Uncertainty damping in kinetic traffic models by driver-assist controls

# Uncertainty damping in kinetic traffic models by driver-assist controls

Andrea Tosin, Mattia Zanella
Department of Mathematical Sciences “G. L. Lagrange”
Dipartimento di Eccellenza 2018-2022
Politecnico di Torino, Torino, Italy
andrea.tosin@polito.it, mattia.zanella@polito.it
###### Abstract

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

Keywords: kinetic traffic model, Boltzmann-type equation, non-Maxwellian particles, Fokker-Planck equation, uncertainty quantification

Mathematics Subject Classification: 35Q20, 35Q70, 35Q84, 35Q93, 90B20

## 1 Introduction

In recent times, the challenge of vehicle automation has inspired new paradigms for the management and the governance of traffic and has imposed a deep reflection about the behavioural aspects involved in vehicle dynamics, see e.g. [3, 15]. On one hand, the emerging technologies constitute a potentially high innovation in the realm of the smart cities, for example with clear benefits in terms of emission reduction and of mitigation of road risk factors. On the other hand, the time horizon of their actual full implementation is still uncertain and questionable. For this reason, recent advances in the modelling of driver-assist/autonomus vehicles have focused, in particular, on the assessment of the effectiveness of Advanced Driver-Assistance Systems (ADAS) in a mixed scenario, with few automated vehicles embedded in a stream of human-manned vehicles [34]. In this context, it is particularly interesting to address the design of suitable control protocols, which are sufficiently robust to cope with some behavioural aspects of human drivers, and then to study their aggregate impact on the traffic stream, see e.g. [30].

In this work, we study a kinetic traffic model with structural uncertainty in the vehicle-to-vehicle interactions. Such an uncertainty, represented by a parameter whose value is not known deterministically, accounts, for example, for the heterogeneous response of different types of vehicles to the speed variations. Kinetic models for traffic flow have a quite long history, going back to the pioneering works [25, 26] up to more recent advances, such as e.g. [13, 18]. The kinetic approach has the clear advantage of linking, in a self-consistent way, the different scales of the problem: from the microscopic one of the interactions among the vehicles to the mesoscopic one of the aggregate distribution of observable quantities, such as the fundamental diagrams, and further also to the macroscopic one of the density waves flowing along a road [34]. For instance, in [27, 28] a kinetic approach has been developed to explain the scattering of the fundamental diagram of traffic, typically observed at high density after the phase transition from free to congested flow. Those works are based on multi-population/mixture models, which take into account the heterogeneous composition of the traffic stream. In this work, we show that a kinetic explanation of the scattering of the fundamental diagram in the congested traffic regime can be effectively obtained also by introducing, in a single-population model, an uncertain parameter characterising the microscopic interactions. On one hand, this method leads to an increased dimensionality of the kinetic problem, owing to the necessity to handle the uncertain parameter in addition to the standard microscopic variables defining the state of the vehicles. On the other hand, however, it avoids the necessity to deal with systems of kinetic equations, thereby making the whole approach more amenable to further developments in terms of modelling and analysis.

Along this line, in this work we exploit the potential offered by the uncertain kinetic setting in order to further address a control problem, aimed at reducing the scattering of the fundamental diagram. Specifically, we design two possible control protocols, which, consistently with the aforementioned ADAS technologies, are applied at the level of the microscopic interactions among the vehicles and, in particular, act in such a way to align the speed of the vehicles to a prescribed congestion-dependent optimal value. Since, in a realistic scenario, all vehicles are not equipped with the ADAS technology, such controls are active only on a possibly small percentage of vehicles, corresponding to the so-called ADAS penetration rate. Our main result is that such microscopic control strategies are able to dampen the structural uncertainty present in the vehicle-to-vehicle interactions, thereby reducing the macroscopic scattering of the fundamental diagram in a way precisely linkable to the penetration rate of the ADAS technology. An immediate consequence of this is a more ordered, hence predictable, macroscopic flow of the vehicles, with clear advantages for the traffic governance at large scale.

From the methodological point of view, the proposed control setting takes advantage of recently introduced methods for the binary control of Boltzmann-type kinetic equations, see [1, 2] and also [33, 34] for the specific application to vehicular traffic. First, we derive an uncertain control, namely one obtained from a microscopic cost functional depending pointwise on the uncertain parameter of the interactions. As a second case, we derive a deterministic control, namely one obtained from a microscopic cost functional averaged with respect to the uncertain parameter. These two microscopic controls lead to two different Boltzmann-type models. The one associated with the uncertain control is a Maxwellian model, since the corresponding Boltzmann equation features a unitary interaction kernel. Conversely, the one associated with the deterministic control is a non-Maxwellian model, because the corresponding Boltzmann equation needs a non-constant interaction kernel in order to discard some microscopic interactions possibly violating the physical bounds of the microscopic variable. We discuss the analytical properties of both models and, in particular, we show that, in the asymptotic regime of the quasi-invariant interactions [8, 31], they can be described by the same Fokker-Planck equation, which leads to explicitly computable and realistic steady states highlighting the reduction of the scattering of the fundamental diagram discussed above. We also propose several numerical tests, which, by appealing to specific numerical methods for Uncertainty Quantification (UQ) for kinetic and mean field equations, cf. e.g. [7, 11, 14, 16, 39, 40], visualise the theoretical results and allow us to consider various probability distributions of the uncertain parameter, spanning also interesting cases in which explicit computations are not possible.

In more detail, the paper is structured as follows: in Section 2, we discuss a basic microscopic model of the interactions among the vehicles without control and we stress, in particular, the role of the uncertain parameter. Then we pass to a Boltzmann-type kinetic description, whence we obtain explicitly the asymptotic trend of the mean speed, which we use to define the fundamental diagram together with its scattering induced by the uncertain parameter. Finally, in the quasi-invariant interaction limit, we recover a Fokker-Planck-type description, whence we deduce an explicit form of the equilibrium distribution of the speed of the vehicles along with its uncertainty. In Section 3, we introduce the two microscopic control strategies to be applied to the interactions among the vehicles. Specifically, in Section 3.1, we discuss the uncertain control, which leads to a Boltzmann-type kinetic description for Maxwellian-like particles and allows us to prove a precise result concerning the reduction of the scattering of the fundamental diagram in terms of the penetration rate of the control strategy. In Section 3.2, we discuss instead the deterministic control, which is more realistically implementable in practice but requires to deal with a more difficult Boltzmann-type kinetic description for non-Maxwellian-like particles. However, we prove that, in the quasi-invariant interaction regime, such a description yields the same limit equations for both the mean speed and the statistical speed distribution as the previous description, whence we recover the validity of all the results proved before. In Section 4, we present several numerical tests supporting the theoretical findings of the previous sections. Finally, in Section 5 we summarise the results of the paper and we briefly outline possible research developments.

## 2 Scattering of the fundamental diagram from uncertain binary interactions

### 2.1 Description of the microscopic interactions with uncertainty

In accordance with the general approach of the kinetic theory, a kinetic description of traffic flow is based on the identification of microscopic interaction rules for pairs of vehicles. We assume, in particular, that these interactions modify the speed of the vehicles in consequence of accelerations and decelerations. Therefore, we characterise the microscopic state of a generic vehicle by means of a variable representing its (dimensionless) speed. Denoting by the speed of the leading vehicle, we describe the speed variation in a binary interaction as

 v′=v+γI(v,v∗;z)+D(v)η,v′∗=v∗. (1)

In (1), is a proportionality parameter and is the interaction function, which describes the acceleration/deceleration dynamics mentioned above. This function depends on the pre-interaction speeds , of the interacting vehicles and on an uncertain parameter, viz. random variable, with known probability distribution , i.e.

 P(z≤¯z)=∫¯z−∞Ψ(z)dz.

Such an uncertain parameter represents a structural uncertainty in the binary rule modelled by , due to the fact that the physics of the interactions among the vehicles is inevitably partly heuristic.

Getting inspiration from [34], we consider the following interaction function:

 I(v,v∗;z)=P(ρ;z)(1−v)+(1−P(ρ;z))(P(ρ;z)v∗−v), (2)

where is the probability of acceleration. The function (2) expresses the fact that with probability the -vehicle accelerates towards the maximum speed, while with probability it adapts to the fraction of the speed of the -vehicle. The probability of acceleration depends on the (dimensionless) density of the vehicles , in such a way that the higher the lower , because a dense traffic hinders accelerations. Moreover, depends also on the uncertain parameter . A possible form is:

 P(ρ;z)=(1−ρ)z,z>0. (3)

From Figure 1, we observe that the mapping expresses the same qualitative trend ( decreasing with ) for all . Nevertheless, it may induce quantitatively different dynamics, because such a trend is either concave, linear or convex depending on the specific value of . It is therefore reasonable to regard the exponent as an uncertain parameter, considering that there are apparently no a priori motivations to opt for a specific value of in the heuristic model (3). Moreover, since , we notice that the higher the lower the probability of acceleration at all densities . This suggests that the values of may be associated with different classes of vehicles characterised by a different motility in the traffic stream. We will come back to this interpretation in Section 2.2.

Returning to (1), the term describes stochastic fluctuations caused by the intrinsic randomness of the driver behaviour. Specifically, is a centred random variable, i.e. one with zero mean:

 ⟨η⟩=0,⟨η2⟩=σ2, (4)

where denotes the average with respect to the probability distribution of and is the standard deviation of . Moreover, is a speed-dependent non-negative diffusion coefficient modulating the amplitude of the stochastic fluctuation. We stress that, unlike , the random variable does not describe a structural uncertainty of the model but rather a stochastic completion of the deterministic dynamics expressed by the interaction function .

Finally, we notice that (1) prescribes that the -vehicle does not change speed during an interaction. This is because vehicle interactions are normally anisotropic, the leading vehicle being unperturbed by the rear vehicle.

#### 2.1.1 Physical admissibility of the interaction rules

In order to be physically admissible, the interaction rules

 v′=v+γ[P(ρ;z)(1−v)+(1−P(ρ;z))(P(ρ;z)v∗−v)]+D(v)ηv′∗=v∗ (5)

have to guarantee for every . This is actually obvious for , while it is more delicate for .

Let us assume . We begin by observing that, since and , a sufficient condition for is

 (1−γ)v+D(v)η≥0,

which is certainly satisfied if there exists a constant such that

 η≥c(γ−1),cD(v)≤v.

Likewise, since , a sufficient condition for is

 (γ−1)(1−v)+D(v)η≤0,

which is satisfied if

 η≤c(1−γ),cD(v)≤1−v.

On the whole, the physical admissibility of (5) is guaranteed by the following sufficient conditions:

 |η|≤c(1−γ),cD(v)≤min{v,1−v}, (6)

where is an arbitrary constant. These conditions imply that the stochastic fluctuation is bounded and that .

### 2.2 Boltzmann-type aggregate description and traffic diagrams

A statistical description of the aggregate dynamics resulting from the superposition of many binary interactions (5) may be obtained by introducing the distribution function , where is the time. Specifically, is such that gives the probability that, at time , a vehicle travels with a speed comprised between and , given the uncertain parameter . Since, under (6), all the binary interactions (5) are physically admissible, the distribution function evolves according to the following Boltzmann-type equation for Maxwellian-like particles, here written in weak form:

 ddt∫10φ(v)f(t,v;z)dv=12∫10∫10⟨φ(v′)−φ(v)⟩f(t,v;z)f(t,v∗;z)dvdv∗ (7)

for every observable quantity , namely any quantity which may be expressed as a function of the microscopic state of the vehicles. This equation states that the time variation of the mean of (left-hand side) is due to the variation, on average, of in the binary interactions (right-hand side). The coefficient depends on the fact that the mean variation of in a single binary interaction is actually , because from (1) it results .

We have stressed that is parametrised by the uncertain parameter , because its evolution depends, among other things, on the uncertain interaction function contained in in (7). Consequently, (7) is a stochastic kinetic equation, whose solution may be regarded, for fixed , , as a random variable given as a function of the uncertain parameter .

Letting in (7), we discover

 ddt∫10f(t,v;z)dv=0,

therefore if is initially a probability density it remains so at all successive times, independently of . With and recalling (4), we obtain from (7) the evolution of the mean speed

 V(t;z):=∫10vf(t,v;z)dv,

i.e.:

 dVdt=12∫10∫10I(v,v∗;z)f(t,v;z)f(t,v∗;z)dvdv∗=γ2[P(ρ;z)(1−V)−(1−P(ρ;z))2V]. (8)

This equation can be solved explicitly to find the mapping parametrised by . Here, we are in particular interested in the asymptotic value reached by for , say , which describes the mean speed emerging when interactions are in equilibrium:

 V∞(ρ;z):=P(ρ;z)P(ρ;z)+(1−P(ρ;z))2. (9)

Notice that, due to the uncertain parameter , this is in turn a stochastic quantity. We may compute the expectation of with respect to as

 ¯V∞(ρ):=Ez(V∞(ρ;z))=∫R+V∞(ρ;z)Ψ(z)dz (10)

and its variance as

 ς2∞(ρ):=Varz(V∞(ρ;z))=∫R+V2∞(ρ;z)Ψ(z)dz−¯V2∞(ρ). (11)

Next, we observe that we may use the mapping to define the fundamental diagram of traffic, namely the equilibrium relationship between the traffic density and the macroscopic flux of the vehicles. Together with its -standard deviation , it produces the following set

 {(ρ,q)∈[0,1]×R+:q∈[ρ¯V∞(ρ)−ρς∞(ρ),ρ¯V∞(ρ)+ρς∞(ρ)]} (12)

in the density-flux plane, where most of the random flux values lie. The set (12) explains the scattering of the fundamental diagram, typically found in empirical measurements of the flow of vehicles, as the result of the superposition of different microscopic dynamics produced by different values of and weighted by the corresponding probability measure .

For a general probability distribution , the exact computation of (10), (11) may be non-trivial and one often needs to rely on numerical quadrature formulas. However, for very special classes of probability distributions , such as those considered below, analytical results can be obtained.

Let us consider the case in which is a discrete random variable with law

 P(z=zk)=αk∈[0,1],n∑k=1αk=1,

so that

 Ψ(z)=n∑k=1αkδ(z−zk),

where is the Dirac delta distribution centred in . Then

 ¯V∞(ρ)=n∑k=1αkV∞(ρ;zk),ς2∞(ρ)=n∑k=1αkV2∞(ρ;zk)−(n∑k=1αkV∞(ρ;zk))2.

We observe that , , is the result of the microscopic dynamics (5) with . If we interpret the ’s as characteristic values of certain classes of vehicles (for instance, cars, lorries, motorcycles, …) which may travel along the road, the formulas above show that , originate from the average superposition of the macroscopic dynamics produced by each of such classes. Within this interpretation, the ’s can be understood as the proportions of the various classes of vehicles present in the traffic stream. This provides a point of view on multi-class traffic models based on structural uncertainties in the microscopic composition of the traffic “mixture”. We mention that other multi-class traffic models are already present in the literature, see e.g. [4, 27], which also explain the scattering of the fundamental diagram by appealing to similar physical motivations but different mathematical formalisations.

Let us assume, for instance, that takes only the two values , with probability , and , with probability . The largest value of , i.e. , may represent e.g., lorries, which accelerate less especially at high traffic density, cf. (3). Conversely, the smallest value of , i.e. , may represent e.g., cars. We consider two different compositions of the traffic stream: first, one with of cars and of lorries; then, the opposite one with of cars and of lorries. In Figure 2, we see that both the fundamental diagram and the region (12) change realistically according to the composition of the traffic stream. In both cases, we notice that the scattering of the fundamental diagram is quite limited at low density, i.e. in the so-called free flow regime, when the flux grows almost linearly with . It becomes more marked at higher density, i.e. in the so-called congested flow regime, when the flux decreases non-linearly with . This is very nicely in agreement with the typical experimental observations, see e.g. [17, Chapter 2] and also [12, 29].

As a second example, let us consider , i.e. the case in which is a continuous random variable uniformly distributed in the interval with . Hence

 Ψ(z)=1b−aχ(a≤z≤b),

where denotes the characteristic function (specifically, if while otherwise). Using (3) and (9), from (10) we compute:

 ¯V∞(ρ)=2√3(b−a)log(1−ρ)⎡⎣arctan(2x−1√3)∣∣∣x=(1−ρ)bx=(1−ρ)a⎤⎦.

Also the variance can be given an explicit representation, indeed from (11) we obtain:

 ς2∞(ρ)=13(b−a)log(1−ρ)[√x−2x−√x+1+2√3arctan(2√x−1√3)∣∣ ∣∣x=(1−ρ)2bx=(1−ρ)2a−¯V2∞(ρ).

Figure 3 shows the area defined by (12) for , i.e. and in the formulas above. Also in this case, we notice that the computed scattering of the fundamental diagram reproduces correctly the qualitative empirical features mentioned in the previous example.

### 2.3 Fokker-Planck description and equilibria

Besides the fundamental diagram of traffic, from the kinetic description of the microscopic dynamics (5) we may also obtain information on the statistical distribution of the speed of the vehicles emerging when interactions are in equilibrium, i.e. for large times ( in the limit). Such a distribution corresponds to the Maxwellian distribution of the classical kinetic theory of gases.

By definition, the equilibrium distribution, say , makes the right-hand side of (7) vanish:

 ∫10∫10⟨φ(v′)−φ(v)⟩f∞(v;z)f∞(v∗;z)dvdv∗=0

for all observable quantities . Nevertheless, it is in general difficult to determine by tackling directly this equation, at least from the analytical point of view. If, however, the interactions (5) are quasi-invariant, i.e. if they produce each time a small change of speed, then it is possible to approximate asymptotically the Boltzmann-type integro-differential equation (7) with a Fokker-Planck partial differential equation, which may allow for an explicit computation of the steady states.

Quasi-invariant interactions are reminiscent of the grazing collisions introduced in the classical kinetic theory [35, 36]. In our context, they amount to assuming in (5), so that both the deterministic part of the interactions and the stochastic fluctuations are small. Since the interest is in the large time behaviour of the system, it is convenient to introduce parallelly the new time scale

 τ:=γ2t, (13)

which is indeed much larger than the characteristic -scale of the binary interactions and therefore more suited to catch directly the asymptotic trend. Denoting by

 g(τ,v;z):=f(2γτ,v;z) (14)

the time-scaled distribution function, from (7) we obtain that satisfies the equation

 ddτ∫10φ(v)g(τ,v;z)dv=1γ∫10∫10⟨φ(v′)−φ(v)⟩g(τ,v;z)g(τ,v∗;z)dvdv∗.

Next, assuming that is sufficiently smooth and expanding in Taylor’s series up to the third order about (because in the quasi-invariant regime), we get

 ddτ∫10φ(v)g(τ,v;z)dv =∫10∫10φ′(v)I(v,v∗;z)g(τ,v;z)g(τ,v∗;z)dvdv∗ =+σ22γ∫10φ′′(v)D2(v)g(τ,v;z)dv+Rγ,σ2φ(g,g)(τ;z),

where is given by (2) and is a remainder which depends on and on .

Let us now consider the quasi-invariant limit . From (13), we observe that, for each fixed , if is small then is large, hence the limit describes the large time trend of the system. Moreover, we assume that , so as to observe, in the limit, a balanced contribution of the deterministic part of the interactions and of the stochastic fluctuation. Since it can be proved that for , see [34] for details, we have that the asymptotic equation satisfied by is

 ddτ∫10φ(v)g(τ,v;z)dv =∫10∫10φ′(v)I(v,v∗;z)g(τ,v;z)g(τ,v∗;z)dvdv∗ =+λ2∫10φ′′(v)D2(v)g(τ,v;z)dv,

i.e., integrating by parts the right-hand side, assuming that the boundary terms produced by the integration-by-parts vanish [34] and invoking the arbitrariness of :

 ∂τg=λ2∂2v(D2(v)g)−∂v[(P(1+(1−P)U)−v)g], (15)

where we have substituted the expression (2) of and where

 U(τ;z):=∫10vg(τ,v;z)dv=V(2γτ;z) (16)

is the time-scaled mean speed, cf. (14).

Equation (15) is the announced Fokker-Planck equation, which describes the large time trend of the system in the quasi-invariant regime. Since, owing to (14), the equilibrium distribution is well approximated by , we look for the stationary solutions to (15), which solve

 λ2∂v(D2(v)g∞)−(V∞−v)g∞=0,

where is the asymptotic mean speed (9). In particular, if we choose111The function (17) does not actually comply with the requirement in (6), due to the vertical tangents at . Nevertheless, such a may be obtained, for , as the uniform limit of a sequence of functions complying with (6), cf. [31]. Therefore, its use is justified in (15) after performing the quasi-invariant limit.

 D(v):=√v(1−v) (17)

we obtain that the unique stationary solution with unitary mass is

 g∞(v;z)=v2V∞(ρ;z)λ−1(1−v)2(1−V∞(ρ;z))λ−1Beta(2V∞(ρ;z)λ,2(1−V∞(ρ;z))λ), (18)

where is the beta function. We notice that (18) is a beta probability density function with mean , as expected, and variance . Interestingly, beta probability densities have been found to fit particularly well the experimental data on the speed distribution of the vehicles [19, 20].

Actually, (18) is a family of equilibrium distributions parametrised by . In order to quantify the intrinsic uncertainty in this description of the equilibrium, we may refer to the mean equilibrium distribution:

 ¯g∞(v):=Ez(g∞(v;z))=∫R+g∞(v;z)Ψ(z)dz

and to its variance

 Varz(g∞(v;z)):=∫R+g2∞(v;z)Ψ(z)dz−¯g2∞(v).

Notice, from (10), that

 ¯V∞(v)=∫10v¯g∞(v)dv.

Figures 45 show , together with the uncertainty bands determined by its standard deviation , for various discrete and continuous probability distributions of . The curves have been computed numerically by means of suitable quadrature formulas in , starting from the analytical expression (18). We observe that, depending on the distribution of , in the mean distribution several local peaks may appear, which suggest a slightly multi-modal average trend of the equilibrium speed distribution. Such a trend is actually not found in any of the distributions (18) for fixed but, interestingly, it is documented in experimental data [19, 20].

## 3 Uncertainty damping by driver-assist control

In this section, we tackle the problem of damping the scattering of the fundamental diagram, i.e., owing to (12), of reducing the variance . Although the scattering is a macroscopic manifestation of the traffic, its origin relies in the structural uncertainty affecting the microscopic interactions (5). Therefore, we will proceed to control a priori (some of) the microscopic interactions (5), assessing then the aggregate impact of such a control, rather than trying to constrain a posteriori the macroscopic flow of the vehicles as a whole. Indeed, in our view, actually feasible ways of controlling a multi-agent system, such as vehicular traffic, make use of bottom-up strategies, which target the simple microscopic interactions, rather than of strategies pointing directly to the aggregate behaviour. In the context of vehicular traffic, such bottom-up strategies refer to the control of the local responses given by the driver-vehicle system to the external stimuli coming from other vehicles by means of Advanced Driver-Assistance Systems (ADAS).

Let us consider, therefore, the following controlled binary interaction:

 v′=v+γ(I(v,v∗;z)+Θu)+D(v)η,v′∗=v∗, (19)

where is given by (2), is a control to be determined from the optimisation of a suitable functional and is a Bernoulli random variable taking into account whether the randomly selected -vehicle is or is not equipped with the ADAS technology. In particular, the law of is

 P(Θ=1)=p,P(Θ=0)=1−p,

where is the so-called penetration rate, i.e. the percentage of ADAS vehicles in the traffic stream.

We propose, in the following, two different functionals, whose minimisation yields an instantaneous optimal control to be used in (19). These functionals correspond, in principle, to different control strategies, which however, as we will see, lead to the same macroscopic effect. Remarkably, one of such strategies has an eminently theoretical interest, whereas the other is closer to an actual implementability.

### 3.1 Pointwise uncertainty control

Let be an optimal speed determined as a function of the vehicle density and suggested to the ADAS vehicles by e.g. some sensors along the road, which are able to detect the level of traffic congestion. Then, an optimal control may be deduced with the aim of keeping the post-interaction speed of an ADAS vehicle as close as possible to the optimal speed . This is obtained by minimising e.g. the following functional:

 J1(v′,u;z):=12⟨(v0(ρ)−v′)2+νu2⟩

subject to the constraint (19). Notice that depends pointwise on the uncertain parameter through . Moreover, , with , is a quadratic penalisation term, which expresses the cost of the control.

The minimisation of may be carried out by forming the Lagrangian:

 L1(v′,u,λ;z):=J1(v′,u;z)+λ⟨v′−v−γ(I(v,v∗;z)+Θu)−D(v)η⟩,

where is the Lagrange multiplier, and then by considering the optimality conditions:

 ⎧⎨⎩∂v′L1=λ−⟨v0(ρ)−v′⟩=0∂uL1=νu−γΘλ=0∂λL1=⟨v′−v−γ(I(v,v∗;z)+Θu)−D(v)η⟩=0,

whence we deduce the optimal control

 u∗(v,v∗;z)=γΘν+γ2Θ2(v0(ρ)−v−γI(v,v∗;z)). (20)

Notice that, for each , this is a feedback control, because it is expressed as a function of the pre-interaction speeds , . Plugged into (19), it produces the controlled binary interactions

 (21)

#### 3.1.1 Physical admissibility of the interactions (21)

The physical admissibility of the interactions (21) requires for all . Considering the expression (2) for , together with the observations made in Section 2.1.1 plus and , we see that a sufficient condition for is

 ν(1−γ)ν+γ2v+D(v)η≥0,

which is certainly satisfied if there exists a constant such that

 η≥cν(γ−1)ν+γ2,cD(v)≤v.

Likewise, since , a sufficient condition for is

 ν(γ−1)ν+γ2(1−v)+D(v)η≤0,

which is certainly satisfied if

 η≤cν(1−γ)ν+γ2,cD(v)≤1−v.

On the whole, (21) is physically admissible if there exists such that

 |η|≤cν(1−γ)ν+γ2,cD(v)≤min{v,1−v}. (22)

Again, this implies that the random variable has to be bounded and that the diffusion coefficient vanishes at . Notice that if , i.e. if the control is so penalised that the optimum is not to control the interactions (, cf. (20)), then we recover exactly (6).

#### 3.1.2 Boltzmann-type kinetic description

Since, under (22), all the interactions (21) are physically admissible, the evolution of the distribution function is described by a Boltzmann-type equation for Maxwellian-like particles analogous to (7). The only difference is that, in the “collisional” term appearing on the right-hand side of (7), we here need to take into account a further average with respect to the random variable . Thus we write:

 ddt∫10φ(v)f(t,v;z)dv=12EΘ[∫10∫10⟨φ(v′)−φ(v)⟩f(t,v;z)f(t,v∗;z)dvdv∗], (23)

where is given by (21).

In particular, we are interested in the trend of the mean speed , which we obtain with and considering furthermore that appearing in (21) is a Bernoulli random variable with parameter , because so is by definition. Hence:

 dVdt=γ2⋅ν+(1−p)γ2ν+γ2[P(ρ;z)(1−V)−(1−P(ρ;z))2V]+γ2⋅pν+γ2(v0(ρ)−V). (24)

In order to better characterise the role of the various parameters in the large time trend of the mean speed, it is useful to refer to the quasi-invariant interaction regime introduced in Section 2.3. In this case, besides sending and to zero, we need to take in such a way that , so as to observe, in the limit, also a balanced contribution of the controlled part of the interactions. From (24), we deduce that the equation satisfied by the time-scaled mean speed , cf. (16), is

 dUdτ=P(ρ;z)(1−U)−(1−P(ρ;z))2U+p∗(v0(ρ)−U), (25)

where

 p∗:=pκ

is what we call the effective penetration rate. Hence, we deduce that the equilibrium mean speed is

 U∞(ρ;z):=P(ρ;z)+p∗v0(ρ)P(ρ;z)+(1−P(ρ;z))2+p∗. (26)

Since , we have222We use the notation to mean that there exists a constant , whose exact value is unimportant, such that .

 |U∞(ρ;z)−v0(ρ)|=∣∣P(ρ;z)+(P(ρ;z)+(1−P(ρ;z))2)v0(ρ)∣∣P(ρ;z)+(1−P(ρ;z))2+p∗≲1p∗, (27)

which shows that the instantaneous control (20) has the effect of keeping the average traffic stream close to the suggested optimal speed. From this, we easily deduce the important consequence:

 ς2∞(ρ)=Varz(U∞(ρ;z))=Varz(U∞(ρ;z)−v0(ρ))≤Ez((U∞(ρ;z)−v0(ρ))2)≲1(p∗)2, (28)

meaning that the control reduces indeed the scattering of the fundamental diagram caused by the uncertain parameter . The magnitude of such a reduction is estimated by the inverse of , hence it is higher for either a higher penetration rate or a lower control penalisation .

Interestingly, this result does not actually depend on the specific form (2) of the interaction function . A fruitful generalisation is enlightened by the following theorem:

###### Theorem 3.1.

In (21), let the interaction function be bounded, i.e. let a constant exist such that

 |I(v,v∗;z)|≤Imax∀v,v∗∈[0,1],z∈R+.

Assume, moreover, that is such that the interactions (21) are all physically admissible in the sense discussed in Section 3.1.1. Then

 ς2∞(ρ)≲1(p∗)2.
###### Proof.

We observe that it suffices to prove an estimate like (27), for then the thesis follows immediately from (28), which holds independently of the specific form of .

Since all interactions (21) are physically admissible by assumption, we may use the Boltzmann-type equation (23). For we find then

 dVdt=γ2[∫10∫10(1−pγ2ν+γ2)I(v,v∗;z)f(t,v;z)f(t,v∗;z)dvdv∗+pγν+γ2(v0(ρ)−V)],

which, in the quasi-invariant limit, becomes

 dUdτ=∫10∫10I(v,v∗;z)g(τ,v;z)g(τ,v∗;z)dvdv∗+p∗(v0(ρ)−U).

Since is constant in time, we have , therefore we may write

 ddτ(U−v0(ρ))+p∗(U−v0(ρ))=∫10∫10I(v,v∗;z)g(τ,v;z)g(τ,v∗;z)dvdv∗,

whence, integrating in time,

 U(τ;z)−v0(ρ) =e−p∗τ(U0(z)−v0(ρ)) =+∫τ0e−p∗(τ−s)∫10∫10I(v,v∗;z)g(s,v;z)g(s,v∗;z)dvdv∗ds,

with .

From the boundedness of we further obtain

 ∣∣∣∫10∫10I(v,v∗;z)g(s,v;z)g(s,v∗;z)dvdv∗∣∣∣≤Imax,

thus we discover:

 |U(τ;z)−v0(ρ)| ≤e−p∗τ(U0(z)−v0(ρ))+Imax∫τ0e−p∗(τ−s)ds =e−p∗τ(U0(z)−v0(ρ))+Imaxp∗(1−e−p∗τ),

whence finally

 |U∞(ρ;z)−v0(ρ)|=limτ→+∞|U(τ;z)−v0(ρ)|≤Imaxp∗

and we are done. ∎

In Figure 6, we show the fundamental diagram , cf. (26), and its standard deviation , cf. (11), for both a discrete (binomial) and a continuous (uniform) probability distribution of the uncertain parameter and increasing values of the effective penetration rate . The curves have been obtained numerically by means of suitable quadrature formulas in , starting from the analytical expression (26). As predicted by (28), and more in general by Theorem 3.1, the results display a clear damping of the uncertainty at the macroscopic level, thanks to the action of the microscopic control. This is particularly evident in Figure 6(c)-(d), which may be compared with Figure 3 illustrating the related uncontrolled case.

#### 3.1.3 Fokker-Planck description and equilibria

Similarly to Section 2.3, we may obtain more details on the large time trend emerging from the controlled interactions (21) by studying the quasi-invariant limit of the full Boltzmann-type equation (23). We recall that, in this case, this amounts to considering together with and . Proceeding like in Section 2.3, we determine the Fokker-Planck equation

 ∂τg=λ2∂2v(D2(v)g)−∂v[(P(1+(1−P)U)+p∗v0−(1+p∗)v)g], (29)

where is the solution to (25), whence, at the steady state,

 λ2∂v(D2(v)