Mathematics – Optimization and Control
Scientific paper
2003-06-30
Mathematics
Optimization and Control
21 pages, 2 figures
Scientific paper
We study a simple but compelling model of $n$ interacting agents via time-dependent, unidirectional communication. The model finds wide application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from other agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, extending recent results reported in the literature. Unlike previous, related studies, the approach of the present paper does not rely on algebraic graph theory and is of a completely nonlinear nature. It is rather surprising that with these nonlinear tools, extensions may be obtained even for the linear cases discussed in the literature. The proof technique consists of a blend of graph-theoretic and system-theoretic tools integrated within a formal framework of set-valued Lyapunov theory and may be of independent interest. Further, it is also observed that more communication does not necessarily lead to better convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
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