Mathematical Models of Nonlinear Uniform Consensus II
DOI:
https://doi.org/10.5890/JAND.2018.03.008Abstract
This paper is a continuation of our previous studies on nonlinear consensus. We have considered a nonlinear protocol for a structured time-invariant and synchronous multi-agent system. We present an opinion sharing dynamics of the multi-agent system as a trajectory of a polynomial stochastic operator associated with a stochastic multidimensional hyper-matrix. We provide a criterion for a uniform consensus in the multi-agent system. Particularly, the uniform consensus is achieved in the multi-agent system if all entries of the stochastic multidimensional hyper-matrix are positive. Some numerical results are also presented to support our theoretical results.References
[1] DeGroot, M.H. (1974), Reaching a consensus, Journal of the American Statistical Association, 69, 118-121.
[2] Vicsek, T., Czirok, A., Ben-Jacob, E., Cohen, I., and Shochet, O. (1995), Novel type of phase transition in a system of self-driven particles, Physical Review Letters, 75, 1226-1229.
[3] Olfati-Saber, R. and Murray, R.M. (2004), Consensus problems in networks of agents with switching topology and time-delays, IEEE Transactions on Automatic Control, 49, 1520-1533.
[4] Liu, X., Chen, T., and Lu, W. (2009), Consensus problem in directed networks of multi-agents via nonlinear protocols, Physics Letters A, 373, 3122-3127.
[5] Wen, G., Duan, Z., Yu, W., and Chen, G. (2013), Consensus of multi-agent systems with nonlinear dynamics and sampled-data information: a delayed-input approach, International Journal of Robust and Nonlinear Control, 23, 602-619.
[6] Yu, W., Chen, G., and Cao, M. (2011), Consensus in directed networks of agents with nonlinear dynamics, IEEE Transactions on Automatic Control, 56, 1436-1441.
[7] Saburov, M. and Saburov, Kh. (2014), Mathematical models of nonlinear uniform consensus, ScienceAsia, 40(4), (2014), 306-312.
[8] Saburov, M. and Saburov, Kh. (2014), Reaching a nonlinear consensus: polynomial stochastic operators, International Journal of Control, Automation and Systems, 12(6), 1276-1282.
[9] Saburov, M. and Saburov, Kh. (2016), Reaching a nonlinear consensus: a discrete nonlinear time-varying case, International Journal of Systems Science, 47(10), 2449-2457.
[10] Berger, R.L. (1981), A necessary and sufficient condition for reaching a consensus using DeGroot's method, Journal of the American Statistical Association, 76, 415-418.
[11] Chatterjee, S. and Seneta, E. (1977), Towards consensus: some convergence theorems on repeated averaging, Journal of Applied Probability, 14, 89-97.
[12] Lyubich, Y.I. (1992), Mathematical Structures in Population Genetics, Springer-Verlag: Berlin Heidelberg.
[13] Ganikhodjaev, N., Saburov, M., and Jamilov, U. (2013), Mendelian and non-Mendelian quadratic operators, Applied Mathematics & Information Sciences, 7, 1721-1729.
[14] Ganikhodjaev, N., Saburov, M., and Nawi, A. M. (2014), Mutation and chaos in nonlinear models of heredity, The Scientific World Journal, 2014, 1-11.
[15] Ganikhodzhaev, R., Mukhamedov, F., and Rozikov, U. (2011), Quadratic stochastic operators and processes: results and open problems, Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14, 279-335.
[16] Mukhamedov, F. and Saburov, M. (2010), On homotopy of volterrian quadratic stochastic operator, Applied Mathematics & Information Sciences, 4, 47-62.
[17] Mukhamedov, F. and Saburov, M. (2014), On dynamics of Lotka-Volterra type operators, Bulletin of the Malaysian Mathematical Sciences Society, 37, 59-64.
[18] Mukhamedov, F., Saburov, M., and Qaralleh, I. (2013), On ξ(s)−quadratic stochastic operators on twodimensional simplex and their behavior, Abstract and Applied Analysis, 2013, 1-12.
[19] Saburov, M. (2013), Some strange properties of quadratic stochastic volterra operators,World Applied Sciences Journal, 21, 94-97.
[20] Krause, U. (2009), Compromise, consensus, and the iteration of means, Elemente der Mathematik, 64, 1-8.
[21] Krause, U. (2009), Markov chains, Gauss soups, and compromise dynamics, Journal of Contemporary Mathematical Analysis, 44, 59-66.
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