## 2 Related Work
### Stability Analysis for Crowd Flow  
Investigations on crowd flows began later than those on traffic flows; as such, there are relatively few studies on crowd stability, most of which are based on theories of traffic flow stability analysis, such as boundary control, sliding mode control, and feedback control [15; 16; 17]. However, owing to the stochastics of crowd movement without expected lanes in vehicle traffic, it is more complex to study crowd flow stability than only traffic flow stability. The existing research methods of macro crowd flow stability remain in the initial stages and can be mainly divided into four categories: methods based on statistical analysis, methods based on computer vision, methods based on control theory, and methods based on mathematical calculation, as shown below. (1) Methods based on statistical analysis. These methods rely on data collection, literature retrieval, and statistical analysis of past crowd disaster events, aiming to explore the unstable factors of the crowds and to provide advice on crowd management. The Guide to Safety at Sports Grounds (the \"Green Guide\") in the United Kingdom established a safety standard for pedestrian crowds based on experience analysis. It considered a crowd with a density between 4 p/m\({}^{2}\) and 7 p/m\({}^{2}\) as a crowd in a critical state [18]. [PERSON] [PERSON] [19] analyzed the stampede in Mexico City on 20 June 2008. The management oversight risk tree technique and the FIST model have also been proposed and can be used to analyze and extract the unstable and fatal factors in a disaster. In 2021, [PERSON] [20] systematically reviewed 145 studies and found that although the instability factors of pedestrian behaviors in traditional emergency situations (i.e., free traffic accidents) have been widely studied, new high-risk situations (such as earthquakes, terrorist attacks, and stampedes) have not received sufficient attention. (2) Methods based on computer vision. Using optical flow theory, the movement characteristics of a crowd can be according to abnormal behaviors. At present, a variety of anomaly detection algorithms have been proposed for crowded scenes, such as abnormal crowd behavior detection [21; 22], human abnormal action detection [23], and traffic event detection [24]. However, this type of abnormal event detection method is only suitable for small-and medium-sized crowds, the scene equipment cost is large, and the calculations are complex. (3) Methods based on control theory. A group interaction matrix has been used to qualitatively predict group behaviors, and the stability of the system was analyzed to determine whether the system structure was reasonable. In general, if a system has high stability, it is reasonable, and if the system has low stability, it is necessary to adjust the system structure to reduce the conflict(s) in the evacuation process. [PERSON] [25] established a cold dynamics model based on a conservation law, and the relationship between density and speed was expressed using a diffusion model. The feedback linearization method of a partial differential equation was used to design a feedback controller for controlling crowd stability so that pedestrians could evacuate in a certain direction and at a fixed speed. [PERSON] [26] designed nonlinear feedback controllers for providing convection, diffusion, and convection/diffusion control, and discussed the control stability using partial differential equations. Based on a study of fluid dynamics and traffic flow, [PERSON] [27] designed a time-limited controller for solving the finite time-tracking problem, and for making the density converge to a given steady-state density. (4) Methods based on mathematical deduction. In one example of these methods, based on fluid dynamics and traffic flow theory, a conservation equation for macro crowd flow dynamics is established, and the stability of a crowd flow is analyzed using the system stability judgment theory.