Stability of a Nonlocal Traffic Flow Model
for Connected Vehicles^††thanks: Submitted to the editors DATE. \fundingThis work is supported in part by the NSF DMS-2012562, DMS-1937254 and ARO MURI Grant W911NF-15-1-0562.

A common block of macroscopic traffic flow models is the continuity equation:

that describes the conservation of vehicles, where $\rho(x,t)$ and $u(x,t)$ denote the aggregated traffic density and velocity and $x$ and $t$ are spatial and temporal coordinates. The Lighthill-Whitham-Richards (LWR) model [43, 45] is the most extensively used macroscopic traffic flow model. It assumes a fundamental relation:

between traffic density and velocity, meaning that the driving speed of a vehicle is determined only by the instantaneous density at the vehicle’s current location. The function $U(\cdot)$ is also referred to as the desired speed function. The LWR model follows from (1)(2) as a scalar conservation law:

The LWR model (3) may produce shock wave solutions even with smooth initial data. Such shock wave solutions qualitatively explain the formation and propagation of traffic jams.

The main objective of this paper is to consider the asymptotic stability of a nonlocal extension to the LWR model (3), proposed by [6, 32]. The basic assumption underneath such a nonlocal model is that each vehicle perceives traffic density information in a road segment of length $\delta>0$ ahead of the vehicle’s current location. The driving speed of the vehicle is then based on an weighted average of density within the road segment:

where the nonlocal kernel $w_{\delta}(\cdot)$ characterizes the nonlocal effect. (1)(4) lead to the following nonlocal LWR model:

In the existing studies, some theoretical and numerical results have been developed on the scalar nonlocal conservation law (5), see Section 1.3 for a review. However, existing studies on the asymptotic stability of the model are still limited. Under some suitable assumptions, we show that the stability is closely related to the nonlocal kernel $w_{\delta}(\cdot)$. In particular, we prove that the solution of the model exponentially converges to a constant density as $t\to\infty$ when the kernel $w_{\delta}(\cdot)$ is non-increasing and non-constant. Meanwhile, a constant kernel may lead to traffic waves that persist in time.

To interpret the significance of the mathematical findings made in this work, let us note their connections to issues that are important in real traffic situation. In the traffic research community, it is widely recognized that the presence of traffic waves could result in elevated risks for traffic safety, an increase in vehicle fuel consumption, as well as a reduction in total traffic throughput [49]. Thus, the dissipation of traffic waves and the stability of constant density states are features that can offer benefits to both drivers of individual vehicles and the whole traffic ecosystem. The particular mathematical results established here, in plain words, provide further evidence to the following natural principle when designing driving algorithms for connected vehicles: it can be beneficial to utilize nonlocal interactions between connected vehicles for traffic decisions, and while doing so, suitable forms and ranges of nonlocality should be adopted with nearby information deserving more attention.

The nonlocal LWR model (5) was first proposed in [6, 32], where the existence, uniqueness and maximum principle of the weak entropy solution were proved using the Lax-Friedrichs numerical approximation. The entropy condition is adopted to ensure the solution uniqueness. In subsequent works, [13] proved the same results for a generalized model of (5):

and [10] developed high-order numerical schemes to solve (6). In a related work, [37] studied a family of nonlocal balance laws:

which include (5) as a special case. The existence and uniqueness of the weak solution were proved using the method of characteristics and a fixed-point argument. The latter also leads to solution uniqueness without the use of the entropy condition.

In the existing studies, some analytic properties of the nonlocal LWR model (5) were discussed. In terms of solution regularity, [8] showed that the solution of (5) has the same regularity as the initial data when: (i) the nonlocal kernel $w_{\delta}(\cdot)$ is $\mathbf{C}^{1}$ smooth and non-increasing on $[0,+\infty)$ with the zero extension on $[\delta,+\infty)$; (ii) the desired speed function $U(\cdot)$ is $\mathbf{C}^{2}$ smooth and $U^{\prime}\leq-c<0$ for some constant $c$. In contrast, the local LWR model (3) can develop shock solutions from smooth initial data whenever the characteristics impinge each other. In terms of relations between the local and nonlocal models, one fundamental question is whether the solution of the nonlocal model (5) converges to that of the local model (3) when $\delta\to 0$, i.e., the vanishing nonlocality limit. In [18], it was shown that such convergence is in general false with a demonstration based on an example associated with the desired speed function $U(\rho)=\rho$ and discontinuous initial data. Nevertheless, convergence results were given in [38] when $U(\cdot)$ is a decreasing function and the initial data is monotone. [8, 7] considered a special case where the nonlocal interaction range is infinite and the nonlocal kernel is exponential, which leads to

In this case, the nonlocal-to-local convergence as $\delta\to 0$ was proved for uniformly positive initial data and any desired speed function $U(\cdot)$ that is $\mathbf{C}^{2}$ smooth and $U^{\prime}\leq-c<0$ for some constant $c$. [17] extended the convergence results for exponentially decaying kernels and a family of decreasing desired speed functions, but required the initial data to be uniformly positive and have no negative jumps. It is still an open problem what are sufficient and necessary conditions on the desired speed function, nonlocal kernel and initial data for the vanishing nonlocality limit to be true. In terms of asymptotic behavior, [46] gave a class of monotone stationary solutions of (5) on an infinitely long road and showed that those solutions are asymptotic local attractors of (5). [36] studied a generalized model of (5) where a nudging (or “look behind”) term is added to the nonlocal velocity:

Under the assumptions that: (i) the model is solved on a ring road; (ii) $U(\cdot)$ is decreasing and $\tilde{U}(\cdot)$ is increasing; (iii) $w_{\delta}(s)=1/\delta$; (iv) $\tilde{\delta}$ is the length of the ring road and $\tilde{w}_{\tilde{\delta}}(s)=(\tilde{\delta}-s)/\tilde{\delta}$, the local exponential stability of uniform equilibrium flows as $t\to\infty$ was proved.

Let us also briefly mention other relevant studies. The nonlocal LWR model (5) has been generalized to the case for 1-to-1 junctions [12] and of multi-class vehicles [14, 16]. There are also nonlocal traffic flow models other than (5). In [48], a model based on Arrhenius “look-ahead” dynamics was proposed where the nonlocal velocity:

[40, 41, 42] analyzed shock formation criteria of the model; In [11], a nonlocal extension to the traditional Aw-Rascle-Zhang model [3] was proposed and the micro-macro limit was demonstrated. More broadly, nonlocal models have been drawing increasing attention in our connected world [24]. Nonlocal conservation laws, in particular, have been studied in many other applications, e.g., pedestrian traffic [19, 9], sedimentation [5] and material flow on conveyor belts [33, 47], see [20] for a review. [27, 26, 25] discussed nonlocal conservation laws inspired from discrete descriptions of local conservation laws. Some more analytical and numerical studies on nonlocal conservation laws can be found in [1, 2, 21, 31, 15, 4].

Before the rigorous statement of the main results of this paper, let us specify the set-up of the model problem. First of all, we consider the problem on a ring road. Mathematically, we use the spatial domain $x\in[0,1]$ to represent the ring road and assume the periodic boundary condition for the equation (5).

• (A1)

  $\rho(0,t)=\rho(1,t)$, $\forall t\geq 0$.

The periodicity assumption is common in stability analysis of traffic flow models and fits the scenarios in field experiments [50, 49]. The nonlocal LWR model (5) is solved with the periodic boundary condition and the following initial condition:

where $\rho_{0}$ is a nonnegative density distribution in $\mathbf{L}^{\infty}([0,1])$. We denote:

the average density of all vehicles on the ring road. Given $\bar{\rho}$, there is a constant solution of (5):

This constant solution, which is an equilibrium of the dynamics described by the nonlocal LWR model (5), represents the uniform flow in traffic where all vehicles are uniformly distributed and drive at the same speed.

We then make the following assumptions on the desired speed function $U(\cdot)$ and the nonlocal kernel $w_{\delta}(\cdot)$.

• (A2)

  $U(\rho)=1-\rho$.

The linear desired speed function $U(\rho)=1-\rho$, usually referred to as the Greenshields speed-density relationship [34], is widely used in traffic flow modeling. We make the assumption (A2) to simplify the problem because in this case (5) can be rewritten as:

where:

The equation (11) defines the one-sided nonlocal gradient operator $\mathcal{D}_{x}^{\delta}$ [30, 28]. In (11), the integration is defined with respect to periodicity of the density function. The formulation (10) reinterprets the nonlocal LWR model (5) as the local one (3) with an additional term that may provide some form of nonlocal diffusion for a suitably chosen kernel $w_{\delta}(\cdot)$. A sufficient condition is provided in the following assumption.

• (A3)

  $w_{\delta}(\cdot)$ is a $\mathbf{C}^{1}$ function defined on $[0,\delta]$, satisfying

  $$w_{\delta}(s)\geq 0,\;\forall s\in[0,\delta]\quad\text{ and }\quad\int_{0}^{\delta}w_{\delta}(s)\,ds=1.$$

  In addition, $w_{\delta}(\cdot)$ is non-increasing and non-constant on $[0,\delta]$.

The assumption (A3) is the key to the main findings of this paper. It is the mathematical reformulation of the natural design principle that density information of nearby vehicles should deserve more attention. Under this assumption, we can deduce that the nonlocal LWR model (5) indeed adds appropriate nonlocal diffusion effect to the local one (3) through a direct spectral analysis, see Section 2.1. More precisely, we will show the following linear stability result.

Naturally, for the nonlinear nonlocal system, it is interesting to see if we can extend the linear stability to get global nonlinear stability. For this, we make one additional assumption on the initial data.

• (A4)

  There exist $0<\rho_{\text{min}}\leq\rho_{\text{max}}\leq 1$ such that:

  $\displaystyle\rho_{\text{min}}\leq\rho_{0}(x)\leq\rho_{\text{max}},\quad\forall x\in[0,1].$

With all above assumptions, we are ready to state the well-posedness of the weak solution as defined below.

The well-posedness theorem follows from [38]. Even though the spatial domain considered in that work is set to be the real line $\mathbb{R}$, the same arguments work with little modifications for the periodic case.

Although the weak solution always exists, it can be discontinuous. In this paper, the stability analysis is based upon an energy estimate. To do that, we make a regularity assumption on the weak solution of (5).

• (A5)

  The weak solution $\rho\in\mathbf{C}^{1}\left([0,1]\times[0,\infty)\right)$.

The assumption (A5) is equivalent to say that $\rho$ is the classical solution of (5). When the assumptions (A1) - (A4) are true, [8] proved a sufficient condition for the assumption (A5): the initial data $\rho_{0}$ is $\mathbf{C}^{1}$ smooth, and the nonlocal kernel $w_{\delta}(\cdot)$ is $\mathbf{C}^{1}$ smooth on $[0,+\infty)$ with the zero extension $w_{\delta}(s)=0$ on $s\in[\delta,+\infty)$.

Now we are in position to state the main results of this paper.

The remainder of this paper is organized as follows: Section 2 is devoted to stability analysis of (5) and the proofs of Theorem 1 and Theorem 3. Section 3 provides numerical experiments to illustrate the results. Conclusions and future research directions follow in Section 4.

Given the assumption (A1), we consider the Fourier series expansion of any real-valued periodic function $\rho(x)$ for $x\in[0,1]$:

For the local gradient operator $\partial_{x}$ and the nonlocal gradient operator $\mathcal{D}_{x}^{\delta}$ defined by (11), a straightforward calculation gives:

where

As a corollary, the spectrum of the nonlocal diffusion operator $\partial_{x}\mathcal{D}_{x}^{\delta}$ is given by the discrete set of eigenvalues $\{-2\pi kb_{\delta}(k)+2\pi ikc_{\delta}(k)\}_{k\in\mathbb{Z}}$. The following lemma gives an estimate on the real parts of the eigenvalues $\{-2\pi kb_{\delta}(k)\}_{k\in\mathbb{Z}}=\{0\}\cup\{-2\pi kb_{\delta}(k)\}_{k\geq 1}$.

We now present the proof of the linear stability given in Theorem 1.

Suppose $\rho(x,t)$ is any $\mathbf{C}^{1}$ solution to the nonlocal LWR model (5). The conservation property gives:

We define the following energy function (aka a Lyapunov functional):

It is straightforward to get:

Note that (10) is equivalent to (5), substituting (10) into (18) yields:

Apply the Newton-Leibniz rule and integration by parts, we obtain:

All boundary terms vanish because of the periodic boundary condition.

We present two lemmas to estimate the right hand side of (19). Then the conclusion of the theorem follows from the estimate of the energy function $E(t)$.

A special case of Lemma 3.2 is when $w_{\delta}(\cdot)$ is a rescaled kernel: $w_{\delta}(s)=w_{1}(s/\delta)/\delta$. That is, the family of kernels $\{w_{\delta}(\cdot)\}_{\delta\in(0,1]}$ is generated from the kernel $w_{1}(\cdot)$ defined on $[0,1]$. In this case, it is worthwhile to mention that the constant $\alpha$ in (20) is independent of the nonlocal range $\delta$.

Next, we present an inequality similar to (20) to deal with the presence of nonlinearity. Actually, Lemma 3.4 stated below bridges between linear and nonlinear diffusion in the nonlocal setting. Its proof uses the following Hardy-Littlewood rearrangement inequality on a periodic domain. A similar inequality is used in [8] to prove that the local limit of nonlocal solutions of (5) satisfies the entropy condition.