# Local Stabilization with Arbitrary Decay of Ring-Road Traffic Using a Single Autonomous Vehicle

Sébastien Fueyo, Carlos Canudas de Wit

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Sébastien Fueyo, Carlos Canudas de Wit. Local Stabilization with Arbitrary Decay of Ring-Road Traffic Using a Single Autonomous Vehicle. 2025. hal-05350535

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# Local Stabilization with Arbitrary Decay of Ring-Road Traffic Using a Single Autonomous Vehicle

S. Fueyo and C. Canudas de Wit

**Abstract**— Controlling road traffic is challenging due to complex interactions among vehicles, often studied with microscopic models like the Optimal Velocity-Follow the Leader framework on ring roads where stop-and-go waves emerge naturally. While it is known that a single autonomous vehicle (AV) using simple proportional feedback can stabilize the traffic, the achievable convergence rate is limited. In this paper, we investigate the potential of general linear state feedbacks, depending on the full system state including positions and velocities of all vehicles. We prove that there exists a linear feedback law capable of achieving arbitrarily fast local stabilization using a single AV, although very fast stabilization requires large feedback gains. Numerical simulations illustrate these theoretical results, and a simple example shows that involving multiple AVs can reduce the required gains and improve overall stability.

**Keywords:** Ring-road traffic, autonomous vehicles, linear feedback control, stabilization.

## I. INTRODUCTION

Modeling and controlling road traffic is challenging due to the intricate interactions among numerous vehicles. To tackle this complexity, researchers often rely on microscopic models, which describe the dynamics of individual vehicles through systems of first- or second-order ordinary differential equations (ODEs). These models explicitly account for vehicle-to-vehicle interactions, with well-known examples including the Optimal Velocity-Follow the Leader (OV-FTL) framework [2], [7].

Traffic exhibits complex collective phenomena that emerge from driver behavior, such as spontaneous traffic jams and stop-and-go waves. These oscillations can significantly disrupt traffic flow, a phenomenon first documented experimentally by [16]. Contemporary traffic control research focuses on mitigating such instabilities by influencing driver actions via measures like variable speed limits, ramp metering, or direct control interventions [4], [10], [15].

Closed-loop ring roads have become a standard testbed for studying traffic dynamics and evaluating control strategies. Due to their periodic structure, ring-road systems naturally develop stop-and-go waves, which are key contributors to congestion. Unlike open-road settings, the circular geometry enforces global coupling between all vehicles: the end of the road connects back to the beginning, which can reinforce oscillatory behavior and complicate stabilization. Despite this, ring roads are particularly valuable for benchmarking traffic control techniques, as they combine structural simplicity with

rich dynamical features, allowing for systematic analysis before applying strategies to more complex or open networks [9], [17], [14], [3].

In this context, recent studies have demonstrated that traffic flow can be stabilized through the control of a single autonomous vehicle. In particular, [8], [3] analyzed the local exponential stability of the microscopic OV-FTL model by introducing a feedback law acting on one autonomous car and a simple proportional feedback. The trade-off of the proportional feedback is that, beyond a certain point, the exponential decay rate is limited by the intrinsic structure of the drivers' behavior described by the OV-FTL model. Consequently, increasing the feedback gain no longer improves the convergence rate or the stability.

The goal of this paper is to take a step further and investigate what can be achieved by controlling a single autonomous vehicle using a general linear feedback law based on the full system state. Standard proportional feedback relies only on the velocity of the autonomous vehicle, which limits its effect on collective traffic dynamics. By designing a linear feedback that also incorporates the positions and velocities of the surrounding vehicles, we can exploit additional information from the system and potentially achieve improved stabilization.

The first contribution of this work is theoretical. We prove that the OV-FTL model can be locally stabilized with an arbitrarily large exponential decay rate by means of a suitably designed linear feedback, as stated in Theorem III.1. This result demonstrates that the system is controllable even with a single autonomous vehicle. However, it comes with an inherent trade-off: achieving a higher decay rate requires increasingly large control gains (see Corollary IV.2). Numerical simulations further illustrate this limitation—enhancing the decay rate demands unrealistically high feedback gains, which may be impractical due to actuator saturation or others physical constraints. This highlights the fundamental balance between the decay rate and the magnitude of the control action when relying on a single vehicle for stabilization.

Finally, we conduct numerical experiments to analyze how the presence of multiple autonomous vehicles affects the overall system stability and the required feedback gains, thereby providing insights into the potential benefits of multi-vehicle control strategies in mixed traffic conditions.

## II. STABILIZATION ON A RING ROAD WITH ONE AUTONOMOUS VEHICLE

Let us consider a circular road of length $L$ occupied by $N$ human-driven vehicles. We denote the positions and

<footnote>Sébastien Fueyo and Carlos Canudas-de-Wit are with Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, GIPSA-Lab, F-38000 Grenoble, France (emails sebastien.fueyo@gipsa-lab.fr and carlos.canudas-de-wit@gipsa-lab.fr). This work was supported by grant ANR-22-CE48-0011 COCOON.</footnote>
velocities of the human-driven vehicles by $(x_i, v_i)$ for $i = 1, \dots, N$. The dynamics of the human-driven vehicles are modeled using the Bando-Follow-the-Leader framework, where the optimal velocity concept is taken from [2] and the follow-the-leader interaction from [7]. These dynamics can be written as, for $1 \le i \le N$,

$$ \left\{ \begin{array}{l} \dot{x}_i = v_i, \\ \dot{v}_i = a \underbrace{\frac{v_{i+1} - v_i}{(x_{i+1} - x_i)^2}}_{\text{Follow-the-Leader}} + b \underbrace{V(x_{i+1} - x_i) - v_i}_{\text{Optimal Velocity}}, \end{array} \right. \quad (1) $$

where $a, b > 0$ are constant parameters, and $V$ denotes the so-called optimal velocity function. Several functional forms of $V$ have been proposed in the literature to model this behavior, ranging from piecewise linear expressions to smooth sigmoidal curves, see for instance [6]. In this paper, the function $V(\cdot)$ is assumed to be continuously differentiable, nonnegative, strictly increasing, and bounded, thereby capturing the natural driving behavior: as the headway between two vehicles increases, drivers tend to accelerate, but their velocity eventually saturates at a maximal admissible speed.

The uniform traffic equilibrium corresponds to all vehicles moving at a constant speed $\bar{v}$, with cars evenly spaced by a distance of $d = L/N$ as imposed by the ring-road geometry. From (1) one sees that the only possible uniform steady state is

$$ \bar{v} = V(d). $$

On the circular road, the closure is obtained by identifying car $N+1$ with car 1, i.e. $v_{N+1} = v_1$ and $x_{N+1} = x_1$.

Depending on the choice of parameters $(a, b)$ and the traffic density $d^{-1} = N/L$, the uniform traffic equilibrium may be either stable or unstable. To formalize these notions, it is convenient to introduce a measure of deviation from equilibrium. Let $x = (x_1, \dots, x_N)$ and $v = (v_1, \dots, v_N)$ denote the collections of vehicle positions and velocities, respectively. We then consider the functional

$$ D(x, v) = \sum_{i=1}^{N} |x_{i+1} - x_i - d| + \sum_{i=1}^{N+1} |v_i - \bar{v}|, \quad (2) $$

which quantifies how far the current trajectory $(x(t), v(t))$ is from the uniform steady state $(d, \bar{v})$. This functional allows us to provide precise mathematical definitions of local exponential stability and local instability.

**Definition II.1 (Local Exponential Stability).** The uniform traffic equilibrium of System (1) is said to be locally exponentially stable if there exist constants $r > 0$, $M \ge 1$, and $\alpha > 0$ such that, for every initial condition $D(x(0), v(0)) < r$, the corresponding solution exists for all $t \ge 0$ and satisfies

$$ D(x(t), v(t)) \le Me^{-\alpha t} D(x(0), v(0)), \quad \forall t \ge 0. \quad (3) $$

The real number $\alpha$ is referred to as the decay rate.

In contrast, when certain small perturbations grow over time, the equilibrium is said to be unstable.

**Definition II.2 (Local Instability).** The uniform traffic equilibrium is said to be locally unstable if there exists an initial condition arbitrarily close to the equilibrium for which the solution eventually leaves any small neighborhood of the equilibrium.

These definitions allow us to precisely state known results on the instability of the uniform traffic flow. In particular, it has been shown [3] that under suitable parameter conditions, the uniform equilibrium becomes locally unstable:

**Proposition II.3 ([3]).** If

$$ \frac{b}{2} + \frac{a}{d^2} < V'(d), \quad (4) $$

then the uniform solution of (1) is locally unstable.

This phenomenon explains why stop-and-go oscillations are observed both in simulations and in real-world experiments, as first reported by [16] and later confirmed in [15], [14]. To mitigate such oscillations, it was shown in [3] that the inclusion of a single autonomous vehicle can locally exponentially stabilize the traffic flow. This line of research was further developed in [8], where estimates on the exponential decay rate of the system and characterizations of the basin of attraction were obtained.

Following the approach in [8], and to recontextualize their results, we consider a circular road with a single controlled autonomous vehicle (AV), whose dynamics are given by

$$ \dot{x}_{N+1}(t) = v_{N+1}(t), \quad \dot{v}_{N+1}(t) = u(t), \quad (5) $$

where $u(t)$ denotes the control input applied to the AV.

A natural and simple choice for the control input is a proportional feedback of the form

$$ u(t) = k(\bar{v} - v_{N+1}(t)), \quad (6) $$

where $k > 0$ is a gain parameter. It was shown in [8] that such a proportional feedback ensures the local exponential stability of the traffic system.

**Theorem II.4 ([8]).** The controlled system (1)–(5) with the proportional feedback (6) is locally exponentially stable if $k > 0$. The best achievable decay rate is

$$ \gamma_{\max} = \min \left( k, \frac{1}{2} \left( \frac{a}{d^2} + b - \Re \left( \sqrt{\left( \frac{a}{d^2} + b \right)^2 - 4bV'(d)} \right) \right) \right). \quad (7) $$

We briefly comment on the result formally stated in Theorem II.4. It demonstrates that a system of human-driven vehicles, which would otherwise be unstable, can be stabilized by the introduction of a single autonomous vehicle, provided a simple proportional feedback law with a single gain is employed. However, a major limitation of this approach is that the resulting exponential convergence may remain very slow, as the overall dynamics are still largely dominated by the behavior of human drivers, as emphasized in Equation (7). Indeed, selecting

$$ k > \frac{1}{2} \left( \frac{a}{d^2} + b - \Re \left( \sqrt{\left( \frac{a}{d^2} + b \right)^2 - 4bV'(d)} \right) \right) $$
does not improve the exponential stability of the human-
driven vehicles. Because driver-induced instabilities are per-
vasive, simply increasing the feedback gain does not nec-
essarily accelerate stabilization; the influence of the human
drivers ultimately constrains the achievable performance. In
this work, our main interest is to stabilize the system at
a desired, possibly fast, rate. To formalize this notion, we
introduce the following definition.

**Definition II.5** (Local Exponential Stability with Arbitrary Decay Rate). *The uniform traffic equilibrium is said to be locally exponentially stable with arbitrary decay rate if, in the sense of Definition II.1, the decay rate α can be chosen arbitrarily large.*

We now focus on the exponential stability of a single
autonomous vehicle, where the decay rate can be made
arbitrarily large.

III. STABILIZATION WITH ARBITRARY DECAY BY A
SINGLE AV

In this section, we investigate the capability of a single
autonomous vehicle (AV) to achieve stabilization with arbi-
trary decay of the mixed traffic system. We provide a theo-
retical analysis of the linearized dynamics, highlighting the
structural properties of the system and establishing precise
conditions under which stabilization with arbitrary decay can
be achieved.

Let us consider System (1) with a single autonomous
vehicle as defined in (5), and let $(d, \bar{v})$ denote the uniform
equilibrium. Introducing the change of variables

$$
\begin{cases}
y_{2p+1} = x_{p+2} - x_{p+1} - d, & 0 \le p \le N-1, \\
y_{2p} = v_{p+1} - v_p, & 1 \le p \le N, \\
y_{2N+1} = \bar{v} - v_{N+1},
\end{cases}
\quad (8)
$$

the system can be equivalently expressed in terms of the
variables $y$, which preserves the local exponential stability
properties of the controlled system (1)–(5). In these variables,
the dynamics of $y$ is governed by the ordinary differential
equation:

$$
\left\{
\begin{array}{ll}
\dot{y}_{2p+1} = y_{2p+2}, & 0 \le p \le N-1, \\
\dot{y}_{2p} = a \left[ \frac{y_{2p+2}}{(d+y_{2p+1})^2} - \frac{y_{2p}}{(d+y_{2p-1})^2} \right] & \\
\phantom{\dot{y}_{2p} = a \left[ \frac{y_{2p+2}}{(d+y_{2p+1})^2} - \frac{y_{2p}}{(d+y_{2p-1})^2} \right]} & + b[V(d+y_{2p+1}) - V(d+y_{2p-1}) - y_{2p}], \\
& 1 \le p \le N-1, \\
\dot{y}_{2N} = u - a \frac{y_{2N}}{(d+y_{2N-1})^2} & \\
\phantom{\dot{y}_{2N} = u - a \frac{y_{2N}}{(d+y_{2N-1})^2}} & - b[V(d+y_{2N-1}) + y_{2N} + y_{2N+1} - \bar{v}], \\
\dot{y}_{2N+1} = -u.
\end{array}
\right.
\tag{9}
$$

After linearizing System (9) around the uniform equilibrium
($\bar{v}, d$), the resulting linearized dynamics can be written in
standard state-space form as

$$
\dot{y} = Ay + Bu, \quad (10)
$$

where the matrices $A \in \mathbb{R}^{(2N+1) \times (2N+1)}$ and $B \in \mathbb{R}^{(2N+1) \times 1}$ capture the interactions among vehicles and the control input of the autonomous vehicle.

To construct these matrices, we first define the local 2 × 2
building blocks corresponding to a single vehicle:

$$
A_1 = \begin{bmatrix} 0 & 1 \\ -bV'(d) & -(\frac{a}{d^2} + b) \end{bmatrix}, \quad B_1 = \begin{bmatrix} 0 & 0 \\ bV'(d) & \frac{a}{d^2} \end{bmatrix},
$$

where $a, b > 0$ are model parameters, $d > 0$ is the
equilibrium spacing, and $V'(d)$ denotes the derivative of the
optimal velocity function $V(\cdot)$ evaluated at $d$. The global
system matrix $A$ can then be written in a compact block
form as

$$
A = \begin{bmatrix} \tilde{A} & z \\ 0 & 0 \end{bmatrix},
$$

where $\tilde{A} \in \mathbb{R}^{2N \times 2N}$ contains the repeated $A_1$ and $B_1$ blocks:

$$
\tilde{A} = \begin{bmatrix}
A_1 & B_1 & 0 & \cdots & 0 \\
0 & A_1 & B_1 & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & \cdots & 0 & A_1 & B_1 \\
0 & \cdots & 0 & 0 & A_1
\end{bmatrix},
$$

and $z \in \mathbb{R}^{2N \times 1}$ captures the coupling with the autonomous vehicle:

$$
z = [0, \dots, 0, -b]^T,
$$

where $(\cdot)^{\top}$ denotes the transpose operator. Finally, the input
matrix is given by

$$
B = [0, \dots, 0, 1, -1]^T.
$$

To establish the criterion for local stabilization with arbitrary decay rate, we first analyze the eigenvalues of the system matrix $A$. Owing to the block upper-triangular structure of $A$, its eigenvalues coincide with those of the local block $A_1$, together with an additional zero eigenvalue. More precisely, the two eigenvalues of $A_1$ are given by

$$
\lambda_1 = \frac{-(a + bd^2) - \sqrt{(a + bd^2)^2 - 4bV'(d)d^4}}{2d^2},
$$

$$
\lambda_2 = \frac{-(a + bd^2) + \sqrt{(a + bd^2)^2 - 4bV'(d)d^4}}{2d^2},
$$

so that the spectrum of $A$ is

$$
\sigma(A) = \{\lambda_1, \lambda_2, 0\}.
$$

**Theorem III.1.** System (9) can be locally exponentially stabilized with an arbitrary decay rate by means of a linear control u acting on the autonomous vehicle if and only if at least one of the following conditions holds:

1) for every eigenvalue $\lambda \in \sigma(A)$,

$$
\lambda \neq -bV'(d) \frac{d^2}{a} \quad \text{and} \quad \lambda \neq -b,
$$

2) $V'(d) \neq \frac{a}{d^2}$.
Proof. Since our system is sufficiently smooth, according to the pole-shifting theorem [13, Th. 7, p. 134], a nonlinear system can be locally exponentially stabilized with an arbitrary decay rate if and only if its linearization at the equilibrium point is controllable. By the Hautus criterion, this is equivalent to requiring that

$$ \mathrm{rank}[\lambda I_d - A, B] = 2N + 1, \quad \forall \lambda \in \sigma(A), \quad (11) $$

where $I_d$ denotes the identity matrix of dimension $2N$. Hence, it suffices to verify that the Hautus criterion holds whenever either condition 1 or 2 is satisfied.

Assume first that condition 1 holds. Suppose, for contradiction, that the Hautus criterion fails. Then there exist a nonzero row vector $x = (x_1, \dots, x_{2N}, x_{2N+1})^\top$ and some $\lambda \in \sigma(A)$ such that

$$ x^\top A = \lambda x^\top, \quad x^\top B = 0, $$

which leads to the system

$$ \left\{ \begin{array}{l} -bV'(d) x_2 = \lambda x_1, \\ x_1 - \left(\frac{a}{d^2} + b\right) x_2 = \lambda x_2, \\ -bV'(d) x_{2k} + bV'(d) x_{2k-2} = \lambda x_{2k-1}, \quad 2 \le k \le N, \\ -\left(\frac{a}{d^2} + b\right) x_{2k} + x_{2k-1} + \frac{a}{d^2} x_{2k-2} = \lambda x_{2k}, \quad 2 \le k \le N, \\ -b x_{2N} = \lambda x_{2N+1}, \\ x_{2N+1} = x_{2N}. \end{array} \right. \quad (12) $$

Since $\lambda \neq -b$, the last two equations imply $x_{2N+1} = x_{2N} = 0$. Evaluating for $k = N$ and using $\lambda \neq -bV'(d) d^2/a$, we obtain $x_{2N-1} = x_{2N-2} = 0$. Iterating this argument yields $x_{2k-1} = x_{2k-2} = 0$ for $k = 2, \dots, N-1$ and in particular $x_2 = 0$. Finally, the first equation gives $x_1 = 0$. Thus $x = 0$, a contradiction, and the system is controllable.

If condition 1 does not hold while condition 2 is satisfied, two situations must be considered (recall that $b > 0$):

* If $\lambda = -b$, the last two equations in (12) yield $x_{2N+1} = x_{2N}$. By backward propagation, one finds $x_{2N} = x_{2N-2} = \dots = x_2$ and $x_{2N-1} = x_{2N-3} = \dots = x_3 = 0$. Substituting into the first two equations then gives $x_1 = x_2 = 0$, so that $x = 0$, a contradiction.

* If $\lambda = -bV'(d) d^2/a$, analogous computations again lead to $x = 0$, a contradiction.

Similarly, if neither condition 1 nor 2 holds, one can construct a nonzero vector that violates the Hautus criterion, showing that the system is not controllable. Therefore, controllability holds if and only if at least one of the two conditions is satisfied. $\square$

As shown by Theorem II.4, the addition of a single autonomous vehicle is sufficient to stabilize the system, but it is not enough to ensure a rapid convergence toward the equilibrium point by just taking a simple proportional feedback as described in (6). However, Theorem III.1 shows that it is possible to construct a linear feedback to stabilize the

system with arbitrary decays rate, except in some particular cases for some parameters of the system. In another terms, it means in theory, that knowing all the position and speed of the other vehicles, help to achieve a better stabilization of the system.

The next section addresses the inherent practical limitations of relying on a single AV, particularly the trade-off between convergence speed and the implementability of the feedback gains.

## IV. LIMITATIONS OF STABILIZATION WITH ARBITRARY DECAY RATES USING A SINGLE AUTONOMOUS VEHICLE

Theorem III.1 ensures that, in theory, a single autonomous vehicle can be used to stabilize with arbitrary decays the linearized ring-road traffic system. Specifically, the system is controllable, which implies that the closed-loop poles can, in principle, be shifted arbitrarily to achieve any desired convergence rate. In practice, however, there are limitations that affect the achievable stabilization speed. In fact, some modes of the system are inherently more difficult to shift, in particular those associated with vehicles far from the controlled autonomous vehicle. These modes, corresponding to the eigenvalues of $A_1$ (whose smallest real part is given in Equation (7)), cannot be directly influenced by the autonomous vehicle.

We next provide a lower bound on the feedback gain when aiming a decay rate $\alpha > 0$. The proof is inspired by the work of [12], [1], who investigate minimal-energy control of complex networks, focusing on the selection of control nodes and the design of control inputs to achieve desired target states while accounting for network topology and energy limitations.

To fit within the energy-based framework, we first introduce the standard Euclidean norm on $\mathbb{R}^{2N+1}$. Specifically, for the state vector $y = (y_1, \dots, y_{2N+1})^\top$, we define

$$ \|y\|_2 := \sqrt{\sum_{i=1}^{2N+1} y_i^2}. $$

This norm is more convenient for linear system analysis, including controllability and state-feedback design. Since we are in finite dimension, all the norms are equivalent. Consequently, local exponential stability with respect to $\|\cdot\|_2$ is equivalent to local exponential stability with respect to $D(x, v)$, and the decay rate is preserved.

**Theorem IV.1** (Lower bound on the feedback gain). Consider the controllable linear system defined in (10) and assume that the state-feedback law $u(t) = -Ky(t)$ achieves exponential stabilization with decay rate $\alpha > 0$, i.e.,

$$ \|y(t)\|_2 \le \|y(0)\|_2 e^{-\alpha t}, \quad \forall t \ge 0. $$

Then, for every horizon $T > 0$, the feedback gain $K$ must satisfy the lower bound

$$ \|K\|_2 \ge \sqrt{\frac{2\alpha}{T}} \frac{\|e^{AT}\|_2 - e^{-\alpha T}}{\|e^{AT}\|_2 \|B\|_2 \sqrt{1 - e^{-2\alpha T}}}. \quad (13) $$
**Proof.** Define the finite-horizon controllability Gramian as

$$W(T) = \int_{0}^{T} e^{At} BB^{\top} e^{A^{\top}t} dt,$$

which is invertible since $(A, B)$ is assumed controllable. From the minimum-energy control problem (see [11]), for any input $u(\cdot)$ steering $y(0) = y_0$ to $y(T) = y_T$, one has

$$\int_{0}^{T} \|u(t)\|_{2}^{2} dt \geq (e^{AT} y_{0} - y_{T})^{\top} W(T)^{-1} (e^{AT} y_{0} - y_{T}). \quad (14)$$

Under state feedback $u = -Ky$, the terminal state is $y_T = e^{(A-BK)T} y_0$, and therefore

$$\int_{0}^{T} \|Ky(t)\|_{2}^{2} dt \geq ((e^{AT} - e^{(A-BK)T})y_{0})^{\top} \times W(T)^{-1}((e^{AT} - e^{(A-BK)T})y_{0}). \quad (15)$$

Using the exponential decay rate assumption $\|y(t)\|_2 \le \|y_0\|_2 e^{-\alpha t}$ together with the inequality

$$y^{\top} W(T)^{-1} y \geq \sigma_{\min}(W(T)^{-1}) \|y\|_{2}^{2},$$

(where $\sigma_{\min}(M)$ denotes the smallest singular value of a matrix $M$), we deduce from (15) that, for all $y_0 \neq 0$,

$$\|K\|_{2}^{2} \|y_{0}\|_{2}^{2} \int_{0}^{T} e^{-2\alpha t} dt \geq \sigma_{\min}(W(T)^{-1}) \times \|(e^{AT} - e^{(A-BK)T})y_{0}\|_{2}^{2}. \quad (16)$$

Taking the infimum over all $\|y_0\|_2 = 1$ yields

$$\|K\|_2 \geq \frac{\sqrt{\sigma_{\min}(W(T)^{-1})} \|e^{AT} - e^{(A-BK)T}\|_2}{\left(\int_0^T e^{-2\alpha t} dt\right)^{1/2}}. \quad (17)$$

Finally, since $\|e^{(A-BK)T}\|_2 \le e^{-\alpha T}$, the triangle inequality gives

$$\|e^{AT} - e^{(A-BK)T}\|_2 \geq \|e^{AT}\|_2 - e^{-\alpha T},$$

and substituting into (17) yields

$$\|K\|_2 \geq \frac{\sqrt{\sigma_{\min}(W(T)^{-1}) (\|e^{AT}\|_2 - e^{-\alpha T})}}{\left(\int_{0}^{T} e^{-2\alpha t} dt\right)^{1/2}}. \quad (18)$$

Let $\sigma_{\max}(M)$ denote the largest singular value of a matrix $M$, a crude upper bound on the Gramian is

$$\sigma_{\max}(W(T)) \leq T \|e^{AT}\|_2 \|B\|_2^2, \quad (19)$$

so that $\sigma_{\min}(W(T)^{-1}) = 1/\sigma_{\max}(W(T))$ leads directly to the estimate (18). $\square$

As a direct corollary of Theorem IV.1, we obtain the following result:

**Corollary IV.2.** For the linearized vehicle system (10), the feedback gain $\|K\|_2$ required to achieve exponential stabilization with rate $\alpha > 0$ diverges as $\alpha \to \infty$.

**Proof.** Let $T > 0$ be fixed. Using the block-triangular structure of the system matrix

$$A = \begin{bmatrix} \tilde{A} & z \\ 0 & 0 \end{bmatrix},$$

with $\tilde{A}$ having eigenvalues with strictly negative real parts, we have

$$e^{AT} = \begin{bmatrix} e^{\tilde{A}T} & \int_{0}^{T} e^{\tilde{A}s} z \, ds \\ 0 & 1 \end{bmatrix}.$$

Since $e^{\tilde{A}T} \to 0$ as $T \to \infty$ and the improper integral

$$\int_{0}^{\infty} e^{\tilde{A}s} z \, ds$$

converges (for instance, it equals $-\tilde{A}^{-1}z$ if $\tilde{A}$ is invertible). Hence, as $T \to \infty$, the matrix exponential satisfies

$$\lim_{T \to \infty} e^{AT} = \begin{bmatrix} 0 & \int_{0}^{\infty} e^{\tilde{A}s} z \, ds \\ 0 & 1 \end{bmatrix}.$$

Thus

$$\|e^{AT}\|_2 \geq c_0 > 0$$

for some constant $c_0$ independent of $\alpha$. The denominator of the bound satisfies

$$\sqrt{\int_{0}^{T} e^{-2\alpha t} dt} = \sqrt{\frac{1 - e^{-2\alpha T}}{2\alpha}} \to 0 \text{ as } \alpha \to \infty.$$

Substituting these estimates into the lower bound of Theorem IV.1 yields

$$\|K\|_2 \geq \frac{\sigma_{\min}(W(T)^{-1})^{1/2} (c_0 - e^{-\alpha T})}{\sqrt{\int_{0}^{T} e^{-2\alpha t} dt}}. \quad (20)$$

As $\alpha \to \infty$, the right-hand side of (20) diverges, implying that $\|K\|_2 \to +\infty$.

Hence, for any fixed horizon $T > 0$, the feedback gain $\|K\|_2$ diverges when the desired decay rate $\alpha$ increases without bound. $\square$

Although the feedback gain tends to infinity when targeting a larger decay rate $\alpha$, the lower bound established in Theorem IV.1 (see Equation (18)) indicates that increasing the norm of $B$ can decrease this bound, thereby allowing for an improved exponential decay of the system.

In the next section, we numerically illustrate this limitation and confirm the difficulty of stabilizing the system faster than the intrinsic dynamics of these modes allow when using a single autonomous vehicle. Finally, through a numerical illustration, we provide an insight suggesting that increasing the proportion of controlled (autonomous) vehicles contributes to improving the system's stabilization performance. In particular, controlling one out of every two drivers already leads to a noticeable enhancement of the exponential decay rate.
V. NUMERICAL ILLUSTRATION

This section provides numerical illustrations supporting the theoretical developments presented above. We first examine the stabilization of a mixed traffic system composed of one autonomous and several human-driven vehicles on a ring road. Through simulations based on the finite-dimensional linearized dynamics, we compare different feedback strategies and quantify their impact on the closed-loop convergence rate. We then extend the analysis to configurations involving multiple autonomous vehicles. These results offer numerical evidence and intuition—although not a formal proof—that increasing the proportion of controlled vehicles tends to enhance the overall stability and accelerate the exponential decay of the system. Such observations highlight the potential benefits of partial control strategies for improving traffic flow stability in mixed environments.

A. A single autonomous vehicle

To illustrate the theoretical findings from the previous sections, we consider the linearized system (9) around the uniform equilibrium $(d, \bar{v})$ with $N = 5$ human-driven vehicles. We fix the parameter of the linear system to be $a = 1, b = 0.5, d = 1$ and $V'(d) = 0.8$ so that the assumption of Theorem III.1 are satisfied and the system is locally exponentially stabilized with arbitrary decay rates. In particular, it means that the fundamental diagram $V(\cdot)$ satisfies $V(1) = \bar{v}$ and the length of the road is $L = 5$. Using a linear state feedback

$$u(t) = -Ky(t), \quad (21)$$

with $y$ defined in (8), we compute the feedback gain $K \in \mathbb{R}^{1 \times (2N+1)}$ numerically using pole placement.

We study two scenarios for pole placement:

* Faster feedback: poles with real part in $[-1.5, -2.0]$, corresponding to relatively aggressive stabilization.

* Slower feedback: poles uniformly spaced in $[-1.0, -1.5]$, corresponding to less aggressive control.

For comparison, we also consider the simple proportional feedback law given in Equation (6) with a gain $k = 1$. According to Theorem II.4, this feedback stabilizes the system without requiring full-state information, but at the cost of a limited decay rate (see Equation (7)). In particular, the decay rate cannot exceed

$$\frac{1}{2} \left( \frac{a}{d^2} + b - \Re \left( \sqrt{\left( \frac{a}{d^2} + b \right)^2 - 4bV'(d)} \right) \right) \approx 0.34,$$

even if $k$ is increased beyond one.

To illustrate and compare the performance of the proposed feedback strategies, we now present several numerical simulations. All computations were carried out in PYTHON using the NumPy, SciPy, and Matplotlib libraries. The simulations are based on the finite-dimensional representation of the coupled vehicle system introduced in the previous section.

Figure 1 shows the closed-loop eigenvalues for all three strategies. The fast feedback moves the eigenvalues further left, achieving faster convergence. The slow feedback produces smaller gains and slower convergence but still stabilizes the system. Proportional feedback with $k = 1$ stabilizes the system but is less effective than full-state feedback.

<figure><img src="image_2.png" /><figcaption>Fig. 1. Closed-loop eigenvalues for proportional feedback (blue), slow pole placement (green), and fast pole placement (red).</figcaption></figure>

Figure 2 displays the magnitudes of the feedback gains for faster and slower pole placement. Aggressive pole placement requires higher gain values, which may be limited in practice by actuator saturation or measurement noise.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Absolute values of feedback gains $K$ for fast (red) and slow (green) pole placement.</figcaption></figure>

Despite the theoretical possibility of arbitrarily shifting the closed-loop poles, practical limitations arise when using a single autonomous vehicle.

We now provide some insight into how the decay rate can be improved by introducing additional controlled (and thus autonomous) vehicles. The following numerical examples illustrate the construction of linear feedback laws that stabilize the system and demonstrate how the limitation imposed by using a single autonomous vehicle can be overcome. In particular, the present approach introduces multiple autonomous vehicles while preserving the natural dynamics of human drivers, in contrast to fully automated traffic settings such as those considered in [5]. This strategy enables the system to achieve locally faster convergence and enhanced stability.
## B. Single Autonomous Vehicle with One-in-Two Driver Behavior Control

In this section, we consider the scenario in which vehicle $N + 1$ remains autonomous, while a certain number of vehicles positioned between two driver-operated vehicles are actively controlled, as illustrated in Figure 3. For simplicity, we assume that $N$ is odd, which ensures that $N + 1$ is even, facilitating a symmetric arrangement of vehicles around the circular track. Specifically, we control $r/2 < (N + 1)/2$ vehicles located between these two driver-operated vehicles, using control inputs $u_i$ for $i = 1, \dots, r/2$. This setup allows us to investigate the potential stabilizing effects of controlling these intermediate vehicles on the overall traffic system.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Schematic of the scenario where vehicle $N+1$ remains autonomous, and $r/2$ vehicles between two driver-operated vehicles are actively controlled.</figcaption></figure>

Mathematically, this means that the input matrix $B \in \mathbb{R}^{2(N+1) \times (r/2+1)}$ in the controlled system (10) takes the following form. Let $i_j$ denote the index of the $j$-th controlled vehicle among the intermediate vehicles, $j = 1, \dots, r/2$. The entries of $B$ are specified component-wise as

$$ B_{i,j} = \begin{cases} -1, & i = 2(i_j - 1) + 1, \quad j = 1, \dots, r/2, \\ 1, & i = 2(i_j - 1), \quad j = 1, \dots, r/2, \\ -1, & i = 2N + 1, \quad j = r/2 + 1, \\ 1, & i = 2N, \quad j = r/2 + 1, \\ 0, & \text{otherwise.} \end{cases} $$

This definition shows that:

* The first $r/2$ columns correspond to the controlled intermediate vehicles.

* The last column corresponds to the autonomous vehicle $N + 1$, whose states are explicitly included in the input matrix.

Following the intuition that increasing the number of autonomous vehicles enhances the exponential decay rate of the system, this effect is illustrated using the example presented in Section V-A with $N = 5$.

## VI. CONCLUSION

In this paper, we have investigated the potential of a single autonomous vehicle to achieve stabilization with arbitrary

<figure><img src="image_5.png" /><figcaption>Fig. 4. Comparison of absolute values of feedback gains $K$ (in log scale) for $r/2 = 0, 1, 2$. The bars represent the absolute values of the feedback gains for different controlled vehicles, including the autonomous vehicle. The green bars correspond to $r/2 = 0$, red to $r/2 = 1$, and blue to $r/2 = 2$.</figcaption></figure>

decay of ring-road traffic. Building on the classical proportional feedback approach, which ensures local exponential stability but limits the achievable convergence rate, we have considered a general linear state-feedback law based on the full system state.

Our theoretical analysis shows that, under mild conditions, the linearized traffic system is controllable with a single autonomous vehicle, which implies that local exponential stabilization with arbitrary decay is achievable in principle. However, we have also highlighted a fundamental practical limitation: the feedback gain required to achieve a high decay rate increases without bound, making extremely fast stabilization potentially unrealistic in practice.

Numerical simulations confirm these theoretical findings and illustrate the trade-off between convergence speed and control effort. Furthermore, our results suggest that involving multiple autonomous vehicles can alleviate this limitation, allowing for faster stabilization with moderate feedback gains.

Future research should aim to further investigate how the proportion and spatial distribution of autonomous vehicles influence the achievable stabilization performance, and to determine under which configurations such mixed-traffic systems can be efficiently and robustly stabilized.

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