Journal of the
Korean Mathematical Society JKMS

We study the collective behaviors of two second-order nonlinear consensus models with a bonding force, namely the Kuramoto model and the Cucker-Smale model with inter-particle bonding force. The proposed models contain feedback control terms which induce collision avoidance and emergent consensus dynamics in a suitable framework. Through the cooperative interplays between feedback controls, initial state configuration tends to an ordered configuration asymptotically under suitable frameworks which are formulated in terms of system parameters and initial configurations. For a two-particle system on the real line, we show that the relative state tends to the preassigned value asymptotically, and we also provide several numerical examples to analyze the possible nonlinear dynamics of the proposed models, and compare them with analytical results.

Keywords: Barbalat's lemma, bonding control, complete synchronization, Cucker-Smale model, flocking, Kuramoto model

MSC numbers: 37N35, 93C15, 93C95, 93C10, 34H05

Supported by: The work of H. Ahn was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MIST)(2022R1C12007321), and the work of J. Byeon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korean government (MEST): No.2019R1A6A1A10073437. The work of S.-Y. Ha was supported by NRF-2020R1A2C3A01003881, and the work of J. Yoon was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP): NRF-2016K2A9A2A13003815.