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On the Selection of Leaders for the Controllability of Multi-agent Networks

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Abstract

This work studies the controllability property of consensus problem of multi-agent networks in leader–follower framework over the random-walk normalised Laplacian dynamics on a continuous time-scale. The interconnection topology is time-invariant. Graphical interpretations are provided to gain more insight on the interactions between the agents of the network, which also helps in designing the communication protocols. Using the tools of algebraic graph theory and spectral analysis, we propose several necessary and sufficient controllability conditions for the dynamics of followers’ states through the selected leaders. The formula to determine the minimum number of leaders which ensures the controllability of the network (in leader–follower framework) is presented, and is used to find such controlling leaders in some standard networks. It is confirmed that the choice of selecting the minimum number of controlling leaders in the network depends not only on the interconnection topology graph, but also on the underlying dynamics of the network. Furthermore, it is confirmed that the least possible number of controlling leaders which control the dynamics of rest of the agents (the followers) in the network is greater or equals to the least possible number of external agents which control the dynamics of all agents in the network.

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Acknowledgements

The authors are grateful to editor and anonymous referees for their suggestions and comments that helped in the enhancement of the clarity and depth of the interpretation of the results in the paper. The second author wishes to express his gratitude to the Department of Science and Technology, Government of India for the granting of DST-Inspire Fellowship (IF190169) and to the Central University of Karnataka for providing the research facilities.

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Authors and Affiliations

  1. Department of Mathematics, Sri H. D. Devegowda Government First Grade College, Paduvalahippe P. O., Hassan, 573211, India

    Vijayakumar S. Muni

  2. Department of Mathematics, Central University of Karnataka, Kadaganchi P. O., Kalaburagi, 585367, India

    Kallu Vetty Muhammed Rafeek & Gudala Janardhana Reddy

  3. Department of Mathematics, Indian Institute of Space Science and Technology, Valiamala P. O., Thiruvananthapuram, 695547, India

    Raju K. George

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  1. Vijayakumar S. Muni
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This article was funded by Department of Science and Technology, Ministry of Science and Technology (Grant no. C/4084/IFD/2020-21).

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Communicated by Majid Gazor.

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Muni, V.S., Muhammed Rafeek, K.V., Reddy, G.J. et al. On the Selection of Leaders for the Controllability of Multi-agent Networks. Bull. Iran. Math. Soc. 48, 3141–3183 (2022). https://doi.org/10.1007/s41980-022-00683-2

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