# American Institute of Mathematical Sciences

doi: 10.3934/eect.2021023
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## Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance

 College of Liberal Arts and Science, National University of Defense Technology, Changsha, 410073, China

* Corresponding author: Yicheng Liu

Received  September 2020 Revised  March 2021 Early access April 2021

Fund Project: This work was partially supported by the National Natural Science Foundation of China (11671011) and Hunan Provincial Innovation Foundation for Postgraduate (CN) (CX20200011)

In this paper, a generalized Motsch-Tadmor model with piecewise interaction function is investigated, which can be viewed as a generalization of the model proposed in [9]. Our analysis bases on the connectedness of the underlying graph of the system. Some sufficient conditions are presented to guarantee the system to achieve flocking. Besides, we add a stochastic disturbance to the system and consider the flocking in the sense of expectation. As results, some criterions to the flocking solution with exponential convergent rate are established by the standard differential equations analysis.

Citation: Yipeng Chen, Yicheng Liu, Xiao Wang. Exponential stability for a multi-particle system with piecewise interaction function and stochastic disturbance. Evolution Equations & Control Theory, doi: 10.3934/eect.2021023
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##### References:
$\lambda = 3$, $\delta = 0$ and $r = 1$. In this case, only the neighbours attract each other. Because the value of $r$ is small, the system does not achieve flocking
$\lambda = 3$, $\delta = 0.1$ and $r = 1$. In this case, not only the neighbours attract each other but also the distant relatives attract each other in a weak force. Hence unconditional flocking occurs
">Figure 3.  $\lambda = 3$, $\delta = 0$ and $r = 2$. The system achieves flocking because of a bigger value of $r$ than Fig. 1
$\lambda = 3$, $\delta = -0.1$ and $r = 2$. In this case, the neighbours attract each other however the distant relatives repel each other in a weak force. The system also achieves flocking
$\lambda = 3$, $\delta = -0.3$ and $r = 3$. By increasing the repulsive force between distant relatives, the system does not achieve flocking
$\lambda = 3$, $\delta = -0.5$ and $r = 3$. Continue to increase the repulsive force between distant relatives, the system still does not achieve flocking
$\lambda = 3$, $\delta = -0.1$, $\sigma_i = 2(i = 1, 2, ..n)$ and $r = 2$. The system achieves flocking under a moderate stochastic disturbance
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