Mean-field type modeling of nonlocal congestion in pedestrian dynamics

2.4 MB·

by  Alexander Aurell

Mean-field type modeling of nonlocal congestion in
pedestrian crowd dynamics
Alexander Aurell ∗ Boualem Djehiche †
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February 1, 2017
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0
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Abstract: We extend the class of pedestrian crowd models introduced by Lachapelle
n
andWolfram(2011)toallowfornonlocalcongestionandarbitrarilybutfinitelymanyin-
a
J teractingcrowds.Thenewcongestionfeaturegrantspedestriansa’personalspace’where
crowding is undesirable. We treat the model as a mean-field type game which we derive
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from a particle picture. Solutions to the mean-field type game are characterized via a
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Pontryagin-typeMaximumPrinciple.Thebehaviorofacrowdactingundernonlocalcon-

] gestion is illustrated by a numerical simulation.
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O MSC 2010: 49N90, 60G09, 60H10, 60H30, 60K35
.
h Keywords: Crowd dynamics, congestion, mean-field approximation, interact-
t ing populations, optimal control, mean-field type games
a
m
[

1. Introduction
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v
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When moving in a crowd, a pedestrian chooses its path based not only on its desired final
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1 destination but it also takes the movement of other surrounding pedestrians into account. The
9 bullet points below are stated in [14] as typical traits of pedestrian behavior.
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.
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• Will to reach specific targets. Pedestrians experience a strong interaction with the envi-
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7 ronment.
1 • Repulsion from other individuals. Pedestrians may agree to deviate from their preferred
:
v path, looking for free surrounding room.
Xi • Deterministic if the crowd is sparse, partially random if the crowd is dense.
r
a These properties appear in classical particle models. Other authors advocate smart particle
models that follow decision-based dynamics. In [14] some fundamental differences between
classical and smart particle models are outlined. We list a few of them in Table 1.
∗Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden.
E-mail address: [email protected]
†Department of Mathematics, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden.
E-mail address: [email protected]
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Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 2
Classical Smart
Robust - interaction only through collisions Fragile - avoidance of collisions and obstacles
Blindness - dynamics ruled by inertia Vision - dynamics ruled at least partially by decision
Local - interaction is pointwise Nonlocal - interaction at a distance
Table 1
A smart particle model lets pedestrians decide where to walk and with what speed. The choice
is based on some rule that takes the available information into account such as the positioning
and movement of other pedestrians. Although more realistic, this approach has complications.
If pedestrian i moves, all pedestrians accessing information on i’s state might have to adapt
their movements. The large number of connections where information is exchanged within a
crowd is a computational difficulty.
The mean-field approach to modeling congestion in pedestrian crowds was introduced in [12],
where pedestrians are treated as particles, following decision-based dynamics, trying to opti-
mize their path by avoiding local congestion.
InacrowdofN pedestrians,anN-crowdfromnowon,pedestrianiwithpositionXi,N controls
its velocity such that its congestion risk measure, Ji,N, is minimized over a finite time horizon
[0,T]. The risk measure penalizes congestion, energy waste and failure to reach a target area.
In this paper we advocate for the use of the following nonlocal congestion term.
 
EN (cid:90) T N1−1 (cid:88)N φr(cid:16)Xti,N −Xtj,N(cid:17)dt, (1.1)
0
j=1,j(cid:54)=i
where the function φ models the ‘personal space’ of a typical pedestrian and Xi,N −Xj,N is
r t t
the distance between pedestrian i and pedestrian j at time t. The personal space, ϕ , has sup-
r
port within a ball of radius r and hence a pedestrian is not effected by crowding outside it. For
positive r, (1.1) is a weighted average of the crowding within the personal space. Connecting
to the terminology in Table 1, we will refer to this as nonlocal congestion. Taking the limit
r → 0 corresponds to shrinking the personal space to a singleton and only pointwise crowding,
that is collisions, will effect the pedestrian. This is called local congestion.
In emergency situations it is often in the interest of all pedestrians to get to a certain place,
such an exit. In evacuation planning or crowd management at mass gatherings, it is in the
interest of the planner to control the crowd along paths and towards certain areas. Common
to such situations is the conflict between attractive locations and repulsive congestion in the
crowd. Pedestrians acting under nonlocal congestion will order themselves more densely in
such places compared to pedestrians acting under local congestion. This effect is caused by
the larger personal space, the nonlocal congestion term (1.1) is an average over a bigger set
hence allowing for higher densities in attractive areas. Higher densities will in turn allow for
more effective emergency planning when designing for example escape routes. The numerical
simulation in the end of this paper confirms this effect. The pedestrians are allowed to move
freely, but the observed effect will become even more beneficial for a planner when introducing
anenvironmentforthepedestrianstointeractwith.Inreality,crowdmanagementisoftendone
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 3
by the strategic placement of obstacles such as pillars and walls. Furthermore, the pedestrians
acting under nonlocal congestion travel at an overall lower congestion risk than their local
congestion counterpart. This suggests that the crowd acting under nonlocal congestion could
potentially move at a higher velocity than the crowd under local congestion which allows for
faster and more successful evacuations.
In [12] the mean-field optimal control is characterized through a matching argument. This
control is an approximate Nash equilibrium for the N-crowd. It is, for each pedestrian, the
bestresponsetothemovementoftheothers.Furthermore,twoN-crowdsareconsideredwhere
each pedestrian has crowd-specific preferences such as the target location and the congestion
preference. The authors set up a mean-field game and show that it is equivalent to an optimal
controlprobleminthecaseofapotentialgamemodel.Inthispaper,welookatthecrowdfrom
the bird’s-eye view of an evacuation planner. We seek a ‘simultaneous’ optimal strategy for all
the pedestrians involved in the crowd through a mean-field type optimal control approach for
the single-crowd case and a mean-field type game approach for the multiple-crowd case.
The contributions of this paper are the following. We identify a model for the N-crowd that
converges to the mean-field model proposed in [12]. This gives us insight into how the inter-
action between pedestrians in the N-crowd effects the mean-field model and reveals that the
crowd of [12] is acting under local congestion. Our second contribution is a relaxation of the
locality of the pedestrian model by allowing for interaction between pedestrians at a distance.
Each pedestrian is given a personal space where it dislikes crowding, instead of interacting
with other pedestrians only through collisions. This conceptual change is realistic since pedes-
trians do not need to be in physical contact to interact. As was discussed above, the suggested
nonlocal congestion model allows for the following desirable features:
• Higher densities in attractive areas such as exits or escape routes where the pedestrians
have to choose between higher congestion and penalty.
• Lower congestion risk, which implies a potential increase in pedestrian velocity allowing
forfasterexitsandalargerflowofpeople,averyusefulfeatureinthedesignofevacuation
strategies.
Finally, we generalize the model to allow for an arbitrary number of interacting crowds and
we treat it as a mean-field type game, in contrast to the mean-field game in [12]. We link the
mean-field type game to an optimal control problem and prove a sufficient maximum principle.
Treating the problem as a mean-field type game, the solution we find is not a Nash equilibrium
but that of a central planner strategy.
The paper is organized as follows. After a short section of preliminaries, we consider the single-
crowd case in Section 3. In Section 4, the multiple-crowd case is studied. The results derived
in Section 3 generalize to an arbitrary finite number of interacting crowds and the solution to
the mean-field type game is characterized by a sufficient maximum principle. An example that
highlightsthedifferencebetweenlocalcongestionandnonlocalcongestionissolvednumerically
in Section 5. For the sake of clarity, all technical proofs are moved to an appendix.
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 4
2. Preliminaries
Given a general Polish space S, let P(S) denote the space of probability measures on B(S).
For an element s ∈ S, the Dirac measure on s is an element of P(S) and will be denoted by
δ . Let P(S) be equipped with the topology of weak convergence of probability measures. A
s
metric that induces this topology is the bounded 1-Lipschitz metric,
d (µ,ν) := (cid:107)µ−ν(cid:107) = sup (cid:104)µ,f(cid:105)−(cid:104)ν,f(cid:105), (2.1)
P(S) 1
f∈L1
where L is the set of real-valued functions on S bounded by 1 and with Lipschitz coefficient
1
1. With his metric, P(S) is a Polish space. The space of probability measures on B(S) with
finite second moments will be denoted by P (S),
2
(cid:26) (cid:90) (cid:27)
P (S) := ν ∈ P(S) : ∃s ∈ S that satisfies d (s,s )2ν(ds) < ∞ . (2.2)
2 0 S 0
S
Equipped with the topology of weak convergence of measures and convergence of second mo-
ments, P (S) is a Polish space. A compatible complete metric is the square Wasserstein metric
2
d ,
P2(S)
 1/2
(cid:90)
d (ν,ν) :=  inf d (s,s)2α(ds,ds) (2.3)
P2(S) (cid:101) α∈P(S×S) S×S S (cid:101) (cid:101) 
α1=ν,α2=ν(cid:101)
where α (α ) denotes the first (second) marginal of α. From (2.3) we have the two following
1 2
inequalities. For all s ,s ∈ S and for all N ∈ N,
i (cid:101)i
(cid:32) N N (cid:33) N
1 (cid:88) 1 (cid:88) 1 (cid:88)
d2 δ , δ ≤ d (s ,s )2. (2.4)
P2(S) N si N s(cid:101)i N S i (cid:101)i
i=1 i=1 i=1
For X with distribution ν and X(cid:101) with distribution ν(cid:101),
(cid:104) (cid:105)
d2P2(S)(ν,ν(cid:101)) ≤ E |X −X(cid:101)|2 . (2.5)
LetT > 0beafinitetimehorizonandletRd,d ∈ N,beequippedwiththeEuclideannorm.Let
M and M be the spaces of continuous functions on [0,T] with values in P(Rd) and P (Rd)
2 2
respectively,
M := C([0,T];P(Rd)), M := C([0,T];P (Rd)). (2.6)
2 2
Equipped with the uniform metrics d and d given by
M M
d (m,m(cid:48)) := sup d (m ,m(cid:48)), d (m,m(cid:48)) := sup d (m ,m(cid:48)), (2.7)
M P(Rd) t t M2 P2(Rd) t t
t∈[0,T] t∈[0,T]
M and M are Polish spaces. The mathematical results stated above can be found in [16,
2
Chapter 2] and [8, Chapter 14].
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 5
Let A be a compact subset of Rd. Given a filtered probability space (Ω,F,F,P), denote by A
the set of A-valued F-adapted processes such that
(cid:20)(cid:90) T (cid:21)
E |a |2dt < ∞. (2.8)
t
0
An element of A will be called an admissible control. From the context, it will be clear which
stochastic basis the notation A is referring to.
Given a vector x = (x1,...,xN) in the product space SN and an element y ∈ S, we let
x−i := (x1,...,xi−1,xi+1,...,xN),
(2.9)
(y,x−i) := (x1,...,xi−1,y,xi+1,...,xN).
Furthermore, the law of any random quantity X will be denoted by L(X) and any index set
of the form {1,...,N} will be denoted by [[N]].
3. Single-crowd congestion
3.1. The N-crowd model
Let (Ω ,FN,FN,P ) be a complete filtered probability space for each N ∈ N. The filtra-
N N
tion FN is right-continuous and augmented with P -null sets. It carries the independent d-
N
dimensional FN-Wiener processes W1,N,...,WN,N. Let, for each i ∈ [[N]], the FN-measurable
0
Rd-valued random variable ξi,N be square-integrable and independent of (W1,N,...,WN,N).
Given a vector of admissible controls, a¯N = (a1,N,...,aN,N) ∈ AN, consider the system
dXi,N = b(t,Xi,N,ai,N)dt+σ(t,Xi,N)dWi,N, Xi,N = ξi,N, i ∈ [[N]]. (3.1)
t t t t t 0
We make the following assumptions on the coefficients b and σ in order to have a unique strong
solution to (3.1).
(A1) b : [0,T]×Rd×A → Rd and σ : [0,T]×Rd → Rd×d are continuous in all arguments.
(A2) For all x ,x ∈ Rd and a ,a ∈ A, there exists a constant K > 0 independent of
1 2 1 2
(t,x ,x ,a ,a ) such that
1 2 1 2
|b(t,x ,a )−b(t,x ,a )| ≤ K(|x −x |+|a −a |),
1 1 2 2 1 2 1 2
|σ(t,x )−σ(t,x )| ≤ K|x −x |,
1 2 1 2
|b(t,x ,a )|+|σ(t,x )| ≤ K(1+|x |+|a |).
1 1 1 1 1
Proposition 3.1. Under (A1) and (A2) the system (3.1) has a unique strong solution in the
sense that
Xi,N = ξi,N, (3.2)
0
(cid:90) t
(cid:12)(cid:12)b(s,Xi,N,ai,N)(cid:12)(cid:12)+(cid:12)(cid:12)σ(s,Xi,N)(cid:12)(cid:12)2ds < ∞, t ∈ [0,T], P−a.s. (3.3)
s s s
0
(cid:90) t (cid:90) t
Xi,N = ξi,N + b(s,Xi,N,ai,N)ds+ σ(s,Xi,N)dWi,N, t ∈ [0,T]. (3.4)
t s s s s
0 0
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 6
Furthermore, the strong solution Xi,N satisfies the estimate
(cid:34) (cid:35)
E sup |Xi,N|2 ≤ K (cid:0)1+E (cid:2)|ξi,N|2(cid:3)(cid:1) (3.5)
N s t N
s∈[0,t]
for all t ∈ [0,T], for all i ∈ [[N]] and for some positive constant K depending only on t.
t
Proof. A proof can be found in [20, Chapter 1, Theorem 6.16]. Note that K is independent of
t
ai,N by compactness of A.
The process Xi,N models the motion of a pedestrian in the N-crowd who partially controls
its velocity through the control ai,N. Since its control is adapted to the full filtration FN, the
modelallowsforthepedestriantotakeeverymovementinthecrowdintoaccount.Itsmotionis
alsoinfluencedbyexternalforces,suchastherandomdisturbancedrivenbyWi,N.Themotion
of the pedestrian may be modeled more generally than above by introducing an explicit weak
interaction in the drift [7], such as
N
dXi,N = 1 (cid:88)(cid:101)b(t,Xi,N,ai,N,Xj,N)dt+σ(t,Xi,N)dWi,N. (3.6)
t N t t t t
j=1
It is also possible to let a common disturbance effect all pedestrians [10], to model for example
evacuations during an earthquake, a fire, a tsunami etc.
PedestrianievaluatesthestateoftheN-crowd,givenbythecontrolvectora¯N = (a1,N,...,aN,N),
according to its individual congestion risk measure Ji,N,
r
(cid:20)(cid:90) T (cid:18)1 (cid:90) (cid:19) (cid:21)
Ji,N(a¯N) := E |ai,N|2+ φ (Xi,N −y)µ−i,N(dy) dt+Ψ(Xi,N) . (3.7)
r N 2 t r t t T
0 Rd
where µ−i,N is the empirical measure of X−i,N,
t
N
µ−i,N := 1 (cid:88)δ , (3.8)
t N −1 Xj,N
t
j=1
j(cid:54)=i
and X1,N,...,XN,N is the solution of (3.1) given a¯N. The region where crowding has an
influence on the pedestrian’s choice of control, its ’personal space’, is ideally modeled by a
normalized indicator function,
(cid:26) Vol(B )−1, x ∈ B ,
I (x) := r r (3.9)
r 0, x ∈/ B ,
r
where B ⊂ Rd is the ball with radius r > 0 centered at the origin and Vol(B ) is its volume.
r r
The term
(cid:90)
I (Xi,N −y)µ−i,N(dy) (3.10)
r t t
Rd
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 7
then represents the number of pedestrians around Xi,N (except itself) within a distance less
t
than r at time t [18]. To simplify the calculations we will use a smoothed version of I . Let γ
r δ
be a mollifier,
γ (x) := γ(x/δ)/δ, (3.11)
δ
where γ is a smooth symmetric probability density with compact support. For a fixed δ > 0,
we define
φ (x) := γ ∗I (x). (3.12)
r δ r
For convergence estimates later in this section, we assume that the final cost Ψ satisfies the
following condition.
(A3) For all x ,x ∈ Rd there exists a constant K > 0 independent of (x ,x ) such that
1 2 1 2
|Ψ(x )−Ψ(x )| ≤ K|x −x |.
1 2 1 2
The interpretation of the congestion risk measure is the following. The first term penalizes
large controls (energy waste) whereas the second term penalizes congestion within a personal
space. The final cost penalizes deviations from specific target regions at at time T. Typically
the final cost takes large values everywhere except in areas where the pedestrians want to end
up, places like meeting points, evacuation doors, etc.
3.2. The mean-field type control problem
Let (Ω,F,F,P) be a complete filtered probability space such that the filtration is right con-
tinuous and augmented with P-null sets. Let F carry a Wiener process W and let ξ be an
F -measurable and square-integrable Rd-valued random variable independent of W. Given a
0
control a ∈ A, the mean-field type dynamics is
dX = b(t,X ,a )dt+σ(t,X )dW , X = ξ. (3.13)
t t t t t 0
By Proposition 3.1 there exists a unique strong solution to (3.13). The mean-field type con-
gestion risk measure is given by
(cid:20)(cid:90) T 1 (cid:90) (cid:21)
J (a) = E |a |2+ φ (X −y)µ (dy)dt+Ψ(X ) . (3.14)
r 2 t r t Xt T
0 Rd
where µ is the distribution of X .
Xt t
Remark 3.1. The difference between a mean-field type control and a mean-field game is that
in general mean-field games can be reduced to a standard control problem and an equilib-
rium while a mean-field type control problem is a nonstandard control problem ([2],[4]). The
matchingproceduretofindthefixedpoint(equilibrium)foramean-fieldgameispedagogically
described as follows ([7],[13]).
(i) Fix a deterministic function µ : [0,T] → P (Rd).
t 2
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 8
(ii) Solve the standard stochastic control problem
(cid:20)(cid:90) T 1 (cid:90) (cid:21)
aˆ = argmin E |a |2+ φ (X −y)µ (dy)dt+Ψ(X ) , (3.15)
t r t t T
2
a∈A 0 Rd
where X is the dynamics corresponding to a.
(iii) Determine the function µˆ : [0,T] → P (Rd) such that µˆ = L(Xˆ ) for all t ∈ [0,T] where
t 2 t t
Xˆ is the dynamics corresponding to the optimal control aˆ.
In the mean-field type control setting, the measure-valued process (µ ; t ∈ [0,T]) is not
Xt
considered to be a separate variable but is given by the input of the congestion risk. This
non-linearity makes the control problem nonstandard.
3.3. Convergence of the processes
We make the following assumptions about the initial data ξ1,N,...,ξN,N.
(cid:104) (cid:105)
(B1) sup E 1 (cid:80)N |ξi,N|2 < ∞ for all i ∈ [[N]].
N∈N N N i=1
(B2) (ξ1,N,...,ξN,N) is exchangeable for all N ∈ N.
(cid:16) (cid:17)
(B3) lim L 1 (cid:80)N δ = δ in P(P (Rd)).
N→∞ N i=1 ξi,N µ0 2
Under assumption (B1) and (B2), the sequence (ξi,N)N∈N is tight for all i ∈ [[N]] [5]. Under
assumptions (B1), (B2) and (B3) ξi,N converges weakly in P (Rd) to a µ -distributed random
2 0
variable, from now on denoted by ξ, for all i ∈ [[N]] [5]. We make the following assumption
about the controls.
(B4) The controls are of feedback form, ai,N(ω) = aN(t,Xi,N(ω)), where each aN is an A-
t t
valued deterministic function and aN converge uniformly to a as N → ∞. Furthermore,
(cid:20)(cid:90) T (cid:21)
sup E |aN(t,Xi,N)|2 < ∞, ∀ i ∈ [[N]]. (3.16)
N t
N∈N 0
Remark 3.2. Assumption (B4) implies that, while the paths of pedestrians in the N-crowd
may differ, they are outcomes from a symmetric joint probability distribution. By exchange-
ability of (ξi,N,Wi,N)N ,
i=1
(aN(t,Xi,N))N =d (aN(t,Xπ(i),N))N (3.17)
t i=1 t i=1
for all permutations π of [[N]] and the interpretation is that we cannot distinguish pedestrians
in the crowd. The pedestrians are anonymous.
Let X1,N,...,XN,N be the solution of (3.1) given aN and let µN denote the empirical measure
process of these states,
N
1 (cid:88)
µN = δ . (3.18)
N Xi,N
i=1
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 9
Proposition 3.2. The collection of measures {L(µN),N ∈ N} in P(M ) is tight.
2
Proof. The empirical measures are elements of M by Proposition 3.1 together with (B1) and
2
(B2). The proof of tightness is found in Appendix 6.1.
Recall that a sequence {X } of random variables converges weakly to X in a Polish space
n
if and only if {X } is tight and every convergent subsequence of {X } converges to X. The
n n
tightness of the empirical measures implies that along a converging subsequence, µN converges
weakly to the measure-valued process µ that satisfies the integral equation
(cid:90) t(cid:28) 1 (cid:29)
(cid:104)µ ,f(cid:105) = (cid:104)µ ,f(cid:105)+ µ ,b(s,·,a(s,·))·∇f + σ(s,·)∆f ds, ∀f ∈ C2(Rd). (3.19)
t 0 s 2 b
0
Since the strong solution of (3.13) is unique, the weak solution is also unique [19] which is
equivalent to uniqueness of solutions to (3.19) [9]. We have the following result.
Theorem 3.1. Let Xi, i ∈ N, be independent copies of the strong solution of (3.13). Under
assumptions (A1),(A2) and (B1)-(B4) the state process Xi,N for a pedestrian in the N-crowd
converges weakly to Xi as N → ∞.
Proof. Applying the propagation of chaos theorem of Sznitman [17], the theorem follows by
the weak convergence of µN to the deterministic measure µ.
3.4. Convergence of the congestion risk measure
From the previous section we know that Xi,N, the strong solution of (3.1), converges weakly
to X, the strong solution of (3.13), and we know that µN converges weakly to µ . Applying
t Xt
(2.4), we get that
d (µ−i,N,µN) ≤ 2/N, (3.20)
P2(Rd) t t
soµ−i,N convergesweaklytoµ aswell.BySkorokhod’sRepresentationTheorem[8,Theorem
t Xt
3.30] we can represent (up to distribution) all the random variables mentioned above in a
common probability space (Ω(cid:101),F(cid:101),P(cid:101)) where they converge P(cid:101)-almost surely. This allows us to
write
|Jri,N(aN)−Jr(a)| ≤ E(cid:101)(cid:34)(cid:90)0T (cid:12)(cid:12)(cid:12)(cid:12)12|aN(t,Xti,N)|2− 12|a(t,Xt)|2(cid:12)(cid:12)(cid:12)(cid:12)+(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90)Rdφr(Xti,N −y)µ−t i,N(dy)
(cid:12) (cid:35)
(cid:90) (cid:12) (cid:12) (cid:12)
− φ (X −y)µ (dy)(cid:12)dt+(cid:12)Ψ(Xi,N)−Ψ(X )(cid:12) ,
r t Xt (cid:12) (cid:12) T T (cid:12)
Rd (cid:12)
(3.21)
where the expectation is taken under P(cid:101). By compactness of A, the Continuous Mapping The-
orem, (B4) and Dominated Convergence we have
Nl→im∞E(cid:101)(cid:34)(cid:90)0T (cid:12)(cid:12)(cid:12)(cid:12)12|aN(t,Xti,N)|2− 21|a(t,Xt)|2(cid:12)(cid:12)(cid:12)(cid:12)(cid:35) = 0. (3.22)
Aurell, Djehiche/Mean-field type modeling of nonlocal congestion in pedestrian crowd dynamics 10
By (A3), Proposition 3.1 and Dominated Convergence,
(cid:34) (cid:35)
(cid:12) (cid:12)
lim E(cid:101) (cid:12)(cid:12)Ψ(XTi,N)−Ψ(XT)(cid:12)(cid:12) = 0. (3.23)
N→∞
Note that
(cid:20)(cid:90) T (cid:12)(cid:90) (cid:90) (cid:12) (cid:21)
E(cid:101) 0 (cid:12)(cid:12)(cid:12) Rdφr(Xti,N −y)µ−t i,N(dy)− Rdφr(Xt−y)µXt(dy)(cid:12)(cid:12)(cid:12)dt
(cid:20)(cid:90) T (cid:12)(cid:90) (cid:90) (cid:12) (cid:21)
≤ E(cid:101) 0 (cid:12)(cid:12)(cid:12) Rdφr(Xti,N −y)µ−t i,N(dy)− Rdφr(Xti,N −y)µXt(dy)(cid:12)(cid:12)(cid:12)dt (3.24)
(cid:20)(cid:90) T (cid:12)(cid:90) (cid:90) (cid:12) (cid:21)
+E(cid:101) 0 (cid:12)(cid:12)(cid:12) Rdφr(Xti,N −y)µXt(dy)− Rdφr(Xt−y)µXt(dy)(cid:12)(cid:12)(cid:12)dt .
The first term on the right hand side tends to zero as N → ∞ by the definition of weak
convergence while the second tends to zero as N → ∞ by the Continuous Mapping Theorem
and Dominated Convergence. We have proved the following result.
Theorem 3.2. Let a ∈ A and let aN = (a,...,a) ∈ AN. Then
Ji,N(aN) = J (a)+ε , (3.25)
r r N
where ε → 0 as N → ∞.
N
3.5. Solutions to NP-1 and MFT-1
The notion of solutions of the the N-crowd model (NP-1) and the mean-field type control
model (MFT-1) for single-crowd congestion will now be defined.
Definition 3.1 (Solution to NP-1). Let aˆN = (aˆ,...,aˆ) ∈ AN for some fixed aˆ ∈ A and let
aN = (a,...,a) ∈ AN for an arbitrary strategy a ∈ A. Then aˆN is a solution to NP-1 if
Ji,N(aˆN) ≤ Ji,N(aN), ∀a ∈ A, ∀i ∈ [[N]]. (3.26)
r r
If, for a given ε > 0, aˆ satisfies
Ji,N(aˆN) ≤ Ji,N(aN)+ε, ∀a ∈ A, ∀i ∈ [[N]], (3.27)
r r
then aˆN is an ε-solution to NP-1.
Definition 3.2 (Solution to MFT-1). If aˆ ∈ A satisfies
J (aˆ) ≤ J (a), ∀a ∈ A, (3.28)
r r
then aˆ is a solution to MFT-1.
The following result motivates the use of MFT-1 as an approximation to NP-1. It confirms
that we can construct an approximate solution to NP-1 using a solution to MFT-1 and the
approximation improves as the size of the crowd increases.