Multiscale control of generic second order traffic models by driver-assist vehicles
MMultiscale control of generic second order traffic models bydriver-assist vehicles
Felisia Angela Chiarello , Benedetto Piccoli , and Andrea Tosin Abstract
We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order trafficmodel as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introducein the vehicle interactions a binary control modelling the automatic feedback provided bydriver-assist vehicles and we upscale such a new particle description by means of anotherEnskog-based hydrodynamic limit. The resulting macroscopic model is now a Generic SecondOrder Model (GSOM), which contains in turn a control term inherited from the microscopicinteractions. We show that such a control may be chosen so as to optimise global traffic trends,such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By meansof numerical simulations, we investigate the effect of this control hierarchy in some specificcase studies, which exemplify the multiscale path from the vehicle-wise implementation of adriver-assist control to its optimal hydrodynamic design.
Keywords: controlled binary interactions, Enskog-type kinetic description, hydrodynamiclimit, GSOM, instantaneous control
Mathematics Subject Classification:
Vehicular traffic models incorporating the presence of driver-assist or autonomous vehicles aregaining a lot of momentum. The reason is at least twofold: on one hand, Advanced Driver-Assistance Systems (ADAS), like all technological innovations, call naturally for a quantitativemathematical approach to their understanding and design. On the other hand, ADAS pose newtheoretical problems, which motivate interesting developments of mathematical techniques in quitechallenging realms such as the one of Artificial Intelligence.In the literature, several mathematical models at various scales may already be found. Withoutpretending to be exhaustive, we mention that in [31] microscopic vehicle-wise control models arereviewed while in [13] the contribution of adaptive cruise control systems is included in a secondorder hydrodynamic traffic model. The model is then extended in [14] to the case of multilanetraffic. In [21, 29] a hybrid microscopic-macroscopic description, inspired by the one introducedfor moving bottlenecks [15, 27] and crowd dynamics [10], is used to simulate a few individu-ally controlled autonomous vehicles within a continuous traffic stream modelled by the Lighthill-Whitham-Richards traffic equation [30, 34]. In [32, 37, 39] a Boltzmann-type kinetic approach isproposed to account statistically for the presence of driver-assist vehicles in hydrodynamic trafficmodels and study their impact on mesoscopic traffic features, such as e.g., the local mean speedand speed variability.Although mathematically different, these models share and convey the idea that driver-assistand autonomous vehicles do not only enhance driver comfort and safety, which were the primary1 a r X i v : . [ phy s i c s . s o c - ph ] A ug oals for which they were conceived. They also impact in a non-negligible manner on the globaltraffic flow, to such an extent that one may realistically imagine to take advantage of them as inner traffic controllers , as also confirmed by recent field experiments [35]. In a traffic stream composedmostly of human-manned vehicles but including a certain percentage (the so-called penetrationrate ) of driver-assist vehicles, they make possible an effective bottom-up control of traffic trendsby exploiting simply the physiological vehicle-to-vehicle interactions. No particular top-down rulesimposed by outer traffic controllers are required, whose efficiency would strongly depend also onthe hardly controllable voluntary observance by individual drivers.Inspired by these arguments, in this paper we pursue the research line set up in the alreadycited papers [32, 37, 39]. In particular, we aim to derive high order macroscopic traffic modelsaccounting for the presence of driver-assist vehicles to be used as traffic optimisers. The noveltiesof our contribution with respect to the aforementioned literature may be summarised as follows:i) unlike [13, 14], we do not postulate the modifications needed in classic hydrodynamic equationsof traffic to reproduce the impact of driver-assist vehicles. Instead, we derive them rigorouslyfrom an organic upscaling of microscopically controlled particle dynamics; ii) unlike [21, 29], wedo not regard driver-assist vehicles as point particles, viz. singularities, in a continuous trafficstream. Instead, we derive genuinely macroscopic particle-free models, in which the contribution ofdriver-assist vehicles is naturally consistent with the upscaling of the whole system; iii) unlike [32,37, 39], we derive hydrodynamic models of order greater than one, which may better account fortraffic perturbations and instabilities [33], and we design driver-assist control algorithms havingin mind multiscale optimisation criteria.The mathematical literature offers several techniques to derive macroscopic descriptions of mi-croscopic particle systems, many of them dealing just with vehicular traffic. As an example, wemention micro-macro many particle limits [8, 11, 16, 17, 23] and mean-field limits [4, 6]. Never-theless, when it comes to controlled particle systems classical approaches become more delicateand difficult due to smoothness issues in the upscaling of the control, see e.g., [1, 20]. In thispaper, we adopt a “collisional” kinetic technique, which is particularly suited to vehicular trafficand is basically free from the technical difficulties just mentioned. Consistently with the classicalkinetic approach, we describe the interactions among the vehicles by means of binary algebraicrules relating instantaneously the post-interaction states of any two interacting vehicles to theirpre-interaction states. We notice that binary interactions fit well the follow-the-leader particle de-scription classically used in vehicular traffic [22]. With a probability depending on the penetrationrate of driver-assist vehicles, these interactions include furthermore a control term. Therefore, atthe particle level we deal with binary control problems, which can be easily solved in feedbackform: the optimal control can be computed explicitly as a function of the pre-interaction statesof the interacting vehicles out of the optimisation of a binary cost functional related to theirreciprocal distance ( headway ). As a result, we obtain an explicit characterisation of controlledinteractions, that we subsequently upscale taking advantage of the classical statistical approach ofkinetic theory. In doing so, we adopt in particular an Enskog-type kinetic description rather thana more common Boltzmann-type one like in [32, 37, 39]. Indeed, the Enskog description allows usto properly take into account the fact that the interacting vehicles do not occupy the same spatialposition, which is at the basis of the correct reproduction of the anisotropic propagation of trafficwaves in high order macroscopic models [3, 12, 26].The obtained macroscopic description with driver-assist vehicles consists in a second ordermodel belonging to the GSOM class [2, 28], which includes as particular cases also the celebratedAw-Rascle-Zhang model [3, 42] and its generalised version (GARZ) [19]. This model keeps trackof the vehicle-wise control in several aspects but notably in a structural parameter of the control,corresponding to a recommended headway , which enters the hydrodynamic equations. In orderto fix the recommended headway, we propose to set up a further control problem, directly atthe macroscopic scale, where this parameter itself plays the role of a control variable for theoptimisation of certain cost functionals related to macroscopic traffic features, such as e.g., thevehicle flux and the road congestion. The background idea is to investigate the possibility todesign multiscale control algorithms for single vehicles which, once embedded in the collectiveflow, produce bottom-up optimisations of the whole traffic stream.2n more details, the paper is organised as follows. In Section 2, we illustrate the generalprocedure to derive high order macroscopic traffic models from a generic follow-the-leader particledescription via the Enskog-type kinetic approach and its hydrodynamic limit. In Section 3, weintroduce controlled microscopic vehicle interactions and we apply the previous procedure to obtainthe corresponding bottom-up controlled macroscopic description in terms of GSOMs. In Section 4,we tackle the problem of designing the parameters of the vehicle-wise control in such a wayto pursue hydrodynamic optimisations. In Section 5, we show the numerical results producedin some case studies by the macroscopic model with optimally controlled driver-assist vehiclesand we compare them with those produced by the more standard GARZ model. As previouslyanticipated, the latter is in turn a GSOM but in our context we may interestingly interpret it as amodel without driver-assist vehicles or alternatively with driver-assist vehicles which do not obeyany specific hydrodynamic optimisation criterion. Finally, in Section 6, we draw some conclusionsand we briefly sketch future research prospects.
We begin by showing how hydrodynamic traffic models of order higher than 1 can be derived froman elementary description of pairwise interactions among the vehicles using a kinetic formalism.This derivation will be the basis to include subsequently a microscopic binary control in theinteractions and upscale it at the level of the global macroscopic flow of vehicles.
We begin by considering a generic Follow-the-Leader (FTL) formulation of microscopic trafficdynamics: ˙ x i = V (cid:18) x i +1 − x i , ω i (cid:19) , ˙ ω i = 0 , (1)where: (i) x i , x i +1 ∈ R , x i < x i +1 , are the dimensionless positions of two consecutive vehicles inthe traffic stream; (ii) ω i ∈ Ω ⊆ R + is the so-called Lagrangian marker , i.e. a characteristic ofthe driving style of the drivers, which remains constant in time for each driver. In most cases, ω i is interpreted as the maximum speed of the i th driver; (iii) V ∈ [0 ,
1] is the dimensionless speedof a vehicle expressed as a function of the distance from the leading vehicle and the Lagrangianmarker. Denoting s i := x i +1 − x i ∈ R + the headway between the i th and ( i + 1)th vehicles, wecan restate the model as ˙ s i = V (cid:18) s i +1 , ω i +1 (cid:19) − V (cid:18) s i , ω i (cid:19) , ˙ ω i = 0 . (2) Assumption 2.1.
We assume that:(i) ∂ s V ( s , ω ) > ∀ ( s, ω ) ∈ R + × Ω ( );(ii) ∃ C > V (cid:0) s , ω (cid:1) ≤ Cs , ∀ ( s, ω ) ∈ R + × Ω. We point out that, here and henceforth, the notation ∂ s V ( s , ω ) stands for ∂ s V (cid:18) s , ω (cid:19) := ( ∂ s V ) (cid:18) s , ω (cid:19) = − s ( ∂ σ V ) (cid:18) s , ω (cid:19) , where σ denotes the first variable of the function V . In practice, we consider V = V ( σ, ω ) along with the composition σ ( s ) = s and we take the derivatives accordingly. emark . In different derivations of high order macroscopic traffic models from the FTL de-scription (1), see e.g., [7, 19], further assumptions are made on the function V , which however arenot needed in the present context.A function V complying with Assumption 2.1 is V (cid:18) s , ω (cid:19) = ωsa + s (3)with a > ω ∈ Ω := [0 , γ >
0, understood as the reaction time of the drivers. Denoting by s := s i ( t ), s ∗ := s i +1 ( t ) the pre-interaction headways and by s (cid:48) := s i ( t + γ ), s (cid:48)∗ := s i +1 ( t + γ ) the post-interaction headways, and using an analogous notation for the Lagrangian markers, we get s (cid:48) = s + γ (cid:20) V (cid:18) s ∗ , ω ∗ (cid:19) − V (cid:18) s , ω (cid:19)(cid:21) , ω (cid:48) = ω, (4)that we may further complement with s (cid:48)∗ = s ∗ to express the anisotropy of vehicle interactions, inparticular the fact that the leading vehicle is not affected by the rear vehicle.For physical consistency, the interaction (4) has to guarantee s (cid:48) ≥ s, s ∗ ≥ ω, ω ∗ ∈ Ω. Thanks to Assumption 2.1(ii), we easily see that this condition is met if γ ≤ C . The aggregate outcome of the microscopic binary interactions (4) may be investigated througha kinetic approach upon introducing the distribution function f = f ( t, x, s, ω ) ≥ f ( t, x, s, ω ) dx ds dω gives, at time t > , the fraction of vehicles located in the interval [ x, x + dx ]with headway comprised in [ s, s + ds ] and Lagrangian marker in [ ω, ω + dω ].In this work, we assume that f satisfies an Enskog-type kinetic equation rather than a moreclassical Boltzmann-type equation. The inspiration comes from [18, 25, 26], where it is stressedthat traffic models derived from a Boltzmann-type kinetic description cannot reproduce backwardwave propagation because in a Boltzmann-type equation the interacting vehicles are assumedto occupy the same space position. Conversely, in an Enskog-type kinetic description they areassumed to occupy two different positions, which in our case is also particularly consistent withthe fact that their microscopic state includes the headway, viz. the reciprocal distance. We writetherefore: ∂ t f + V (cid:18) s , ω (cid:19) ∂ x f = Q E ( f, f ) , (5)where Q E ( f, f ) is the Enskog collision operator. The precise definition of Q E ( f, f ) is bettergiven in weak form, i.e. through its action on an arbitrary macroscopic observable (test function) φ = φ ( s, ω ):( Q E ( f, f ) , φ ) := 12 (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x + s, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ , (6)where s (cid:48) , ω (cid:48) are given by (4). Notice that the two distribution functions describing the interactingvehicles are computed in x and x + s , respectively. Indeed, if s is the headway of the rear vehiclelocated in x then the leading vehicle is located in x + s .In order to make (5), (6) more amenable to analytical investigations, it is useful to approximate f ( t, x + s, s ∗ , ω ∗ ) ≈ f ( t, x, s ∗ , ω ∗ ) + ∂ x f ( t, x, s ∗ , ω ∗ ) s, (7)4hich, for s sufficiently small, coincides with the first order Taylor expansion of f in x . Then (6)takes the form( Q E ( f, f ) , φ ) = 12 (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ + 12 (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) ∂ x f ( t, x, s ∗ , ω ∗ ) s ds ds ∗ dω dω ∗ =: ( Q ( f, f ) , φ ) + ( Q ( f, s∂ x f ) , φ ) . (8)The first term on the right-hand side, i.e. Q ( f, f ), is now a classical Boltzmann-type collisionoperator with the two distribution functions computed in the same point x . The second term Q ( f, s∂ x f ) is instead a first order correction, which will be fundamental to recover consistentmacroscopic models.The passage from (5) to a macroscopic traffic description is performed via the so-called hydro-dynamic limit . Let 0 < η (cid:28) t → tη , x → xη , (9)which formalises the passage from microscopic to macroscopic time and space scales. Then (5), (8)become ∂ t f + V (cid:18) s , ω (cid:19) ∂ x f = 1 η Q E ( f, f ) (10)and( Q E ( f, f ) , φ ) = 12 (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ + η (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) ∂ x f ( t, x, s ∗ , ω ∗ ) s ds ds ∗ dω dω ∗ = ( Q ( f, f ) , φ ) + η ( Q ( f, s∂ x f ) , φ )(for simplicity, we still denote by f = f ( t, x, s, ω ) the distribution function in the scaled time andspace variables). Hence, Q E ( f, f ) = Q ( f, f ) + ηQ ( f, s∂ x f ), which plugged into (10) yields ∂ t f + V (cid:18) s , ω (cid:19) ∂ x f = 1 η Q ( f, f ) + Q ( f, s∂ x f ) . (11)Owing to the smallness of η , this equation can be split in two contributions. On one hand, local interactions among the vehicles, which take place on a microscopic (quick) time scale and reachrapidly the equilibrium: ∂ t f = Q ( f, f ) (12)(we have scaled the time back to the microscopic scale as t → ηt using the factor η in front of thecollision operator); on the other hand, a transport of the local equilibrium distribution generatedby (12) on a hydrodynamic (slow) time scale: ∂ t f + V (cid:18) s , ω (cid:19) ∂ x f = Q ( f, s∂ x f ) . (13)Here, we use the local Maxwellian , viz. the equilibrium distribution produced by (12), to obtainthe macroscopic evolution of the hydrodynamic parameters locally conserved by the interactions.5 .3 Hydrodynamic limit The first step of the strategy just outlined is the study of the local equilibrium distribution resultingfrom (12). In weak form, (12) reads ∂ t (cid:90) Ω (cid:90) R + φ ( s, ω ) f ( t, x, s, ω ) ds dω = 12 (cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ . (14)Choosing φ ( s, ω ) = 1 and defining the macroscopic density of the vehicles in x as ρ ( t, x ) := (cid:90) Ω (cid:90) R + f ( t, x, s, ω ) ds dω we immediately observe that ρ is conserved in time by the local interactions. Likewise, choosing φ ( s, ω ) = s and defining the mean headway in x as h ( t, x ) := 1 ρ ( t, x ) (cid:90) Ω (cid:90) R + sf ( t, x, s, ω ) ds dω we see that also h is locally conserved in time owing to (4). Finally, choosing φ ( s, ω ) = ω anddefining the mean Lagrangian marker in x as w ( t, x ) := 1 ρ ( t, x ) (cid:90) Ω (cid:90) R + ωf ( t, x, s, ω ) ds dω we obtain from (4) that also w is locally conserved in time. We conclude that φ ( s, ω ) = 1 , s, ω are “collisional invariants” and therefore that the local Maxwellian will be parametrised by thehydrodynamic quantities ρ , h , w .More in general, choosing in (14) a macroscopic observable φ ( s, ω ) = ψ ( ω ) independent of s and using (4) we deduce ∂ t (cid:90) Ω (cid:90) R + ψ ( ω ) f ( t, x, s, ω ) ds dω = 0 , i.e. the whole marginal distribution of ω is locally constant in time. Consequently, the localMaxwellian should be parametrised by all the statistical moments of the ω -marginal. To avoid aninfinite proliferation of hydrodynamic parameters, we assume that the ω -marginal is of the form δ ( ω − w ), where δ denotes the Dirac delta, so that all its moments can be expressed in terms of w . This leads us to consider a distribution function f of the form f ( t, x, s, ω ) := ρ ( x ) g h ( x ) ( t, s ) δ ( ω − w ( x )) , (15)where g h is the marginal of s parametrised by the conserved mean headway h : (cid:90) R + g h ( t, s ) ds = 1 , (cid:90) R + sg h ( t, s ) ds = h ∀ t ≥ . We point out that in (15) we have omitted the dependence of ρ , h , w on t because these hydro-dynamic parameters are constant on the time scale of the microscopic interactions.Plugging (15) into (14) and choosing a macroscopic observable φ ( s, ω ) = ϕ ( s ) independent of ω we deduce the following equation for g h : ddt (cid:90) R + ϕ ( s ) g h ( t, s ) ds = ρ (cid:90) Ω (cid:90) R ( ϕ ( s (cid:48) ) − ϕ ( s )) g h ( t, s ) g h ( t, s ∗ ) δ ( ω − w ) δ ( ω ∗ − w ) ds ds ∗ dω dω ∗ , (16)6hich, in view of (4), admits g ∞ h ( s ) := δ ( s − h ) (17)as an equilibrium distribution. Indeed, a direct calculation shows that such a g ∞ h makes the right-hand side of (16) vanish. In general, (17) may not be the only possible equilibrium distributionof (16) under the interaction rules (4) due to the arbitrariness of the speed function V . In thefollowing we prove however that (17) is the unique equilibrium distribution at least in a particularregime of the parameters of the interactions (4), which allows us to identify a “universal” trendsubstantially independent of V .Let us consider quasi-invariant interactions , namely interactions which induce a small changeof the microscopic state of the vehicles. This concept is inspired by the grazing collisions of theclassical kinetic theory [40, 41] and has been introduced in the kinetic theory of multi-agent systemin [9]. In (4), this is the case if e.g., V ( s ∗ , ω ∗ ) − V ( s , ω ) is small so that s (cid:48) ≈ s . Let us assumethat V is parametrised by a parameter (cid:15) > V (cid:18) s , ω (cid:19) ∼ (cid:15)c ( ω ) s for (cid:15) → + , where c ( ω ) ≥ ω . This implies that there exists a function V (cid:15) ( s ) such that V (cid:15) ( s ) → (cid:15) → + and V (cid:18) s , ω (cid:19) = (cid:15)c ( ω ) s V (cid:15) ( s ) . (18)We will further assume that V (cid:15) ( s ) is bounded for all (cid:15) > s ∈ R + . For example, if we let a = (cid:15) then the function V given in (3) satisfies (18) with c ( ω ) = ω and V (cid:15) ( s ) = (cid:15)s .Obviously, with the sole assumption of small (cid:15) we cannot observe any interesting universal trendof the interactions towards the equilibrium. Indeed, in the limit (cid:15) → + we simply get s (cid:48) = s in (4),which implies definitively a constant solution f to (14) coinciding with the arbitrarily chosen initiallocal distribution. To compensate for the smallness of (cid:15) we increase simultaneously the frequencyof the interactions as (cid:15) , so as to balance the small transfer of microscopic state from one vehicleto another in a single interaction with a high number of such interactions per unit time. Hence,in the quasi-invariant regime we consider (16) in the form ddt (cid:90) R + ϕ ( s ) g h ( t, s ) ds = ρ (cid:15) (cid:90) Ω (cid:90) R ( ϕ ( s (cid:48) ) − ϕ ( s )) g h ( t, s ) g h ( t, s ∗ ) δ ( ω − w ) δ ( ω ∗ − w ) ds ds ∗ dω dω ∗ . (19)Notice that the scaling of the interaction frequency does not affect either the equilibrium distri-butions or the conservation of h . The first statistical moment of g h ( x ) which in general is notconserved by the microscopic interactions is still the second moment, namely the energy E ( t ) := (cid:90) R + s g h ( t, s ) ds, whose trend is provided by (19) with ϕ ( s ) = s : dEdt = γρ(cid:15) (cid:90) R s (cid:20) V (cid:18) s ∗ , w (cid:19) − V (cid:18) s , w (cid:19)(cid:21) g h ( t, s ) g h ( t, s ∗ ) ds ds ∗ + γρ (cid:15) (cid:90) R (cid:20) V (cid:18) s ∗ , w (cid:19) − V (cid:18) s , w (cid:19)(cid:21) g h ( t, s ) g h ( t, s ∗ ) ds ds ∗ . Recalling (18), this yields dEdt = γρc ( w ) (cid:90) R s ( s ∗ V (cid:15) ( s ∗ ) − s V (cid:15) ( s )) g h ( t, s ) g h ( t, s ∗ ) ds ds ∗ γ(cid:15) ρc ( w ) (cid:90) R ( s ∗ V (cid:15) ( s ∗ ) − s V (cid:15) ( s )) g h ( t, s ) g h ( t, s ∗ ) ds ds ∗ and finally, passing to the limit (cid:15) → + by dominated convergence to obtain a universal trend forsmall (cid:15) , dEdt = γρc ( w )( h − E ) . (20)From this equation we deduce E → h for t → + ∞ , thus the variance E − h of the equilibriumdistribution g ∞ h vanishes asymptotically. This proves that (17) is the unique distribution towardswhich the system converges for large times in the quasi-invariant regime.Motivated by these arguments, we finally consider the following local Maxwellian as the resultof the local interaction step (12): M ρ,h,w ( s, ω ) = ρδ ( s − h ) ⊗ δ ( ω − w ) . (21) Macroscopic equations are obtained by plugging the local Maxwellian (21) into (13) to determineevolution equations for the hydrodynamic parameters ρ , h , w : ∂ t M ρ,h,w + V (cid:18) s , ω (cid:19) ∂ x M ρ,h,w = Q ( M ρ,h,w , s∂ x M ρ,h,w ) . (22)We stress that here we need to restore the dependence of the hydrodynamic parameters on timebecause they are in general not constant on the time scale of the hydrodynamic transport.Writing (22) in weak form and using (21) we get ∂ t ( ρφ ( h, w )) + ∂ x (cid:18) ρφ ( h, w ) V (cid:18) h , w (cid:19)(cid:19) = γ ρ h∂ x V (cid:18) h , w (cid:19) ∂ s φ ( h, w ) , whence for φ ( s, ω ) = 1 , ω, s (the collisional invariants) we obtain the third order hydrodynamicsystem ∂ t ρ + ∂ x (cid:18) ρV (cid:18) h , w (cid:19)(cid:19) = 0 ∂ t ( ρw ) + ∂ x (cid:18) ρwV (cid:18) h , w (cid:19)(cid:19) = 0 ∂ t ( ρh ) + ∂ x (cid:18) ρhV (cid:18) h , w (cid:19)(cid:19) = γ ρ h∂ x V (cid:18) h , w (cid:19) . (23)The first two equations express a classical conservative transport of the density of the vehiclesand of their mean Lagrangian marker by the velocity field V . The third equation deserves insteada couple of further comments. First, this additional equation is present because the microscopicinteractions (4) conserve locally also h . Second, it expresses a balance and not a conservation,i.e. the right-hand side is not zero, because of the non-local correction to the vehicle interactionsincluded in the Enskog collision operator (8). Third order models were already occasionally pro-posed in the traffic literature, see [24] for an example, however not within an organic derivationfrom microscopic principles like in this case.System (23) can be written in quasilinear vector form as ∂ t U + A ( U ) ∂ x U = , with U := ( ρ, w, h ) T and A ( U ) := V ( h , w ) ρ∂ ω V ( h , w ) ρ∂ s V ( h , w )0 V ( h , w ) 00 − γ ρh∂ ω V ( h , w ) V ( h , w ) − γ ρh∂ s V ( h , w ) , ∂ s V . The eigenvalues λ , λ , λ and eigen-vectors r , r , r of this matrix are λ = λ = V (cid:18) h , w (cid:19) with r = (1 , , , r = (cid:18) , ∂ s V (cid:18) h , w (cid:19) , ∂ ω V (cid:18) h , w (cid:19)(cid:19) and λ = V (cid:18) h , w (cid:19) − γ ρh∂ s V (cid:18) h , w (cid:19) with r = (cid:16) , , − γ h (cid:17) . Since the eigenvalues are real and A ( U ) is diagonalisable, system (23) is hyperbolic. Nevertheless,since λ = λ it is not strictly hyperbolic. Furthermore, under Assumption 2.1(i) it results λ < λ = λ = V , therefore no characteristic speed is greater than the flow speed. Hence (23)complies with the Aw-Rascle consistency condition [3]. The first and second characteristic fields arelinearly degenerate: ∇ λ · r = ∇ λ · r = 0, thus the associated waves are contact discontinuities.Conversely, the third characteristic field is genuinely nonlinear: ∇ λ · r (cid:54) = 0, hence the associatedwaves are either shocks or rarefactions. In this section, we take advantage of the procedure illustrated in Section 2 to derive similarmacroscopic traffic models incorporating the presence of driver-assist vehicles. At the microscopicscale, the latter are regarded as special vehicles equipped with automatic feedback controllers,which respond locally to the actions of the human drivers with the aim of optimising a certain costfunctional in each binary interaction. We anticipate that the introduction of controlled vehicleswill give rise to second (rather than third) order hydrodynamic models.
To implement the presence of driver-assist vehicles, we restate the interaction rules (4) as follows: s (cid:48) = s + γ (cid:20) V (cid:18) s ∗ , ω ∗ (cid:19) − V (cid:18) s , ω (cid:19) + Θ u (cid:21) , ω (cid:48) = ω. (24)Here, u ∈ R denotes the control applied to the dynamics of a driver-assist vehicle and Θ ∈ { , } is a Bernoulli random variable expressing the fact that a randomly chosen vehicle may or may notbe equipped with a driver-assist technology with a certain probability. In particular, by lettingProb(Θ = 1) = p, Prob(Θ = 0) = 1 − p we mean that p ∈ [0 ,
1] is the percentage of driver-assist vehicles in the traffic stream, namely theso-called penetration rate .Aiming at collision avoidance , the control u is chosen so as to minimise the following costfunctional: J ( s (cid:48) , u ) := 12 (cid:16) ( s d ( ρ, w ) − s (cid:48) ) + νu (cid:17) , (25)where s d ( ρ, w ) ≥ ρ, w and ν > u tries to align the headway of the vehicle to therecommended one, thereby implementing a form of collision avoidance. The optimal control u ∗ ischosen as u ∗ := arg min u ∈U J ( s (cid:48) , u )subject to (24), where U = { u ∈ R : s (cid:48) ≥ } is the set of the admissible controls.9lugging the constraint (24) into (25) and equating to zero the derivative with respect to u ,we deduce the following optimality condition: γ Θ (cid:26) s − s d ( ρ, w ) + γ (cid:20) V (cid:18) s ∗ , ω ∗ (cid:19) − V (cid:18) s , ω (cid:19)(cid:21)(cid:27) + (cid:0) ν + γ Θ (cid:1) u ∗ = 0yielding u ∗ = Θ γν + Θ γ ( s d ( ρ, w ) − s ) − Θ γ ν + Θ γ (cid:20) V (cid:18) s ∗ , ω ∗ (cid:19) − V (cid:18) s , ω (cid:19)(cid:21) . (26)Notice that u ∗ is a feedback control because it is a function of the pre-interaction states s , s ∗ , ω , ω ∗ of the vehicles. This allows us to plug it straightforwardly into (24), whence we obtain thefollowing controlled binary interactions: s (cid:48) = s + γν + Θ γ (cid:26) ν (cid:20) V (cid:18) s ∗ , ω ∗ (cid:19) − V (cid:18) s , ω (cid:19)(cid:21) + Θ γ ( s d ( ρ, w ) − s ) (cid:27) , ω (cid:48) = ω. (27)Finally, we check that u ∗ ∈ U , which amounts to checking the physical admissibility of thecontrolled interaction (27). Recalling Assumption 2.1(ii) and considering that 0 ≤ Θ ≤
1, weeasily see that the condition s (cid:48) ≥ ν ≥ γ − Cγ under the further restriction γ ≤ C already established in Section 2.1. This condition implies thatthere is a physiological lower bound on the cost of the implementation of the driver-assist control,which cannot be assumed too cheap. Remark . If ν → + ∞ then u ∗ = 0. In this case, from (27) we recover the uncontrolledinteraction rules (4). Another case in which we obtain (4) from (27) is if Θ = 0, which correspondsto a vehicle without driver-assist control. The Enskog-type description is the same as the one discussed in Section 2.2 but for the fact thatthe collision operator Q E ( f, f ) takes now into account also the presence of the random parameterΘ in the interaction rules (27). Specifically, the generalisation of (6) to the present case reads( Q E ( f, f ) , φ ) = 12 (cid:42)(cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x + s, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ (cid:43) , where (cid:104)·(cid:105) denotes the expectation with respect to the law of Θ.The same expansion (7) followed by the hyperbolic scaling (9) leads again to (11), where theBoltzmann-type collision operator Q ( f, f ) includes in turn the expectation with respect to Θ:( Q ( f, f ) , φ ) = 12 (cid:42)(cid:90) Ω (cid:90) R ( φ ( s (cid:48) , ω (cid:48) ) − φ ( s, ω )) f ( t, x, s, ω ) f ( t, x, s ∗ , ω ∗ ) ds ds ∗ dω dω ∗ (cid:43) . Choosing φ ( s, ω ) = 1 and φ ( s, ω ) = ψ ( ω ) (a function of ω alone) and using (27) we see that( Q ( f, f ) ,
1) = ( Q ( f, f ) , ψ ( ω )) = 0 , hence the mass of the vehicles as well as any statistical moment of the ω -marginal are locallyconserved by the controlled interactions. Conversely, choosing φ ( s, ω ) = s we discover( Q ( f, f ) , s ) = pγ ρ ν + γ ) ( s d ( ρ, w ) − h ) , dhdt = pγ ρ ν + γ ) ( s d ( ρ, w ) − h ) . (28)We point out that in this equation we are omitting the dependence of ρ, h, w on x for brevity,considering that for local interactions x is a parameter. Moreover, here ρ, w have to be regarded asconstant with respect to t in view of the conservations discussed above. From (28) we deduce that h is no longer conserved by the interactions (27) and, in particular, that it converges exponentiallyfast in time to s d ( ρ, w ) at a rate proportional to the penetration rate p .Out of these arguments, we conclude that an admissible form of the kinetic distribution functionin the local interaction step is f ( t, x, s, ω ) = ρ ( x ) g ( t, s ) δ ( ω − w ( x )) , where the ω -marginal is chosen based on the same considerations as in Section 2.3.1. Conversely,the distribution g now satisfies only the normalisation condition (cid:90) R + g ( t, s ) ds = 1 ∀ t ≥ g can then be written in the form ddt (cid:90) R + ϕ ( s ) g ( t, s ) ds = ρ (cid:42)(cid:90) Ω (cid:90) R ( ϕ ( s (cid:48) ) − ϕ ( s )) g ( t, s ) g ( t, s ∗ ) δ ( ω − w ) δ ( ω ∗ − w ) (cid:43) ds ds ∗ dω dω ∗ for an arbitrary macroscopic observable ϕ depending only on the headway s . From here, we easilycheck that g ∞ ρ,w ( s ) = δ ( s − s d ( ρ, w ))is a possible equilibrium distribution, which, consistently with the discussion set forth above, hasmean s d ( ρ, w ). To prove that this is actually the only possible equilibrium distribution, at least inthe quasi-invariant regime, we perform a quasi-invariant scaling inspired by that of Section 2.3.1.In particular, we assume (18) and we observe that in order for interactions (27) to be quasi-invariant we also need to ensure that the additional term proportional to s d ( ρ, w ) − s gives asmall contribution when the scaling parameter (cid:15) is small. To this end, we may further scale either ν = (cid:15) or p = (cid:15) . In both cases, letting ϕ ( s ) = s we find that the trend of the energy in the quasi-invariant limit (cid:15) → + is ruled exactly by (20), which, together with (28), implies E → s d ( ρ, w )for t → + ∞ .In conclusion, the local Maxwellian that we consider is M ρ,w ( s, ω ) = ρδ ( s − s d ( ρ, w )) ⊗ δ ( ω − w ) . Notice that in this case it is parametrised only by the hydrodynamic quantities ρ, w . As a con-sequence, from the transport step (13) we expect a second order macroscopic traffic model withstate variables ρ, w . Indeed, proceeding like in Section 2.3.2 with φ ( s, ω ) = 1 , ω (the collisionalinvariants) we end up with ∂ t ρ + ∂ x (cid:18) ρV (cid:18) s d ( ρ, w ) , w (cid:19)(cid:19) = 0 ∂ t ( ρw ) + ∂ x (cid:18) ρwV (cid:18) s d ( ρ, w ) , w (cid:19)(cid:19) = 0 , (29)11amely a Generic Second Order Model (GSOM) of the type introduced in [2, 28].A few remarks about model (29) are in order. First, it is strictly hyperbolic provided ∂ ρ s d (cid:54) = 0and complies with the Aw-Rascle consistency condition if ∂ ρ s d ≤
0, indeed its eigenvalues are λ = V (cid:18) s d ( ρ, w ) , w (cid:19) + ∂ s V (cid:18) s d ( ρ, w ) , w (cid:19) ∂ ρ s d ( ρ, w ) , λ = V (cid:18) s d ( ρ, w ) , w (cid:19) . Second, we stress again that, unlike (23) and despite the analogous derivation, it is a secondorder model, the ultimate reason being that the introduction of the control in the microscopicinteractions destroys the local conservation of the mean headway. In particular, when locally inequilibrium the mean headway becomes a function of ρ, w , thus it no longer enters the macroscopicequations. Interestingly, the kinetic derivation of the hydrodynamic models (23), (29) unveils themicroscopic origin of their structural differences. Third, we observe that the penetration rate p of the driver-assist vehicles does not appear explicitly in (29). The reason is again linked to thenon-conservation of the local mean headway: as (28) shows, p affects the rate of convergence of h to its local equilibrium but not the local equilibrium itself. However, it is clear that the time scaleseparation between local interactions and transport, which is at the basis of the hydrodynamiclimit leading to (29), is more or less valid depending on the speed of convergence of the interactionsto the local equilibrium. Thus, (29) is implicitly valid only for a sufficiently high penetration rate p . In other words, it describes universal macroscopic trends of a traffic stream with a large enoughpercentage of driver-assist vehicles. Fourth, with the particular choice s d ( ρ, w ) = 1 ρ , (30)which satisfies ∂ ρ s d < (cid:40) ∂ t ρ + ∂ x ( ρV ( ρ, w )) = 0 ∂ t ( ρw ) + ∂ x ( ρwV ( ρ, w )) = 0 , (31)i.e. the Generalised Aw-Rascle-Zhang (GARZ) model proposed in [19]. Apart from this particularcase, the design of the recommended headway s d ( ρ, w ) will be the specific object of the nextsection. The recommended headway s d appears in the hydrodynamic model (29) in consequence of thefeedback control (26) implemented in the microscopic interaction rules (24) and subsequentlyupscaled via the Enskog-type kinetic description. The idea is now to understand the function s d ( ρ, w ) as a control in the hydrodynamic equations and to design it so as to optimise macroscopic traffic trends, such as the global flux or the global congestion of the vehicles. This correspondsto a multiscale traffic control, which is explicitly implemented at the scale of single vehicles andfinally produces a hydrodynamic optimisation. Remark . In this work, we do not investigate the local mesoscopic (statistical) effects producedby a generic s d . Instead, we refer the interested readers to [32, 37] for thorough analyses of thisaspect.Assume that the space domain of (29) is the interval [ − L, L ], L >
0, with periodic boundaryconditions. This simulates a circular track, a setting often used in real experiments on trafficflow [35, 36]. We consider the following macroscopic functionals to be optimised:i) to maximise the global flux of vehicles we look for a control u = u ( t, x ) which maximises J ρV ( u ) := (cid:90) T (cid:90) L − L (cid:18) ρV (cid:18) u , w (cid:19) − µF ( u ) (cid:19) dx dt (32)12ubject to (29), where, as anticipated, we understand s d ( ρ, w ) as u . Notice that, oncedetermined as u ∗ := arg max J ρV ( u ), the optimal control u ∗ will be expressed in feedbackform as a function of ρ, w , thus it will be suited to play the role of the recommended headway s d ( ρ, w );ii) to minimise the global traffic congestion we look for a control u = u ( t, x ) which minimises J ρ ( u ) := (cid:90) T (cid:90) L − L ( ρ α + µF ( u )) dx dt (33)subject to (29) with the same relationship between u and s d ( ρ, w ) set forth above. Thistime, however, the optimal control is determined as u ∗ = arg min J ρ ( u ).In both (32) and (33) T > F ( u ) is a convexpenalisation function (cost of the control) and µ > α > u represents s d ( ρ, w ), the admissible controls are non-negative functions: u ( t, x ) ≥ t ≥ x ∈ [ − L, L ]. Therefore, the optimisation of the functionals J ρV and J ρ should beperformed under the further constraint u ≥
0, which however typically increases the technicalityof the problem with no particular added value to the model itself. For this reason, we prefer totake into account the non-negativity of the control by choosing a penalisation function definedonly for u ≥
0, so that on the whole both functionals (32), (33) are not defined for u <
0. Aconvex function F complying with this requirement is F ( u ) = u (log u −
1) + 1 , (34)which is also continuous on R + up to letting F (0) := lim u → + F ( u ) = 1 and such that F (cid:48) ( u ) = log u for u > Consistently with the instantaneous response of the driver-assist vehicles to the actions of thehuman drivers, it is reasonable to understand the recommended headway as an instantaneouscontrol strategy . In other words, s d ( ρ, w ) should be defined in terms of the instantaneous valuesof ρ, w , that a driver-assist vehicle can readily detect and use, rather than on their time historyover a long time horizon.We implement this idea by considering first the functional (32). Let ∆ t > t, t + ∆ t ]: J ρV ( u ) = ∆ t (cid:90) L − L (cid:18) ρ ( t + ∆ t, x ) V (cid:18) u ( t, x ) , w ( t + ∆ t, x ) (cid:19) − µF ( u ( t, x )) (cid:19) dx (35)subject to the following discrete-in-time version of (29): ρ ( t + ∆ t, x ) = ρ ( t, x ) − ∆ t∂ x (cid:18) ρ ( t, x ) V (cid:18) u ( t, x ) , w ( t, x ) (cid:19)(cid:19) w ( t + ∆ t, x ) = w ( t, x ) − ∆ tV (cid:18) u ( t, x ) , w ( t, x ) (cid:19) ∂ x w ( t, x ) . (36)Plugging these values of ρ ( t + ∆ t, x ), w ( t + ∆ t, x ) into (35) we obtain J ρV ( u ) = ∆ t (cid:90) L − L (cid:18) ρV (cid:18) u , w (cid:19) − µF ( u ) (cid:19) dx + o (∆ t ) , where we have omitted the variables ( t, x ) of the quantities ρ, w, u for brevity. Here, o (∆ t ) denoteshigher order terms in ∆ t that we may formally neglect under the assumption of small time horizon.13o find the optimality condition associated with the maximisation of J ρV we consider u = u ∗ + εv ,where u ∗ is the (unknown) optimal control, v is an arbitrary test function and ε > J ρV at u ∗ : ddε J ρV ( u ∗ + εv ) (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = 0 , we find the equation (cid:90) L − L (cid:18) ρ∂ s V (cid:18) u ∗ , w (cid:19) − µF (cid:48) ( u ∗ ) (cid:19) v dx + o (1) = 0 , which in the limit ∆ t → + and owing to the arbitrariness of v implies ρ∂ s V (cid:18) u ∗ , w (cid:19) − µF (cid:48) ( u ∗ ) = 0 . (37)From (37), solving for u ∗ we get the instantaneously optimal control in terms of ρ , w , whichrepresents the recommended headway s d ( ρ, w ) for the maximisation of the flux of vehicles. Forinstance, if V, F are given respectively by (3), (34) we obtain( a + u ∗ ) log u ∗ = aµ ρw, (38)which admits a unique solution u ∗ ≥ u from R + to R .Let us repeat now these arguments for the functional (33). Its discrete-in-time version over atime horizon [ t, t + ∆ t ] with ∆ t > J ρ ( u ) = ∆ t (cid:90) L − L ( ρ α ( t + ∆ t, x ) + µF ( u ( t, x ))) dx subject to (36). Using these constraints we determine in particular ρ α ( t + ∆ t, x ) = ρ α ( t, x ) − α ∆ tρ α − ( t, x ) ∂ x (cid:18) ρ ( t, x ) V (cid:18) u ( t, x ) , w ( t, x ) (cid:19)(cid:19) + o (∆ t )= ρ α ( t, x ) − α ∆ t∂ x (cid:18) ρ α ( t, x ) V (cid:18) u ( t, x ) , w ( t, x ) (cid:19)(cid:19) + ( α − V (cid:18) u ( t, x ) , w ( t, x ) (cid:19) ∂ x ρ α ( t, x ) + o (∆ t )and we observe that ∂ x (cid:0) ρ α V (cid:0) u , w (cid:1)(cid:1) integrates to zero on [ − L, L ] because of the periodic boundaryconditions. Hence we obtain J ρ ( u ) = ∆ t (cid:90) L − L (cid:18) ρ α + ( α − tV (cid:18) u , w (cid:19) ∂ x ρ α + µF ( u ) (cid:19) dx + o (∆ t ) , which, imposing ddε J ρ ( u ∗ + εv ) (cid:12)(cid:12)(cid:12)(cid:12) ε =0 = 0for an arbitrary test function v , produces the optimality condition (cid:90) L − L (cid:18) ( α − t∂ s V (cid:18) u ∗ , w (cid:19) ∂ x ρ α + µF (cid:48) ( u ∗ ) (cid:19) v dx + o (∆ t ) = 0 . If the penalisation coefficient µ is independent of ∆ t then in the limit ∆ t → + we get F (cid:48) ( u ∗ ) = 0,namely an equation for the optimal control independent of ρ , w . If instead we scale the penalisation14oefficient as µ = κ ∆ t with κ >
0, meaning that the cost of the control is proportional to thelength of the time horizon of the optimisation, then for ∆ t → + we have( α − ∂ s V (cid:18) u ∗ , w (cid:19) ∂ x ρ α + κF (cid:48) ( u ∗ ) = 0 , (39)whence we get in general a richer instantaneous optimal control, viz. recommended headway s d ,depending on ρ , w . Notice however that for α = 1, i.e. if the goal is to minimise ρ itself, (39)reduces in turn to F (cid:48) ( u ∗ ) = 0. On the other hand, if α > V, F are given by (3), (34)then (39) yields ( a + u ∗ ) log u ∗ = (1 − α ) aκ w∂ x ρ α , (40)which admits a unique solution u ∗ ≥
0. In particular, for α = 1 this solution is u ∗ = 1, viz. aconstant unitary recommended headway. The Aw-Rascle-Zhang (ARZ) model is a very popular traffic model of the form (31) with V ( ρ, w ) = w − p ( ρ ) ,p : R + → R + being a monotonically increasing function called the traffic pressure . This model wasproposed by Aw and Rascle [3], and independently by Zhang [42], to overcome some drawbacksof second order macroscopic traffic models pointed out by Daganzo [12]. The traffic pressure isusually taken of the form p ( ρ ) = ρ δ , δ > . (41)Recalling that, in the present context, (31) is obtained from (29) with the choice (30), we canrecast the ARZ model in the controlled setting (29) by letting V (cid:18) s , ω (cid:19) = ω − p (cid:18) s (cid:19) , (42)then we can exploit the results of Section 4.1 to deduce instantaneous optimal controls for fluxmaximisation and congestion minimisation.Specifically, condition (37) for the maximisation of the flux becomes ρ ( u ∗ ) p (cid:48) (cid:18) u ∗ (cid:19) − µF (cid:48) ( u ∗ ) = 0 , which for F, p like in (34), (41) produces( u ∗ ) δ log u ∗ = δµ ρ. (43)This equation admits a unique solution u ∗ ≥ u from [1 , + ∞ ) to R + . Notice that the resulting recommended headway s d ( ρ, w ) = u ∗ is actually independent of w .On the other hand, condition (39) for the minimisation of the traffic congestion becomes α − u ∗ ) p (cid:48) (cid:18) u ∗ (cid:19) ∂ x ρ α + κF (cid:48) ( u ∗ ) = 0 , which with F, p like in (34), (41) yields( u ∗ ) δ log u ∗ = (1 − α ) δκ ∂ x ρ α . − α ) ∂ x ρ α ≤
0. Indeed, the mapping u (cid:55)→ u δ log u is decreasingfor 0 < u < e − δ , increasing for u > e − δ and reaches the absolute minimum − δ ) e at u = e − δ . Consequently, if − δ ) e ≤ (1 − α ) ∂ x ρ α ≤ − α ) ∂ x ρ α < − δ ) e there is no solution. Remark . The speed function (42), together with the choice (41) of the traffic pressure, compliesneither with Assumption 2.1(ii) nor with (18). Therefore, the inclusion of the ARZ model amongthe particular cases obtainable from (29) is only formal, being not strictly supported by thederivation performed in Sections 2, 3. We point out that a genuine Enskog-type kinetic derivationof the ARZ model with uncontrolled speed-based vehicle interactions may instead be found in therecent paper [18].
We exemplify now the results of Section 4 through selected numerical tests. In more detail, we solvenumerically the hydrodynamic model (29) with s d chosen out of the instantaneous optimisationof either functional (32), (33) and we compare the results with those obtained by fixing a priori s d like in (30), which produces the GARZ model (31).We consider both the speed function (3), motivated by FTL microscopic dynamics, and thespeed function (42), directly suggested by the ARZ macroscopic model.In all cases, we solve the hydrodynamic model by means of an upwind scheme coupled witha non-linear algebraic solver of (37), (39) at each grid point ( x, t ). Consistently with the theorydeveloped in Section 4, we take as spatial domain the interval [ − ,
1] with periodic boundaryconditions, which simulates a circular track. As initial conditions ρ ( x ) := ρ (0 , x ), w ( x ) := w (0 , x ), we prescribe ρ ( x ) = (cid:40) . x ≤
00 if x > , w ( x ) = (cid:40) .
55 if x ≤ . x > , which mimic a platoon of vehicles filling initially one half of the circular track.The first three columns from the left of Figure 1 show the density profiles (solid lines) at thethree successive computational times t = 1 , . , V givenby (3) in the cases of flux maximisation and congestion minimisation. The flux maximisation (firstcolumn) is ruled by the optimality condition (38) with µ = 0 . a = 1 whereas the congestionminimisation (second and third columns) is ruled by (40) with κ = 0 . a = 1 and α = 1 , s d given by (30), i.e. withno specific optimisation. It is clear that the optimal s d ’s operate so as to keep the platoon ofvehicles compact. In particular, they avoid the formation of a rarefaction wave responsible for thespreading of the density across the whole domain. This effect is further emphasised by the wavediagrams in the xt -plane shown in Figure 2(a). Finally, Figure 3(a) shows the instantaneous valuesof the functionals J ρV , J ρ with the optimal s d ’s (solid line) and with s d given by (30) (dashedline). It is interesting to observe that, starting approximately from the computational time t = 4,the functionals take the same values both in the optimised and in the non-optimised cases. Thisis probably a consequence of the periodic boundary conditions, which, in the long run, tend tomake the integral values of the flux and the density uniform despite persisting differences in thecorresponding pointwise profiles.The fourth column from the left of Figure 1 compares the density profiles with (solid line) andwithout (dashed line) flux maximisation obtained from the ARZ model, i.e. the GSOM (29) with V given now by (42). In this case, the flux maximisation is ruled by (43) with µ = 0 . δ = 3while the non-optimised case is again obtained taking s d like in (30). We observe that the fluxmaximisation is achieved through a redistribution of the vehicles in the platoon. Initially theyare slowed down, whereby their density diminishes and the rear part of the platoon elongates.16igure 1: Density profiles at three successive computational times obtained from the numericalsolution of the GSOM (29) with the speed function (3) (first three columns from the left) andthe speed function (42) (fourth column from the left). Solid lines: optimal choice of s d for theoptimisations indicated on the top of the columns. Dashed lines: “standard” choice s d = ρ ,cf. (30).Subsequently, the platoon remains compact and recovers essentially the same speed as in the non-optimised case, cf. the wave diagrams in Figure 2(b). From Figure 3(b) we observe that, unlikethe previous cases, the instantaneous values of the non-optimised functional J ρV (dashed line)depart more and more consistently from those of the optimised one (solid line), probably as aconsequence of a much higher implementation cost (viz. penalisation) of the non-optimal s d . In this paper, we have derived generic high order macroscopic traffic models from a feedback-controlled particle description via an Enskog-type kinetic approach.At the microscopic scale, we have considered a class of generic Follow-the-Leader (FTL) modelswhich include a Lagrangian marker, i.e. a label attached to each vehicle representing a constant-in-time driving characteristic, such as e.g., the maximum speed. We have shown that the cor-responding natural macroscopic description is provided by a third order hyperbolic system ofconservation/balance laws for the density of vehicles, their mean Lagrangian marker and themean headway among them. These are the hydrodynamic parameters conserved by the FTLinteractions, or in classical kinetic terms the “collisional” invariants.Next, we have included a feedback control in the FTL interaction rules, which mimics theaction of a driver-assistance system trying to maintain a recommended distance s d from the leadingvehicle. We have modelled s d as a parameter depending on the local traffic congestion and thelocal mean Lagrangian marker. Moreover, we have taken into account that all vehicles may notbe equipped with such a controller. For this, we have assumed that a randomly selected vehicleis controlled with a certain probability p understood as the penetration rate of the driver-assisttechnology in the traffic stream. In the regime of sufficiently high p , we have upscaled the controlled17 a) GSOM(b) Aw-Rascle-Zhang Figure 2: Wave diagrams in the xt -plane corresponding to: (a) the first three columns from theleft of Figure 1; (b) the fourth column from the left of Figure 1.FTL model to a macroscopic model by taking the hydrodynamic limit of the corresponding Enskog-type kinetic description.We have shown that the resulting hydrodynamic model describes universal traffic trends forlarge enough penetration rates. Indeed, p does not parametrise the macroscopic equations butaffects the convergence rate of the microscopic interactions to their local equilibrium. Remarkably,this hydrodynamic model turns out to be a second order one belonging to the GSOM class. Theorder reduction with respect to the uncontrolled case has its origin in the fact that the introductionof the driver-assist control destroys the local conservation of the mean headway among the vehicles.Furthermore, this model is parametrised by the recommended distance s d , which we have proposedto understand as a further control to be fixed in such a way to optimise macroscopic trafficdynamics. Using the technique of the instantaneous control, which is particularly meaningful fordriver-assist vehicles, we have proved that there exist instantaneously optimal choices of s d (i.e.optimal s d ’s based on the instantaneous values of the hydrodynamic variables describing the trafficstream) which e.g., maximise the flow of vehicles or minimise the traffic congestion. Apart fromthese two examples, the technique that we have proposed is quite general and may also be appliedto other macroscopic functionals to be optimised.Summarising, in this paper we have ultimately performed a multiscale control and optimisation of traffic. Indeed, starting from a microscopic control, which optimises the interaction of a singlevehicle with its leading vehicle, we have shown that it is possible to design explicitly the controlparameters so as to optimise global traffic trends. This also suggests that vehicle-wise automaticdecision algorithms may successfully turn driver-assist vehicles into bottom-up traffic controllers ,provided their penetration rate in the traffic stream is sufficiently high. On the other hand, webelieve that the conceptual scheme we have proposed in this paper may be fruitfully applied also18 a) GSOM(b) Aw-Rascle-Zhang Figure 3: Instantaneous values of the functionals J ρV , J ρ corresponding to: (a) the first threecolumns from the left of Figure 1 and to Figure 2(a); (b) the fourth column from the left ofFigure 1 and to Figure 2(b). Solid lines: optimal choice of s d for the optimisations indicated onthe top of the pictures. Dashed lines: “standard” choice s d = ρ , cf. (30).to the multiscale control of several other multi-agent systems, such as e.g., human crowds or socialsystems, in which desired collective trends cannot be simply obtained by top-down impositionsbut need rather to emerge spontaneously from suitably controlled individual interactions. Acknowledgements
This research was partially supported by the Italian Ministry for Education, University and Re-search (MIUR) through the “Dipartimenti di Eccellenza” Programme (2018-2022), Departmentof Mathematical Sciences “G. L. Lagrange”, Politecnico di Torino (CUP: E11G18000350001) andthrough the PRIN 2017 project (No. 2017KKJP4X) “Innovative numerical methods for evolution-ary partial differential equations and applications”.F.A.C. acknowledges support from “Compagnia di San Paolo” (Torino, Italy)F.A.C. and A.T. are members of GNFM (Gruppo Nazionale per la Fisica Matematica) ofINdAM (Istituto Nazionale di Alta Matematica), Italy.The research of B.P. is based upon work supported by the U.S. Department of Energy’s Officeof Energy Efficiency and Renewable Energy (EERE) under the Vehicle Technologies Office awardnumber CID DE-EE0008872. The views expressed herein do not necessarily represent the viewsof the U.S. Department of Energy or the United States Government.19 eferences [1] G. Albi, Y.-P. Choi, M. Fornasier, and D. Kalise. Mean field control hierarchy.
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