $\partial_t \tilde{\rho}$ converge exponentially to $\bar{\rho}$ and $\partial_t \bar{\rho}$ respectively, there exist constants $\gamma > 0$ and $t_0 > 0$ such that, for all $t \ge t_0$ and $x \in [0, 2\pi]$, we have

$$-\alpha + \frac{\partial_t \rho(t, x)}{\rho(t, x)} \le -\gamma. \qquad (34)$$

Combining (20), the uniform boundedness assumption on $\partial_x \bar{v}$, and Theorem 3, we conclude that $h$ converges uniformly and exponentially to zero. Applying Grönwall's inequality to Equation (23b) then establishes the desired result.

In the equilibrium case, the condition of Theorem 4 is automatically satisfied, allowing us to recover the result established in [6].

## 5 Discretization of the control and numerical illustration

We now demonstrate how the continuous control law proposed in (22) can be discretized to design a feedback strategy for the microscopic model defined in (1). The objective is to steer the system toward a prescribed velocity and density profile $(\bar{\rho}, \bar{v})$. We conclude this section with a numerical example that illustrates the effectiveness of the proposed control scheme. We now turn to the discretization of the control procedure.

### 5.1 Discretization of the Control

To formulate a control law at the microscopic level, we begin by discretizing the key quantities involved in (22), namely: $\dot{x}_i, \rho v, \partial_x(\rho v)$, and $\partial_x \rho$. From the approximation (13), we obtain the following discrete expressions:

$$v \approx \dot{x}_i, \quad \rho v \approx \frac{\dot{x}_i}{x_{i+1} - x_i}. \qquad (35)$$

In addition, using the approximation $\rho \approx 1/(x_i - x_{i-1})$, we derive the expression according to the time derivative:

$$\partial_t \rho \approx - \frac{\dot{x}_{i+1} - \dot{x}_i}{(x_{i+1} - x_i)^2}. \qquad (36)$$
Substituting this into the first equation of (17a) yields a discrete expression for the spatial derivative of the momentum:

$$ \partial_x(\rho v) \approx \frac{\dot{x}_{i+1} - \dot{x}_i}{(x_{i+1} - x_i)^2}. \quad (37) $$

The spatial derivative of the density, $\partial_x\rho$, is not specified by the continuation procedure. Its discretization affects the structure of the resulting control law, particularly in terms of vehicle connectivity. In this sense, our approach can be viewed as a generic controller that accommodates a family of intervehicle control laws with varying connectivity patterns. For illustration, and following the scheme introduced in [6], we adopt a symmetric stencil that incorporates both backward and forward intervehicle spacings. Nevertheless, alternative discretizations of $\partial_x\rho$ remain possible. The chosen approximation reads:

$$ \partial_x \rho \approx \frac{1}{x_{i+1} - x_i} - \frac{1}{x_i - x_{i-1}}. \quad (38) $$

Combining the approximations from (35) to (38), and recalling that the target velocity and density are denoted by $\bar{v}$ and $\bar{\rho}$, we obtain the following discretized control law:

$$ u_i = -f(\Delta x_i, \Delta \dot{x}_i, \dot{x}_i) + \Delta x_i \left[ \dot{x}_i \frac{\Delta \dot{x}_i}{(\Delta x_i)^2} - \bar{v}(t, x_i) \partial_x(\bar{\rho}(t, x_i)\bar{v}(t, x_i)) + \alpha \bar{\rho}(t, x_i) \bar{v}(t, x_i) \right. \\ \left. - \alpha \frac{\dot{x}_i}{\Delta x_i} - \beta \left( \frac{1}{\Delta x_i} - \frac{1}{\Delta x_{i-1}} \right) \right], \quad (39) $$

$$ \text{where } \Delta x_i = x_{i+1} - x_i, \quad \Delta \dot{x}_i = \dot{x}_{i+1} - \dot{x}_i, \quad (40) $$

for all $i \in \{1, \dots, N\}$.

## 5.2 Design guidances and Numerical examples

For instance, our main results, stated in Theorem 3 and Theorem 4, provide a control strategy that drives the system toward a configuration where all vehicles travel at a common constant speed, while their spatial distribution matches a prescribed density profile. Without loss of generality, we normalize the target velocity to $\bar{v} = 1$. In this case, the reference density satisfies the linear transport equation

$$ \partial_t \bar{\rho} + \partial_x \bar{\rho} = 0, \quad (41) $$

with initial condition $\bar{\rho}(0, x) = \rho_0(x)$ and periodic boundary conditions $\bar{\rho}(t, 0) = \bar{\rho}(t, 2\pi)$. By periodically extending $\rho_0$ over $\mathbb{R}$ with period $2\pi$, the solution to (41) is explicitly given by the method of characteristics as

$$ \bar{\rho}(t, x) = \rho_0(x - t). \quad (42) $$

Given any desired initial density profile $\rho_0$, the control law defined in (39) guarantees that the vehicle distribution in the microscopic model (1) closely follows the transported target density $\bar{\rho}(t, x)$, while maintaining a uniform velocity $\dot{x}_i(t) = 1$, provided the assumptions of Theorem 4 are satisfied. We begin by discussing how to appropriately choose the target density profile $\rho_0$.

### 5.2.1 Admissible targets

The density profile $\rho_0$ must satisfy certain constraints imposed by the ring-road geometry. In particular, the continuation method highlights that $\rho_0$ corresponds to the reciprocal of the distance between consecutive vehicles. Consequently, we cannot have $\rho_0 < \frac{1}{2\pi}$, as this would imply an intervehicle distance greater than $2\pi$, which is impossible on a ring.

Conversely, if the minimum of $\rho_0$ approaches $\frac{1}{2\pi}$, then $\rho_0$ must be sufficiently large elsewhere, since the total sum of all intervehicle distances must remain equal to $2\pi$. In other words, only a few vehicles can be separated by distances close to $2\pi$. Naturally, the larger the number of vehicles $N$, the higher the density must be to accommodate them on the ring.

We illustrate this with the following two examples:

**Example 1 (N = 10) We consider the initial density**

$$ \rho_0(x) = 5 \sin \left( \frac{x}{2} \right) + 1. \quad (43) $$

It is straightforward to verify that the assumptions of Theorem 4 are satisfied. This choice yields higher density in certain regions (with intervehicle spacing close to 1/6), and lower density elsewhere (with spacing approaching 1), reflecting the sinusoidal modulation.

**Example 2 (Case N = 40) We now increase the number of vehicles and consider a more oscillatory target profile:**

$$ \rho_0(x) = 10 \sin \left( \frac{x}{2} \right) + 5. \quad (44) $$

This corresponds to a tenfold amplification of the sinusoidal term used in the previous example. Again, the assumptions of Theorem 4 are satisfied. The result is a denser vehicle clustering in high-density regions (with spacing around 1/15), producing a sharper contrast in the spatial configuration.

It remains to choose the parameters $\alpha$ and $\beta$ appropriately in order to ensure convergence.

### 5.2.2 Role of $\alpha$ and $\beta$

Successful convergence of the control strategy requires appropriate tuning of the parameters $\alpha$ and $\beta$. Improper
choices may degrade performance or result in undesirable effects such as negative vehicle speeds, which we seek to avoid. This is illustrated in Example 1, where we set $\beta = 1$ and $\alpha = 2.5$ (see Fig. 3).

<figure><img src="image_3.png" /><figcaption>Fig. 3. Speed of the vehicles in the first numerical example with $\alpha = 2.5$</figcaption></figure>

This undesirable behavior can be better understood by analyzing the proof of Theorem 3. Indeed, certain choices of $\alpha$ and $\beta$ can induce oscillatory dynamics due to the spectral properties of the system. On the other hand, the only way to counteract these oscillations is to choose $\alpha$ large enough. This observation supports the intuition that, in such regimes, the system converges toward the desired density profile.

By setting $\alpha = 10$ and keeping $\beta = 1$, the simulations shown in Fig. 4 and Fig. 5 confirm that the proposed control strategy effectively synchronizes the vehicle velocities and preserves the target density profile as it is transported around the ring, in accordance with Example 1 and Example 2. In particular, in Example 1, all vehicle speeds become strictly positive, unlike in the case with $\alpha = 2.5$. This highlights that increasing $\alpha$ facilitates convergence toward the desired density and velocity profiles while ensuring positive speeds.

From Fig. 5, we can observe that the leading vehicle (in green), labeled as vehicle 1, initially reduces its speed to nearly zero in order to allow the vehicles behind it to cluster. Once the vehicles are sufficiently grouped, vehicle 1 starts to accelerate again, while the last vehicle in the chain (vehicle *N* in red) maintains a constant velocity throughout the process. This dynamic illustrates how the control law effectively synchronizes the speeds of all vehicles, driving the system towards the desired configuration. The system progressively aligns the vehicle velocities while respecting the target density distribution as it is transported along the ring, demonstrating the effectiveness of the control in achieving both velocity synchronization and density tracking.

## 6 Conclusion

This paper introduces a control strategy for stabilizing traffic on a ring road using autonomous vehicles. Un-

like traditional approaches, our method enables convergence toward time-varying density and velocity profiles. Relying on a continuation-based framework, we derived a macroscopic control law from a second-order microscopic model and proposed a procedure to synthesize the required control inputs.

A numerical example illustrates the flexibility of the method to accommodate time-varying patterns and heterogeneous intervehicle spacing. This paves the way for new traffic control strategies adapted to dynamic and complex environments.

Future work will focus on incorporating communication delays, sensing imperfections, and vehicle heterogeneity, as well as validating the approach in real-world settings with autonomous platforms. Another line of investigation will explore the integration of driver behavior models into the control design, with the aim of extending the framework to non-autonomous vehicles.

## Acknowledgements

This work was supported by grant ANR-22-CE48-0011 COCOON.

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<figure><img src="image_4.png" /><figcaption>Fig. 4. Comparison of the system evolution under the control law (39) for $N = 10$ (top row) and $N = 40$ (bottom row). We set $\alpha = 10, \beta = 1$. Each row shows the initial density $\rho_0(x)$, the intervehicle distances $x_{i+1}(t) - x_i(t)$, and the vehicle speeds $\dot{x}_i(t)$. In both cases, the control law drives the system toward the desired velocity $\bar{v} = 1$ and the transported density $\bar{\rho}(t, x) = \rho_0(x-t)$.</figcaption></figure>

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