<p style='text-indent:20px;'>The collision-avoidance and flocking of the Cucker–Smale-type model with a discontinuous controller are studied. The controller considered in this paper provides a force between agents that switches between the attractive force and the repulsive force according to the movement tendency between agents. The results of collision-avoidance are closely related to the weight function <inline-formula><tex-math id="M1">\begin{document}$ f(r) = (r-d_0)^{-\theta } $\end{document}</tex-math></inline-formula>. For <inline-formula><tex-math id="M2">\begin{document}$ \theta \ge 1 $\end{document}</tex-math></inline-formula>, collision will not appear in the system if agents' initial positions are different. For the case <inline-formula><tex-math id="M3">\begin{document}$ \theta \in [0,1) $\end{document}</tex-math></inline-formula> that not considered in previous work, the limits of initial configurations to guarantee collision-avoidance are given. Moreover, on the basis of collision-avoidance, we point out the impacts of <inline-formula><tex-math id="M4">\begin{document}$ \psi (r) = (1+r^2)^{-\beta } $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ f(r) $\end{document}</tex-math></inline-formula> on the flocking behaviour and give the decay rate of relative velocity. We also estimate the lower and upper bound of distance between agents. Finally, for the special case that agents moving on the 1-D space, we give sufficient conditions for the finite-time flocking.</p>
Collision-avoidance and flocking in the Cucker–Smale-type model with a discontinuous controller
