Finite-Time Control of One Dimensional Crowd Evacuation Systemdownloads.hindawi.com/journals/jat/2019/6597360.pdf · 2019-08-05 · Finite-Time Control of One Dimensional Crowd Evacuation

Research ArticleFinite-Time Control of One Dimensional CrowdEvacuation System

Wei Qin 12 Baotong Cui12 and Zhengxian Jiang 3

1Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education) Jiangnan University Wuxi 214122 China2School of IoT Engineering Jiangnan University Wuxi 214122 China3School of Science Jiangnan University Wuxi 214122 China

Correspondence should be addressed to Wei Qin weiqinvipjiangnaneducn

Received 26 February 2019 Revised 18 June 2019 Accepted 3 July 2019 Published 5 August 2019

Academic Editor Eneko Osaba

Copyright copy 2019 Wei Qin et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper pertains to the study of finite-time control of one dimensional crowd evacuation system Benefiting from the research offluid dynamics and vehicle traffic a one dimensional crowd evacuation system is constructed whose density-velocity relationshipis represented by a diffusionmodel In order to deal with the nondirectionality of crowdmovement the free flow speed is chosen asa control variable Since the control variable is included in a partial derivative it increases the difficulty of designing the controllerIn this paper finite-time controller is designed which not only guarantees the effective evacuation but also obtains the estimationof evacuation time Then finite-time tracking problem is solved which makes the density converge to a given density Finallynumerical examples illustrate the effectiveness of the controllers

1 Introduction

In everyday life crowds would gather in many places forexample subway stations stadiums and cinemas Effectivemeasures should be taken to ensure the safety and comfortof pedestrians so how to evacuate people when emergenciesoccur has been a challenging job In the early 1990s the Inter-national Conference on Engineering for Crowd Safety [1] hasshown the importance of this topic Physicists sociologistspsychologists computer scientists and traffic scientists havebeen conducting depth-going researches on this topic fromtheir research fields

Modeling the crowd dynamics is the primary task ofcrowd evacuation but due to the complexity and uncertaintyof crowd dynamics it is very difficult to build a modelthat suits all situations Therefore various models have beendeveloped such as the continuummodel the network-basedmodel [2] agent-based models [3 4] game-theoretic models[5] cellular automata models [6 7] and the fractional model[8]

In this paper the continuum model a macroscopicsimulation model is recommended At medium and high

densities the motion of pedestrian crowds shows somestriking analogies with the motion of fluids [9] so thetheory describing fluid dynamics is introduced to describepedestrian dynamics Pedestrians are treated as a collectionrather than individuals Average density and velocity at agiven location are proposed to describe the crowd dynam-ics Based on three hypotheses a first-order continuummodel was developed to describe the pedestrian dynamicsin [10] Huang et al [11] provided an efficient method tosolve Hughesrsquo model Appert-Rolland et al [12] extendeda macroscopic vehicle traffic model to pedestrian trafficThe cell transmission approach and the continuum approachwere combined in [13] to predict densities and travel timesSome continuum models were compared by making use ofnumerical method in [14] All above focus on modeling thepedestrian dynamics in different situations but few litera-tures present strategies for controlling the crowd dynamicsWadoo [15] designed advective diffusive and advective-diffusive controllers for crowd dynamics in one dimensionSliding mode control method was applied to crowd dynamicmodels for the synthesis of robust controllers in [16] Donget al [17] designed feedback control law for two dimensional

HindawiJournal of Advanced TransportationVolume 2019 Article ID 6597360 9 pageshttpsdoiorg10115520196597360

2 Journal of Advanced Transportation

crowd model Robin Neumann and Dirichlet boundarycontrol laws are designed for a disturbed crowd evacuationsystem in [18]

In this paper the problem of finite-time evacuation isstudied In some special circumstances such as fires andterrorist attacks pedestrians need to be evacuated as soonas possible Evacuation strategy cannot be used until itseffectiveness is tested because the cost involves not onlyproperty damage but also pedestrian injury or death There-fore it is particularly important to estimate the evacua-tion time The estimated evacuation time can be used toevaluate the effectiveness of the evacuation strategy andthe corresponding evacuation strategy can be adjusted tominimizing the loss Benefiting from Orlovrsquos research [19]finite-time controller is designed to evacuate pedestriansin finite time and then the finite-time control method isextended to deal with the tracking problem which makesthe crowd density follow a given reference density Someof the latest research on advanced control such as adaptivecontrol [20ndash22] guaranteed cost control [23] 119867infin control[24] and tracking control [25 26] gave us inspiration in thedesign of the controllerThe control and stability problem areformulated directly in the framework of a distributed modelof partial differential equations (PDEs) which can avoiderrors introduced by spatial discretization The developmentof hardware technology has made distributed sensors andactuators realistic but we mainly explore the crowd manage-ment strategies on theoretical interest in this paper and theimplementation needs further study in the future

The rest of this paper is organized as follows One dimen-sional crowd evacuation dynamic is modeled in Section 2Section 3 presents a feedback controller to evacuate pedes-trians in finite time Section 4 designs a finite-time controllerto make the crowd density profile track a reference densitySimulation examples are given in Section 5 to illustrate theeffectiveness of the controllers Conclusions and future workare discussed in Section 6

NotationThenotation is used throughout the paper1198672(0 119871)denotes the infinite-dimensional Hilbert space on interval[0 119871] 119871 gt 0 represents the interval length with 1198712 norm

1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 = [int119871

01205882 (119909 119905) d119909]12

120588 (119909 119905) isin 1198672 (0 119871) (1)

For notational convenience we denote

120588119905 (119909 119905) = 120597120588 (119909 119905)120597119905

120588119909 (119909 119905) = 120597120588 (119909 119905)120597119909

120588119909119909 (119909 119905) = 1205972120588 (119909 119905)1205971199092

(2)

2 Mathematical Modeling

The Lighthill-Whitham-Richards (LWR) model [27 28]based on the conservation law of mass is recommended inthis paper to represent the crowd evacuation dynamics in onedimension implying that the number of pedestrians comingin and going out of a corridor section account for the changeof crowd density on that section The LWRmodel is given by

120588119905 (119909 119905) + 119902119909 (119909 119905) = 0 (119909 119905) isin Ω (3)

where Ω = (0 119871) times (0 +infin) 119909 and 119905 are space and timevariables respectively 120588(119909 119905) is the average crowd densityand 120588119905(119909 119905) is the partial derivative of the density 120588(119909 119905) withrespect to time 119905 at position 119909 119902(119909 119905) denotes the flux ofthe crowd and 119902119909(119909 119905) is the partial derivative of the flux119902(119909 119905) with respect to position 119909 at time 119905 The flux 119902(119909 119905) isa function of 120588(119909 119905) and the average crowd speed V(119909 119905) asshown below

119902 (119909 119905) = 120588 (119909 119905) V (119909 119905) (119909 119905) isin Ω (4)

Various models have been developed to mimic thevelocity-density relationship such as Greenshield modelDrew model Greenberg model Pipes Munjal model andUnderwood model [29] Here the diffusion model is recom-mended which is given as

V (119909 119905) = V119891 [1 minus 120588 (119909 119905)120588119898 ] minus 119863120588119909 (119909 119905)120588 (119909 119905) (119909 119905) isin Ω (5)

where V119891 denotes the free flow speed that is the maximummoving speed when the density is zero 120588119898 is the maximumcrowd density and 119863 denotes diffusion coefficient which isa positive constant given by 119863 = 120591V2119903 where V119903 is a randomvelocity and 120591 is a relaxation parameter

Remark 1 The diffusion model (5) is an extension of theGreenshieldrsquosmodel where the speed depends not only on thetraffic density but also on the density gradient The diffusionterm demonstrates the fact that pedestrians can adjust theirmovement speed in real time based on the density aheadTheadjustment makes the change of their speed gradual ratherthan abrupt in response to the shock wave which creates thedependence of speed on density gradient [29]

Combining the LWR model (3) with the diffusion model(5) yields

120588119905 (119909 119905) = 119863120588119909119909 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 ) V119891]

(119909 119905) isin Ω(6)

Journal of Advanced Transportation 3

By choosing the free flow speed V119891 as the distributed controlvariable denoted by u(xt) one can derive

120588119905 (119909 119905) = 119863120588119909119909 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(7)

where 119906(119909 119905) isin [minusV119898 V119898] is the controller and V119898 is themaximum velocity The model is subject to the followinginitial condition and boundary condition

120588 (119909 1199050) = 1205880 (119909) forall119909 isin (0 119871) 120588 (0 119905) = 0120588 (119871 119905) = 0

forall119905 isin (0infin) (8)

Remark 2 The paper [30] has suggested that pedestriantraffic can be handled in the similar way as the vehicle trafficBut there is a main difference between pedestrian traffic andvehicle traffic In the vehicle traffic the car on a lane isunidirectional so its speed can be fixed by the traffic densityusing the diffusion model While in the pedestrian trafficpeople can move in both directions its density cannot fixthe speedwith any velocity-density relationshipmodel so thefree flow speed V119891 is chosen as the control variable and withthe actuation system people can be told to change their speed

3 Finite-Time Control

In this section by virtue of the finite-time control theory adistributed controller is designed to make the state convergeto zero in finite time The stability of the crowd evacuationsystem under the finite-time controller is analyzed using theLyapunov method

The important finite-time control theory ([19] Lemma43) is stated by the following lemma

Lemma 3 (see [19]) Let an everywhere nonnegative function119882(119905)meet the differential inequality

(119905) le minus2120574119882120572 (119905) (9)

for all 119905 ge 0 and for some constants 120574 ge 0 and 120572 isin (0 1) Then119882(119905) = 0 for all119905 ge [2120574 (1 minus 120572)]minus1119882(1minus120572) (0) (10)

The following lemma is an important inequality used inour proof which can be considered as a special case ofHolderintegral inequality

Lemma 4 (see [31]) Consider an arbitrary real coefficient 119901 ge1 and let 120588(sdot 119905) isin 119871119901(0 119871) where 119871119901(0 119871) is p-th integrableBanach space defined on interval (0 119871) Then the followinginequality holds

[int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816 119889119909]119901 le int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816119901 119889119909 (11)

In order to stabilize the crowd evacuation system (7) infinite time the following distributed controller is designed

119906 (119909 119905) = 1205821120588119898120588119898 minus 120588 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 (119909 119905) isin Ω (12)

where 1205821 is a constant control coefficient 1205721 isin (0 1) is aconstant

Remark 5 The distributed control mentioned in [32 33]means that the controller of a system can use informationof the connected systems to construct the control strategyHowever the distributed controller mentioned in this paperis the controller of distributed parameter systems (as opposedto a lumped parameter system) whose state space is infinite-dimensional

Theorem 6 The crowd evacuation system (7) subject tothe initial and boundary conditions (8) with the distributedcontroller (12) achieves the attainment of 120588(119909 119905) = 0 in 1198712norm when 119905 ge (21205821(1 minus 1205721))(int1198710 12058820 (119909)119889119909)(1minus1205721)2Proof Consider the Lyapunov functional

119882(119905) = 12 int119871

01205882 (119909 119905) d119909 119905 ge 0 (13)

Computing the time derivative of119882(119905) for 119905 ge 0 yields

(119905) = int1198710120588 (119909 119905) 120588119905 (119909 119905) d119909 = int

119871

0120588 (119909 119905)

sdot 119863120588119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] d119909

(14)

Substituting the controller (12) into (14) for 119905 ge 0 and usingthe Leibniz integral rule

119889119889119909 (int

119887(119909)

119886(119909)119891 (119909 119905) 119889119905)

= 119891 (119909 119887 (119909)) 119889119889119909119887 (119909) minus 119891 (119909 119886 (119909))

119889119889119909119886 (119909)

+ int119887(119909)119886(119909)

120597120597119909119891 (119909 119905) 119889119905

(15)


one can derive

(119905) = int1198710120588 (119909 119905) 119863120588119909119909 (119909 119905)

minus 1205821120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

minus 1205821120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 = 119863int119871

0120588 (119909 119905)

sdot 120588119909119909 (119909 119905) d119909 minus 1205821 int119871

0120588 (119909 119905) 120588119909 (119909 119905)

sdot int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus 1205821 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

(16)

Integrating by parts the first term of (16) and considering theboundary condition (8)

119863int1198710120588 (119909 119905) 120588119909119909 (119909 119905) d119909

= 119863120588 (119909 119905) 120588119909 (119909 119905)10038161003816100381610038161198710 minus 119863int11987101205882119909 (119909 119905) d119909

= minus119863int11987101205882119909 (119909 119905) d119909 119905 ge 0

(17)

As to the second term of (16) by using the same manipula-tions one can derive

int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= [1205882 (119909 119905) int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585]100381610038161003816100381610038161003816100381610038161003816119871

0

minus int1198710[120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

+ 120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172] 120588 (119909 119905) d119909 = minusint119871

0120588 (119909 119905)

sdot 120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909119905 ge 0

(18)

that is

int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= minus12 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 119905 ge 0(19)

Substituting (17) (19) into (16) the time derivative of119882(119905) for119905 ge 0 becomes

(119905) = minus119863int11987101205882119909 (119909 119905) d119909 minus 12058212 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

= minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212int1198710[1205882 (119909 119905)](2+1205721)2 d1199091003817100381710038171003817120588 (119909 119905)10038171003817100381710038172

(20)

As to the second term of (20) by virtue of Lemma 4 equation(20) can be rewritten in the form

(119905) le minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

le minus12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

= minus2(1205721minus1)21205821119882(1205721+1)2 (119905) 119905 ge 0

(21)

According to Lemma 3 when 119905 ge (21205821(1 minus1205721))(int1198710 12058820 (119909)d119909)(1minus1205721)2 119882(119905) converges to zero that isthe distributed controller (12) makes the crowd evacuationsystem (7)-(8) achieve the attainment of 120588(119909 119905) = 0 in 1198712norm

4 Finite-Time Tracking Control

In this section a finite-time tracking controller is designedto make the crowd density 120588(119909 119905) follow a given referencedensity 119877(119909 119905)Assumption 7 Thereference density119877(119909 119905) is smooth enoughand its spatial derivatives up to the second order are squareintegrable in 1198712 norm Also it satisfies the initial andboundary conditions

119877 (119909 0) = 1198770 (119909) forall119909 isin (0 119871) 119877 (0 119905) = 0119877 (119871 119905) = 0

forall119905 isin (0 +infin) (22)

Define the tracking error as 119890(119909 119905) = 120588(119909 119905) minus119877(119909 119905) (119909 119905) isin Ω By using (7) the error dynamic is givenas

119890119905 (119909 119905) = 119863119890119909119909 (119909 119905) + 119863119877119909119909 (119909 119905) minus 119877119905 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(23)


and the initial and boundary conditions are

119890 (119909 0) = 1205880 (119909) minus 1198770 (119909) forall119909 isin (0 119871) 119890 (0 119905) = 0119890 (119871 119905) = 0


In order to stabilize the error dynamic systems (23) infinite time the distributed controller is designed as

119906 (119909 119905) = 120588119898120588 (119909 119905) (120588119898 minus 120588 (119909 119905)) [1205822 int119909

0

1198901205722+1 (120585 119905)1003817100381710038171003817119890 (120585 119905)10038171003817100381710038172 d120585

minus int1199090119877119905 (120585 119905) d120585 + 119863int119909

0119877119909119909 (120585 119905) d120585] (119909 119905) isin Ω

(25)

where 120582 is a constant control coefficient and 120572 isin (0 1) is aconstant Then the following result is gotten

Theorem 8 Consider the crowd evacuation system (7) withthe initial and boundary conditions (8) and the reference den-sity 119877(119909 119905) satisfying Assumption 7 Then the error dynamicsystems (23) subject to the boundary conditions (24) can bestabilized to zero in1198712 normwith the distributed controller (25)when 119905 ge (11205822(1 minus 1205722))(int1198710 (1205880(119909) minus 1198770(119909))2119889119909)(1minus1205722)2Proof Consider the Lyapunov functional

119882119890 (119905) = 12 int119871

01198902 (119909 119905) d119909 119905 ge 0 (26)

Computing the time derivative of119882119890(119905) for 119905 ge 0 yields119890 (119905) = int

119871

0119890 (119909 119905) 119890119905 (119909 119905) d119909 = int

119871

0119890 (119909 119905)

sdot 119863119890119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] minus 119877119905 (119909 119905)

+ 119863119877119909119909 (119909 119905) d119909

(27)

Substituting the distributed controller (25) into (27) for 119905 ge 0and using the Leibniz integral rule (15) one can derive

119890 (119905) = int119871

0119890 (119909 119905) 119863119890119909119909 (119909 119905) minus 1205822 119890

1205722+1 (119909 119905)

119890 (119909 119905)2 d119909

= int1198710119890 (119909 119905) 119890119909119909 (119909 119905) d119909

minus 1205822 int119871

0

1198901205722+2 (119909 119905)119890 (119909 119905)2 d119909

(28)

As to the first term of (28) integrating it by parts andconsidering the boundary condition (24) the equation (28)can be written as

119890 (119905) = minus119863int11987101198902119909 (119909 119905) d119909

minus 1205822int119871

0[1198902 (119909 119905)](1205722+2)2 d119909

119890 (119909 119905)2 (29)

By virtue of Lemma 4 the time derivative of119882119890(119905) for 119905 ge 0is

119890 (119905) le minus1205822 [int119871

01198902 (119909 119905) d119909](1205722+1)2

= minus1205822 (2119882119890 (119905))(1205722+1)2 (30)

According to Lemma 3 the error dynamic system (23) withboundary conditions (24) can be stabilized to zero when 119905 ge(11205822(1 minus 1205722))(int1198710 (1205880(119909) minus 1198770(119909))2d119909)(1minus1205722)2Remark 9 In the application of crowd evacuation the crowddensity can be stabilized to different values to achievedifferent control objectives such as maximizing the evacu-ation flow and maximizing the pedestrian movement speedTherefore the research of tracking control is of great practicalsignificance Meanwhile estimating the time when the crowddensity stabilizes to the reference density can evaluate theeffectiveness of the control strategy

5 Simulation Results

In this section numerical results are given to illustrate theeffectiveness of the finite-time controller (12) and the finite-time tracking controller (25) respectively The numericalmethod is the finite volume method For simulation theinitial density is given by 120588(119909 0) = 119866 exp(minus(119909 minus 119886)2) with119886 = 2 being the center of the Gaussian distribution and119866 = 48 being the highest magnitude of the distribution Theconstant reference density is 119877119888 = 25 and a general referencedensity is given by 119877 = 25 + sin(05120587119909) cos(120587119905) The mainparameters are 119871 = 4 119863 = 01 1205821 = 2 1205721 = 08 1205822 =13 1205722 = 03 120588119898119886119909 = 5 V119898 = 14

When the controller 119906(119909 119905) = 0 the mathematical modelof the crowd evacuation system (7) shown in Figure 1 is adiffusion model 120588119905(119909 119905) = 119863120588119909119909(119909 119905)

Figure 2 illustrates the density response of the crowd evac-uation system with the finite-time controller (12) Because ofthe effect of the advection term the density profile movestowards the exit (119909 = 4) and converges to zero in finite timeFor a clearer demonstration the density evolutions at 119909 = 2and119909 = 4 are shown in Figure 3Thedensity at119909 = 2 becomeszero at about 61 seconds the density at 119909 = 4 becomes zero


01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity

Figure 1 Density response of the uncontrolled crowd evacuationsystem

0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)

Figure 2 Finite-time control of the crowd evacuation system

at about 71 seconds and the crowd evacuation process endsThe evacuation time calculated byTheorem 6 is

119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2

= 22 lowast (1 minus 08) (int

119871

0(48

lowast exp (minus (119909 minus 2)2))2 d119909)(1minus08)2 = 69987

(31)

It can be easily seen that the calculated evacuation time andthe simulated evacuation time are almost equal and the erroris within a reasonable range

Figure 4 demonstrates the density response of the crowdevacuation system with the finite-time tracking controller(25) where 119877(119909 119905) = 25 is chosen as the reference densityThe density evolutions of 119909 = 1 and 119909 = 2 are shown inFigure 5 It can be seen that the density at 119909 = 1 converges

0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)

Figure 3 Density evolution at 119909 = 2 and 119909 = 4 with the finite-timecontroller

2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)

Figure 4 Finite-time tracking control with constant reference 119877119888 =25

to 25 at about 19 seconds and the density at 119909 = 2 convergesto 25 at about 22 secondsThe evacuation time calculated byTheorem 8 is

119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871

0(48 lowast exp (minus (119909 minus 2)2)

minus 25)2 d119909)(1minus03)2 = 25861

(32)

There is a small error between the calculated evacuation timeand the simulated evacuation time which may be caused bythe discretization of simulation but it is within a reasonableerror range

Next a more general reference density 119877 = 25 +sin(05120587119909) cos(120587119905) is selected to show the effectiveness of thefinite-time tracking controller as shown in Figure 6

The density response of the crowd dynamic systemunder the finite-time tracking controller is demonstrated in


0 1 2 3 4 5012345

Time t

Den

sity

(1t)

with controlreference density


0 1 2 3 4 5012345

Time t

Den

sity

(2t)

Figure 5 Density evolution at 119909 = 1 and 119909 = 2 with ConstantReference 119877119888 = 25

24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)

Figure 6 Reference density 119877 = 25 + sin(05120587119909) cos(120587119905)

Figure 7 Figure 8 illustrates the density evolution at119909 = 1 and119909 = 2with reference119877 = 25+sin(05120587119909) cos(120587119905)The densityat119909 = 1 reaches the reference density at about 20 seconds andthe density at 119909 = 2 reaches the reference density at about29 seconds Comparing the evacuation time calculated byTheorem 8

119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)

Figure 7 Finite-time tracking control with reference 119877 = 25 +sin(05120587119909) cos(120587119905)

0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)

Figure 8 Density evolution at 119909 = 1 and 119909 = 2 with reference119877 = 25 + sin(05120587119909) cos(120587119905)

minus 25 minus sin (05 lowast 119901119894 lowast 119909))2 d119909)(1minus03)2

= 27350(33)

and they are almost equalTo sum up the effectiveness of the designed controllers

has been shown by the comparisons and the calculatedevacuation time is almost equal to the simulated evacuationtime so the estimated evacuation time mentioned in thetheorem is feasible


6 Conclusion

In this paper the crowd dynamic model was constructed bycombining the LWR model and the diffusion model Thenfinite-time controllers were designed for the crowd evacua-tion system which solved the problem of nondirectionalityof crowd movement and got the estimation of evacuationtimeThis theoretical research can promote the improvementof practical application but the effect of time delay anddisturbance in implementation needs to be further studied

Data Availability

All the data used to support the findings of this studyare available from the corresponding author uponrequest The email address of the corresponding authoris weiqinvipjiangnaneducn

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by National Natural Science Foun-dation of China (grant number 61807016) and PostgraduateResearch amp Practice Innovation Program of Jiangsu Province(grant number KYCX17 1460)References

[1] R A Smith and J F Dickie ldquoEngineering for crowd safetyrdquo inProceedings of the International Conference on Engineering forCrowd Safety Elsevier Amsterdam The Netherlands 1993

[2] R Guo H Huang and S Wong ldquoCollection spillback anddissipation in pedestrian evacuation a network-basedmethodrdquoTransportation Research Part B Methodological vol 45 no 3pp 490ndash506 2011

[3] N Wagner and V Agrawal ldquoAn agent-based simulation systemfor concert venue crowd evacuation modeling in the presenceof a fire disasterrdquo Expert Systems with Applications vol 41 no6 pp 2807ndash2815 2014

[4] N Chooramun P J Lawrence and E R Galea ldquoAn agent basedevacuation model utilising hybrid space discretisationrdquo SafetyScience vol 50 no 8 pp 1685ndash1694 2012

[5] A Lachapelle andMWolfram ldquoOn ameanfield game approachmodeling congestion and aversion in pedestrian crowdsrdquoTrans-portation Research Part B Methodological vol 45 no 10 pp1572ndash1589 2011

[6] R-Y Guo H-J Huang and S C Wong ldquoRoute choice inpedestrian evacuation under conditions of good and zerovisibility experimental and simulation resultsrdquo TransportationResearch Part B Methodological vol 46 no 6 pp 669ndash6862012

[7] L A Pereira L H Duczmal and F R B Cruz ldquoCongestedemergency evacuation of a population using a finite automataapproachrdquo Safety Science vol 51 no 1 pp 267ndash272 2013

[8] K Cao Y Q Chen D Stuart and D Yue ldquoCyber-physicalmodeling and control of crowd of pedestrians a review and new

frameworkrdquo IEEECAA Journal of Automatica Sinica vol 2 no3 pp 334ndash344 2015

[9] D Helbing and A Johansson ldquoPedestrian crowd and evacua-tion dynamicsrdquo in Extreme Environmental Events pp 697ndash716Springer New York NY USA 2011

[10] R L Hughes ldquoA continuum theory for the flow of pedestriansrdquoTransportation Research Part B Methodological vol 36 no 6pp 507ndash535 2002

[11] L Huang S C Wong M Zhang C-W Shu andW H K LamldquoRevisiting Hughesrsquo dynamic continuum model for pedestrianflow and the development of an efficient solution algorithmrdquoTransportation Research Part B Methodological vol 43 no 1pp 127ndash141 2009

[12] C Appert-Rolland P Degond and S Motsch ldquoTwo-waymulti-lane traffic model for pedestrians in corridorsrdquo Networks andHeterogeneous Media vol 6 no 3 pp 351ndash381 2011

[13] F S Hanseler M Bierlaire B Farooq and T MuhlematterldquoA macroscopic loading model for time-varying pedestrianflows in public walking areasrdquo Transportation Research Part BMethodological vol 69 pp 60ndash80 2014

[14] M Twarogowska P Goatin and R Duvigneau ldquoCompara-tive study of macroscopic pedestrian modelsrdquo TransportationResearch Procedia vol 2 pp 477ndash485 2014

[15] S A Wadoo ldquoSliding mode control of crowd dynamicsrdquo IEEETransactions on Control Systems Technology vol 21 no 3 pp1008ndash1015 2013

[16] S A Wadoo and P Kachroo ldquoFeedback control of crowdevacuation in one dimensionrdquo IEEE Transactions on IntelligentTransportation Systems vol 11 no 1 pp 182ndash193 2010

[17] H Dong X Yang Y Chen and Q Wang ldquoPedestrian evacua-tion in two-dimension via state feedback controlrdquo in Proceed-ings of the 2013 1st American Control Conference ACC 2013 pp302ndash306 USA June 2013

[18] W Qin B Zhuang and B Cui ldquoBoundary control of the crowdevacuation system based on continuum modelrdquo Control andDecision vol 33 no 11 pp 2073ndash2079 2018

[19] Y V Orlov Discontinuous SystemsndashLyapunov Analysis andRobust Synthesis Under Uncertainty Conditions Springer-Verlag Berlin Germany 2009

[20] G Zhang C Huang X Zhang and W Zhang ldquoPracticalconstrained dynamic positioning control for uncertain shipthrough the minimal learning parameter techniquerdquo IET Con-trol Theory amp Applications vol 12 no 18 pp 2526ndash2533 2018

[21] X Zhao X Wang S Zhang and G Zong ldquoAdaptive neuralbackstepping control design for a class of nonsmooth nonlinearsystemsrdquo IEEE Transactions on Systems Man and CyberneticsSystems pp 1ndash12 2018

[22] L Ma X Huo X Zhao B Niu and G Zong ldquoAdaptiveneural control for switched nonlinear systems with unknownbacklash-like hysteresis and output dead-zonerdquo Neurocomput-ing vol 357 pp 203ndash214 2019

[23] X Chang R Huang and J Park ldquoRobust guaranteed cost con-trol under digital communication channelsrdquo IEEE Transactionson Industrial Informatics pp 1ndash9 2019

[24] X Chang R Liu and JH Park ldquoA further study on output feed-back Hinfin control for discrete-time systemsrdquo IEEE Transactionson Circuits and Systems II Express Briefs pp 1-1 2019

[25] G Zhang Y Deng W Zhang and C Huang ldquoNovel DVSguidance and path-following control for underactuated shipsin presence of multiple static and moving obstaclesrdquo OceanEngineering vol 170 pp 100ndash110 2018


[26] X Zhao X Wang L Ma and G Zong ldquoFuzzy-approximation-based asymptotic tracking control for a class of uncertainswitched nonlinear systemsrdquo IEEE Transactions on Fuzzy Sys-tems pp 1-1 2019

[27] M J Lighthill and G BWhitham ldquoOn kinematic waves I flowmovement in long rivers ii a theory of traffic flow on longcrowded roadsrdquo Pharmacology ampTherapeutics vol 53 no 3 pp317ndash345 1955

[28] P I Richards ldquoShock waves on the highwayrdquo OperationsResearch vol 4 no 1 pp 42ndash51 1956

[29] P Kachroo K M and K M Ozbay Feedback RampMetering inIntelligent Transportation Systems Springer Science amp BusinessMedia 2011

[30] A C May Traffic Flow Fundamental PrenticeHall EnglewoodCliffs NJ USA 1990

[31] M Abramowitz and I A Stegun Handbook of MathematicalFunctions with Formulas Graphs and Mathematical TablesDover New York NY USA 1972

[32] X Jin S Wang G Yang and D Ye ldquoRobust adaptive hierar-chical insensitive tracking control of a class of leader-followeragentsrdquo Information Sciences vol 406-407 pp 234ndash247 2017

[33] X Jin S Wang J QinW Zheng and Y Kang ldquoAdaptive fault-tolerant consensus for a class of uncertain nonlinear second-order multi-agent systems with circuit implementationrdquo IEEETransactions on Circuits and Systems I Regular Papers vol 65no 7 pp 2243ndash2255 2018

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Submit your manuscripts atwwwhindawicom


crowd model Robin Neumann and Dirichlet boundarycontrol laws are designed for a disturbed crowd evacuationsystem in [18]

In this paper the problem of finite-time evacuation isstudied In some special circumstances such as fires andterrorist attacks pedestrians need to be evacuated as soonas possible Evacuation strategy cannot be used until itseffectiveness is tested because the cost involves not onlyproperty damage but also pedestrian injury or death There-fore it is particularly important to estimate the evacua-tion time The estimated evacuation time can be used toevaluate the effectiveness of the evacuation strategy andthe corresponding evacuation strategy can be adjusted tominimizing the loss Benefiting from Orlovrsquos research [19]finite-time controller is designed to evacuate pedestriansin finite time and then the finite-time control method isextended to deal with the tracking problem which makesthe crowd density follow a given reference density Someof the latest research on advanced control such as adaptivecontrol [20ndash22] guaranteed cost control [23] 119867infin control[24] and tracking control [25 26] gave us inspiration in thedesign of the controllerThe control and stability problem areformulated directly in the framework of a distributed modelof partial differential equations (PDEs) which can avoiderrors introduced by spatial discretization The developmentof hardware technology has made distributed sensors andactuators realistic but we mainly explore the crowd manage-ment strategies on theoretical interest in this paper and theimplementation needs further study in the future

The rest of this paper is organized as follows One dimen-sional crowd evacuation dynamic is modeled in Section 2Section 3 presents a feedback controller to evacuate pedes-trians in finite time Section 4 designs a finite-time controllerto make the crowd density profile track a reference densitySimulation examples are given in Section 5 to illustrate theeffectiveness of the controllers Conclusions and future workare discussed in Section 6

NotationThenotation is used throughout the paper1198672(0 119871)denotes the infinite-dimensional Hilbert space on interval[0 119871] 119871 gt 0 represents the interval length with 1198712 norm

1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 = [int119871

01205882 (119909 119905) d119909]12

120588 (119909 119905) isin 1198672 (0 119871) (1)

For notational convenience we denote

120588119905 (119909 119905) = 120597120588 (119909 119905)120597119905

120588119909 (119909 119905) = 120597120588 (119909 119905)120597119909

120588119909119909 (119909 119905) = 1205972120588 (119909 119905)1205971199092

(2)

2 Mathematical Modeling

The Lighthill-Whitham-Richards (LWR) model [27 28]based on the conservation law of mass is recommended inthis paper to represent the crowd evacuation dynamics in onedimension implying that the number of pedestrians comingin and going out of a corridor section account for the changeof crowd density on that section The LWRmodel is given by

120588119905 (119909 119905) + 119902119909 (119909 119905) = 0 (119909 119905) isin Ω (3)

where Ω = (0 119871) times (0 +infin) 119909 and 119905 are space and timevariables respectively 120588(119909 119905) is the average crowd densityand 120588119905(119909 119905) is the partial derivative of the density 120588(119909 119905) withrespect to time 119905 at position 119909 119902(119909 119905) denotes the flux ofthe crowd and 119902119909(119909 119905) is the partial derivative of the flux119902(119909 119905) with respect to position 119909 at time 119905 The flux 119902(119909 119905) isa function of 120588(119909 119905) and the average crowd speed V(119909 119905) asshown below

119902 (119909 119905) = 120588 (119909 119905) V (119909 119905) (119909 119905) isin Ω (4)

Various models have been developed to mimic thevelocity-density relationship such as Greenshield modelDrew model Greenberg model Pipes Munjal model andUnderwood model [29] Here the diffusion model is recom-mended which is given as

V (119909 119905) = V119891 [1 minus 120588 (119909 119905)120588119898 ] minus 119863120588119909 (119909 119905)120588 (119909 119905) (119909 119905) isin Ω (5)

where V119891 denotes the free flow speed that is the maximummoving speed when the density is zero 120588119898 is the maximumcrowd density and 119863 denotes diffusion coefficient which isa positive constant given by 119863 = 120591V2119903 where V119903 is a randomvelocity and 120591 is a relaxation parameter

Remark 1 The diffusion model (5) is an extension of theGreenshieldrsquosmodel where the speed depends not only on thetraffic density but also on the density gradient The diffusionterm demonstrates the fact that pedestrians can adjust theirmovement speed in real time based on the density aheadTheadjustment makes the change of their speed gradual ratherthan abrupt in response to the shock wave which creates thedependence of speed on density gradient [29]

Combining the LWR model (3) with the diffusion model(5) yields

120588119905 (119909 119905) = 119863120588119909119909 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 ) V119891]

(119909 119905) isin Ω(6)



120588119905 (119909 119905) = 119863120588119909119909 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(7)


120588 (119909 1199050) = 1205880 (119909) forall119909 isin (0 119871) 120588 (0 119905) = 0120588 (119871 119905) = 0







(119905) le minus2120574119882120572 (119905) (9)




[int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816 119889119909]119901 le int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816119901 119889119909 (11)


119906 (119909 119905) = 1205821120588119898120588119898 minus 120588 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 (119909 119905) isin Ω (12)




119882(119905) = 12 int119871

01205882 (119909 119905) d119909 119905 ge 0 (13)


(119905) = int1198710120588 (119909 119905) 120588119905 (119909 119905) d119909 = int

119871

0120588 (119909 119905)

sdot 119863120588119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] d119909

(14)


119889119889119909 (int

119887(119909)

119886(119909)119891 (119909 119905) 119889119905)

= 119891 (119909 119887 (119909)) 119889119889119909119887 (119909) minus 119891 (119909 119886 (119909))

119889119889119909119886 (119909)

+ int119887(119909)119886(119909)

120597120597119909119891 (119909 119905) 119889119905

(15)


one can derive

(119905) = int1198710120588 (119909 119905) 119863120588119909119909 (119909 119905)

minus 1205821120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

minus 1205821120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 = 119863int119871

0120588 (119909 119905)

sdot 120588119909119909 (119909 119905) d119909 minus 1205821 int119871

0120588 (119909 119905) 120588119909 (119909 119905)

sdot int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus 1205821 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

(16)


119863int1198710120588 (119909 119905) 120588119909119909 (119909 119905) d119909

= 119863120588 (119909 119905) 120588119909 (119909 119905)10038161003816100381610038161198710 minus 119863int11987101205882119909 (119909 119905) d119909

= minus119863int11987101205882119909 (119909 119905) d119909 119905 ge 0

(17)


int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= [1205882 (119909 119905) int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585]100381610038161003816100381610038161003816100381610038161003816119871

0

minus int1198710[120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

+ 120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172] 120588 (119909 119905) d119909 = minusint119871

0120588 (119909 119905)

sdot 120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909119905 ge 0

(18)

that is

int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= minus12 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 119905 ge 0(19)



0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

= minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212int1198710[1205882 (119909 119905)](2+1205721)2 d1199091003817100381710038171003817120588 (119909 119905)10038171003817100381710038172

(20)


(119905) le minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

le minus12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

= minus2(1205721minus1)21205821119882(1205721+1)2 (119905) 119905 ge 0

(21)




119877 (119909 0) = 1198770 (119909) forall119909 isin (0 119871) 119877 (0 119905) = 0119877 (119871 119905) = 0




120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(23)






119906 (119909 119905) = 120588119898120588 (119909 119905) (120588119898 minus 120588 (119909 119905)) [1205822 int119909

0

1198901205722+1 (120585 119905)1003817100381710038171003817119890 (120585 119905)10038171003817100381710038172 d120585

minus int1199090119877119905 (120585 119905) d120585 + 119863int119909

0119877119909119909 (120585 119905) d120585] (119909 119905) isin Ω

(25)



119882119890 (119905) = 12 int119871

01198902 (119909 119905) d119909 119905 ge 0 (26)


119871

0119890 (119909 119905) 119890119905 (119909 119905) d119909 = int

119871

0119890 (119909 119905)

sdot 119863119890119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] minus 119877119905 (119909 119905)

+ 119863119877119909119909 (119909 119905) d119909

(27)


119890 (119905) = int119871

0119890 (119909 119905) 119863119890119909119909 (119909 119905) minus 1205822 119890

1205722+1 (119909 119905)

119890 (119909 119905)2 d119909

= int1198710119890 (119909 119905) 119890119909119909 (119909 119905) d119909

minus 1205822 int119871

0

1198901205722+2 (119909 119905)119890 (119909 119905)2 d119909

(28)


119890 (119905) = minus119863int11987101198902119909 (119909 119905) d119909

minus 1205822int119871

0[1198902 (119909 119905)](1205722+2)2 d119909

119890 (119909 119905)2 (29)


119890 (119905) le minus1205822 [int119871

01198902 (119909 119905) d119909](1205722+1)2

= minus1205822 (2119882119890 (119905))(1205722+1)2 (30)







01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity


0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2


119871

0(48


(31)



0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


minus 25)2 d119909)(1minus03)2 = 25861

(32)





0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




120588119905 (119909 119905) = 119863120588119909119909 (119909 119905)minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(7)


120588 (119909 1199050) = 1205880 (119909) forall119909 isin (0 119871) 120588 (0 119905) = 0120588 (119871 119905) = 0







(119905) le minus2120574119882120572 (119905) (9)




[int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816 119889119909]119901 le int1198710

100381610038161003816100381610038161205882 (119909 119905)10038161003816100381610038161003816119901 119889119909 (11)


119906 (119909 119905) = 1205821120588119898120588119898 minus 120588 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 (119909 119905) isin Ω (12)




119882(119905) = 12 int119871

01205882 (119909 119905) d119909 119905 ge 0 (13)


(119905) = int1198710120588 (119909 119905) 120588119905 (119909 119905) d119909 = int

119871

0120588 (119909 119905)

sdot 119863120588119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] d119909

(14)


119889119889119909 (int

119887(119909)

119886(119909)119891 (119909 119905) 119889119905)

= 119891 (119909 119887 (119909)) 119889119889119909119887 (119909) minus 119891 (119909 119886 (119909))

119889119889119909119886 (119909)

+ int119887(119909)119886(119909)

120597120597119909119891 (119909 119905) 119889119905

(15)


one can derive

(119905) = int1198710120588 (119909 119905) 119863120588119909119909 (119909 119905)

minus 1205821120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

minus 1205821120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 = 119863int119871

0120588 (119909 119905)

sdot 120588119909119909 (119909 119905) d119909 minus 1205821 int119871

0120588 (119909 119905) 120588119909 (119909 119905)

sdot int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus 1205821 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

(16)


119863int1198710120588 (119909 119905) 120588119909119909 (119909 119905) d119909

= 119863120588 (119909 119905) 120588119909 (119909 119905)10038161003816100381610038161198710 minus 119863int11987101205882119909 (119909 119905) d119909

= minus119863int11987101205882119909 (119909 119905) d119909 119905 ge 0

(17)


int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= [1205882 (119909 119905) int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585]100381610038161003816100381610038161003816100381610038161003816119871

0

minus int1198710[120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

+ 120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172] 120588 (119909 119905) d119909 = minusint119871

0120588 (119909 119905)

sdot 120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909119905 ge 0

(18)

that is

int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= minus12 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 119905 ge 0(19)



0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

= minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212int1198710[1205882 (119909 119905)](2+1205721)2 d1199091003817100381710038171003817120588 (119909 119905)10038171003817100381710038172

(20)


(119905) le minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

le minus12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

= minus2(1205721minus1)21205821119882(1205721+1)2 (119905) 119905 ge 0

(21)




119877 (119909 0) = 1198770 (119909) forall119909 isin (0 119871) 119877 (0 119905) = 0119877 (119871 119905) = 0




120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(23)






119906 (119909 119905) = 120588119898120588 (119909 119905) (120588119898 minus 120588 (119909 119905)) [1205822 int119909

0

1198901205722+1 (120585 119905)1003817100381710038171003817119890 (120585 119905)10038171003817100381710038172 d120585

minus int1199090119877119905 (120585 119905) d120585 + 119863int119909

0119877119909119909 (120585 119905) d120585] (119909 119905) isin Ω

(25)



119882119890 (119905) = 12 int119871

01198902 (119909 119905) d119909 119905 ge 0 (26)


119871

0119890 (119909 119905) 119890119905 (119909 119905) d119909 = int

119871

0119890 (119909 119905)

sdot 119863119890119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] minus 119877119905 (119909 119905)

+ 119863119877119909119909 (119909 119905) d119909

(27)


119890 (119905) = int119871

0119890 (119909 119905) 119863119890119909119909 (119909 119905) minus 1205822 119890

1205722+1 (119909 119905)

119890 (119909 119905)2 d119909

= int1198710119890 (119909 119905) 119890119909119909 (119909 119905) d119909

minus 1205822 int119871

0

1198901205722+2 (119909 119905)119890 (119909 119905)2 d119909

(28)


119890 (119905) = minus119863int11987101198902119909 (119909 119905) d119909

minus 1205822int119871

0[1198902 (119909 119905)](1205722+2)2 d119909

119890 (119909 119905)2 (29)


119890 (119905) le minus1205822 [int119871

01198902 (119909 119905) d119909](1205722+1)2

= minus1205822 (2119882119890 (119905))(1205722+1)2 (30)







01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity


0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2


119871

0(48


(31)



0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


minus 25)2 d119909)(1minus03)2 = 25861

(32)





0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



one can derive

(119905) = int1198710120588 (119909 119905) 119863120588119909119909 (119909 119905)

minus 1205821120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

minus 1205821120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 = 119863int119871

0120588 (119909 119905)

sdot 120588119909119909 (119909 119905) d119909 minus 1205821 int119871

0120588 (119909 119905) 120588119909 (119909 119905)

sdot int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus 1205821 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

(16)


119863int1198710120588 (119909 119905) 120588119909119909 (119909 119905) d119909

= 119863120588 (119909 119905) 120588119909 (119909 119905)10038161003816100381610038161198710 minus 119863int11987101205882119909 (119909 119905) d119909

= minus119863int11987101205882119909 (119909 119905) d119909 119905 ge 0

(17)


int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= [1205882 (119909 119905) int1199090

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585]100381610038161003816100381610038161003816100381610038161003816119871

0

minus int1198710[120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585

+ 120588 (119909 119905) 1205881205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172] 120588 (119909 119905) d119909 = minusint119871

0120588 (119909 119905)

sdot 120588119909 (119909 119905) int119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909 minus int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909119905 ge 0

(18)

that is

int1198710120588 (119909 119905) 120588119909 (119909 119905) int

119909

0

1205881205721 (120585 119905)1003817100381710038171003817120588 (120585 119905)10038171003817100381710038172 d120585 d119909

= minus12 int119871

0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909 119905 ge 0(19)



0

1205882+1205721 (119909 119905)1003817100381710038171003817120588 (119909 119905)10038171003817100381710038172 d119909

= minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212int1198710[1205882 (119909 119905)](2+1205721)2 d1199091003817100381710038171003817120588 (119909 119905)10038171003817100381710038172

(20)


(119905) le minus119863int11987101205882119909 (119909 119905) d119909

minus 12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

le minus12058212 [int11987101205882 (119909 119905) d119909](1+1205721)2

= minus2(1205721minus1)21205821119882(1205721+1)2 (119905) 119905 ge 0

(21)




119877 (119909 0) = 1198770 (119909) forall119909 isin (0 119871) 119877 (0 119905) = 0119877 (119871 119905) = 0




120588 (119909 119905)120588119898 )119906 (119909 119905)]

(119909 119905) isin Ω(23)






119906 (119909 119905) = 120588119898120588 (119909 119905) (120588119898 minus 120588 (119909 119905)) [1205822 int119909

0

1198901205722+1 (120585 119905)1003817100381710038171003817119890 (120585 119905)10038171003817100381710038172 d120585

minus int1199090119877119905 (120585 119905) d120585 + 119863int119909

0119877119909119909 (120585 119905) d120585] (119909 119905) isin Ω

(25)



119882119890 (119905) = 12 int119871

01198902 (119909 119905) d119909 119905 ge 0 (26)


119871

0119890 (119909 119905) 119890119905 (119909 119905) d119909 = int

119871

0119890 (119909 119905)

sdot 119863119890119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] minus 119877119905 (119909 119905)

+ 119863119877119909119909 (119909 119905) d119909

(27)


119890 (119905) = int119871

0119890 (119909 119905) 119863119890119909119909 (119909 119905) minus 1205822 119890

1205722+1 (119909 119905)

119890 (119909 119905)2 d119909

= int1198710119890 (119909 119905) 119890119909119909 (119909 119905) d119909

minus 1205822 int119871

0

1198901205722+2 (119909 119905)119890 (119909 119905)2 d119909

(28)


119890 (119905) = minus119863int11987101198902119909 (119909 119905) d119909

minus 1205822int119871

0[1198902 (119909 119905)](1205722+2)2 d119909

119890 (119909 119905)2 (29)


119890 (119905) le minus1205822 [int119871

01198902 (119909 119905) d119909](1205722+1)2

= minus1205822 (2119882119890 (119905))(1205722+1)2 (30)







01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity


0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2


119871

0(48


(31)



0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


minus 25)2 d119909)(1minus03)2 = 25861

(32)





0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







119906 (119909 119905) = 120588119898120588 (119909 119905) (120588119898 minus 120588 (119909 119905)) [1205822 int119909

0

1198901205722+1 (120585 119905)1003817100381710038171003817119890 (120585 119905)10038171003817100381710038172 d120585

minus int1199090119877119905 (120585 119905) d120585 + 119863int119909

0119877119909119909 (120585 119905) d120585] (119909 119905) isin Ω

(25)



119882119890 (119905) = 12 int119871

01198902 (119909 119905) d119909 119905 ge 0 (26)


119871

0119890 (119909 119905) 119890119905 (119909 119905) d119909 = int

119871

0119890 (119909 119905)

sdot 119863119890119909119909 (119909 119905)

minus 120597120597119909 [120588 (119909 119905) (1 minus

120588 (119909 119905)120588119898 )119906 (119909 119905)] minus 119877119905 (119909 119905)

+ 119863119877119909119909 (119909 119905) d119909

(27)


119890 (119905) = int119871

0119890 (119909 119905) 119863119890119909119909 (119909 119905) minus 1205822 119890

1205722+1 (119909 119905)

119890 (119909 119905)2 d119909

= int1198710119890 (119909 119905) 119890119909119909 (119909 119905) d119909

minus 1205822 int119871

0

1198901205722+2 (119909 119905)119890 (119909 119905)2 d119909

(28)


119890 (119905) = minus119863int11987101198902119909 (119909 119905) d119909

minus 1205822int119871

0[1198902 (119909 119905)](1205722+2)2 d119909

119890 (119909 119905)2 (29)


119890 (119905) le minus1205822 [int119871

01198902 (119909 119905) d119909](1205722+1)2

= minus1205822 (2119882119890 (119905))(1205722+1)2 (30)







01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity


0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2


119871

0(48


(31)



0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


minus 25)2 d119909)(1minus03)2 = 25861

(32)





0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



01

23

45

01

23

40

1

2

3

4

5

TimeDistance

Den

sity


0

5

10

01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 21205821 (1 minus 1205721) (int

119871

012058820 (119909) d119909)

(1minus1205721)2


119871

0(48


(31)



0 2 4 6 8 100

2

4

6

Time t

Den

sity

(2t)

0 2 4 6 8 100

1

2

3

Time t

Den

sity

(4t)


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


minus 25)2 d119909)(1minus03)2 = 25861

(32)





0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)


24 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Refe

renc

e den

sity

R(x

t)



119905 ge 11205822 (1 minus 1205722) (int

119871

0(1205880 (119909) minus 1198770 (119909))2 d119909)

(1minus1205722)2


119871


2 4 6 8 10

0 01

23

40

1

2

3

4

5

Time tDistance x

Den

sity

(xt)


0 1 2 3 4 5012345

Time t

Den

sity

(1t)



0 1 2 3 4 5012345

Time t

Den

sity

(2t)



= 27350(33)




6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



6 Conclusion


Data Availability




Acknowledgments







































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia













RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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Finite-Time Control of One Dimensional Crowd Evacuation Systemdownloads.hindawi.com/journals/jat/2019/6597360.pdf · 2019-08-05 · Finite-Time Control of One Dimensional Crowd Evacuation