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June  2022, 11(3): 729-748. doi: 10.3934/eect.2021023

## 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 Published  June 2022 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 and Control Theory, 2022, 11 (3) : 729-748. doi: 10.3934/eect.2021023
##### References:
 [1] S.-M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, Journal of Mathematical Physics, 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895. [2] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [3] F. Cucker and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [4] F. Cucker and E. Mordecki, Flocking in noisy environments, Journal de Mathématiques Pures et Appliquées, 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002. [5] J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim, On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies, SIAM Journal on Control and Optimization, 58 (2020), 2332-2353.  doi: 10.1137/19M1279150. [6] R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM Journal on Applied Mathematics, 76 (2016), 1535-1557.  doi: 10.1137/15M1030467. [7] S.-Y. Ha and J. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Communications in Mathematical Sciences, 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [8] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Communications in Mathematical Sciences, 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [9] Y. Jin, Flocking of the Motsch-Tadmor model with a cut-off interaction function, Journal of Statistical Physics, 171 (2018), 345-360.  doi: 10.1007/s10955-018-2006-0. [10] Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quarterly of Applied Mathematics, 73 (2015), 693-709.  doi: 10.1090/qam/1401. [11] Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, Journal Mathematical Analysis and Applications, 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036. [12] Y. Liu and J. Wu, Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling, Journal Mathematical Analysis and Applications, 444 (2016), 947-956.  doi: 10.1016/j.jmaa.2016.06.049. [13] Y. Liu and J. Wu, Opinion consensus with delay when the zero eigenvalue of the connection matrix is semi-simple, Journal Mathematical Analysis and Applications, 29 (2017), 1539-1551.  doi: 10.1007/s10884-016-9548-0. [14] X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [15] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [16] L. Pedeches, Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete and Continuous Dynamical Systems, 38 (2018), 2731-2762.  doi: 10.3934/dcds.2018115. [17] J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM Journal on Applied Mathematics, 68 (2007), 694-719.  doi: 10.1137/060673254. [18] T. V. Ton, N. T. Lihn and A. yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Analysis and Application, 12 (2014), 63-73.  doi: 10.1142/S0219530513500255. [19] X. Wang, L. Wang and J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Communications in Nonlinear Science and Numerical Simulation, 70 (2019), 80-88.  doi: 10.1016/j.cnsns.2018.10.017.

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##### References:
 [1] S.-M. Ahn and S.-Y. Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, Journal of Mathematical Physics, 51 (2010), 103301, 17pp. doi: 10.1063/1.3496895. [2] F. Cucker and S. Smale, Emergent behavior in flocks, IEEE Transactions on Automatic Control, 52 (2007), 852-862.  doi: 10.1109/TAC.2007.895842. [3] F. Cucker and S. Smale, On the mathematics of emergence, Japanese Journal of Mathematics, 2 (2007), 197-227.  doi: 10.1007/s11537-007-0647-x. [4] F. Cucker and E. Mordecki, Flocking in noisy environments, Journal de Mathématiques Pures et Appliquées, 89 (2008), 278-296.  doi: 10.1016/j.matpur.2007.12.002. [5] J.-G. Dong, S.-Y. Ha, J. Jung and D. Kim, On the stochastic flocking of the Cucker-Smale flock with randomly switching topologies, SIAM Journal on Control and Optimization, 58 (2020), 2332-2353.  doi: 10.1137/19M1279150. [6] R. Erban, J. Haskovec and Y. Sun, A Cucker-Smale model with noise and delay, SIAM Journal on Applied Mathematics, 76 (2016), 1535-1557.  doi: 10.1137/15M1030467. [7] S.-Y. Ha and J. Liu, A simple proof of the Cucker-Smale flocking dynamics and mean field limit, Communications in Mathematical Sciences, 7 (2009), 297-325.  doi: 10.4310/CMS.2009.v7.n2.a2. [8] S.-Y. Ha, K. Lee and D. Levy, Emergence of time-asymptotic flocking in a stochastic Cucker-Smale system, Communications in Mathematical Sciences, 7 (2009), 453-469.  doi: 10.4310/CMS.2009.v7.n2.a9. [9] Y. Jin, Flocking of the Motsch-Tadmor model with a cut-off interaction function, Journal of Statistical Physics, 171 (2018), 345-360.  doi: 10.1007/s10955-018-2006-0. [10] Z. Li and S.-Y. Ha, On the Cucker-Smale flocking with alternating leaders, Quarterly of Applied Mathematics, 73 (2015), 693-709.  doi: 10.1090/qam/1401. [11] Y. Liu and J. Wu, Flocking and asymptotic velocity of the Cucker-Smale model with processing delay, Journal Mathematical Analysis and Applications, 415 (2014), 53-61.  doi: 10.1016/j.jmaa.2014.01.036. [12] Y. Liu and J. Wu, Local phase synchronization and clustering for the delayed phase-coupled oscillators with plastic coupling, Journal Mathematical Analysis and Applications, 444 (2016), 947-956.  doi: 10.1016/j.jmaa.2016.06.049. [13] Y. Liu and J. Wu, Opinion consensus with delay when the zero eigenvalue of the connection matrix is semi-simple, Journal Mathematical Analysis and Applications, 29 (2017), 1539-1551.  doi: 10.1007/s10884-016-9548-0. [14] X. Mao, Stochastic Differential Equations and Applications, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402. [15] S. Motsch and E. Tadmor, A new model for self-organized dynamics and its flocking behavior, Journal of Statistical Physics, 144 (2011), 923-947.  doi: 10.1007/s10955-011-0285-9. [16] L. Pedeches, Asymptotic properties of various stochastic Cucker-Smale dynamics, Discrete and Continuous Dynamical Systems, 38 (2018), 2731-2762.  doi: 10.3934/dcds.2018115. [17] J. Shen, Cucker-Smale flocking under hierarchical leadership, SIAM Journal on Applied Mathematics, 68 (2007), 694-719.  doi: 10.1137/060673254. [18] T. V. Ton, N. T. Lihn and A. yagi, Flocking and non-flocking behaviour in a stochastic Cucker-Smale system, Analysis and Application, 12 (2014), 63-73.  doi: 10.1142/S0219530513500255. [19] X. Wang, L. Wang and J. Wu, Impacts of time delay on flocking dynamics of a two-agent flock model, Communications in Nonlinear Science and Numerical Simulation, 70 (2019), 80-88.  doi: 10.1016/j.cnsns.2018.10.017.
$\lambda = 3$, $\delta = 0$ 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
$\lambda = 3$, $\delta = 0$ 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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