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This paper pertains to the study of finite-time control of one dimensional crowd evacuation system. Benefiting from the research of fluid dynamics and vehicle traffic, a one dimensional crowd evacuation system is constructed, whose density-velocity relationship is represented by a diffusion model. In order to deal with the nondirectionality of crowd movement, the free flow speed is chosen as a control variable. Since the control variable is included in a partial derivative, it increases the difficulty of designing the controller. In this paper, finite-time controller is designed, which not only guarantees the effective evacuation, but also obtains the estimation of evacuation time. Then, finite-time tracking problem is solved, which makes the density converge to a given density. Finally, numerical examples illustrate the effectiveness of the controllers.

In everyday life, crowds would gather in many places, for example, subway stations, stadiums, and cinemas. Effective measures should be taken to ensure the safety and comfort of pedestrians, so how to evacuate people when emergencies occur has been a challenging job. In the early 1990s, the International Conference on Engineering for Crowd Safety [

Modeling the crowd dynamics is the primary task of crowd evacuation, but due to the complexity and uncertainty of crowd dynamics, it is very difficult to build a model that suits all situations. Therefore, various models have been developed, such as the continuum model, the network-based model [

In this paper, the continuum model, a macroscopic simulation model, is recommended. At medium and high densities, the motion of pedestrian crowds shows some striking analogies with the motion of fluids [

In this paper, the problem of finite-time evacuation is studied. In some special circumstances, such as fires and terrorist attacks, pedestrians need to be evacuated as soon as possible. Evacuation strategy cannot be used until its effectiveness is tested, because the cost involves not only property damage but also pedestrian injury or death. Therefore, it is particularly important to estimate the evacuation time. The estimated evacuation time can be used to evaluate the effectiveness of the evacuation strategy, and the corresponding evacuation strategy can be adjusted to minimizing the loss. Benefiting from Orlov’s research [

The rest of this paper is organized as follows. One dimensional crowd evacuation dynamic is modeled in Section

The Lighthill-Whitham-Richards (LWR) model [

Various models have been developed to mimic the velocity-density relationship, such as Greenshield model, Drew model, Greenberg model, Pipes Munjal model, and Underwood model [

The diffusion model (

Combining the LWR model (

The paper [

In this section, by virtue of the finite-time control theory, a distributed controller is designed to make the state converge to zero in finite time. The stability of the crowd evacuation system under the finite-time controller is analyzed using the Lyapunov method.

The important finite-time control theory ([

Let an everywhere nonnegative function

The following lemma is an important inequality used in our proof, which can be considered as a special case of Hölder integral inequality.

Consider an arbitrary real coefficient

In order to stabilize the crowd evacuation system (

The distributed control mentioned in [

The crowd evacuation system (

Consider the Lyapunov functional

In this section, a finite-time tracking controller is designed to make the crowd density

The reference density

Define the tracking error as

In order to stabilize the error dynamic systems (

Consider the crowd evacuation system (

Consider the Lyapunov functional

In the application of crowd evacuation, the crowd density can be stabilized to different values to achieve different control objectives, such as maximizing the evacuation flow and maximizing the pedestrian movement speed. Therefore, the research of tracking control is of great practical significance. Meanwhile, estimating the time when the crowd density stabilizes to the reference density can evaluate the effectiveness of the control strategy.

In this section, numerical results are given to illustrate the effectiveness of the finite-time controller (

When the controller

Density response of the uncontrolled crowd evacuation system.

Figure

Finite-time control of the crowd evacuation system.

Density evolution at

Figure

Finite-time tracking control with constant reference

Density evolution at

Next, a more general reference density

Reference density

The density response of the crowd dynamic system under the finite-time tracking controller is demonstrated in Figure

Finite-time tracking control with reference

Density evolution at

To sum up, the effectiveness of the designed controllers has been shown by the comparisons, and the calculated evacuation time is almost equal to the simulated evacuation time, so the estimated evacuation time mentioned in the theorem is feasible.

In this paper, the crowd dynamic model was constructed by combining the LWR model and the diffusion model. Then, finite-time controllers were designed for the crowd evacuation system, which solved the problem of nondirectionality of crowd movement and got the estimation of evacuation time. This theoretical research can promote the improvement of practical application, but the effect of time delay and disturbance in implementation needs to be further studied.

All the data used to support the findings of this study are available from the corresponding author upon request. The email address of the corresponding author is

The authors declare that they have no conflicts of interest.

This work was supported by National Natural Science Foundation of China (grant number 61807016) and Postgraduate Research & Practice Innovation Program of Jiangsu Province (grant number KYCX