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Robust global synchronization of two complex dynamical networks
7.When the achievement of synchronization with a given scheme depends on the initial conditions of the networks, we say that the scheme provides only local synchronization. This is opposite to schemes that ensure global synchronization, i.e., irrespective of the initial conditions.
12.It should be remarked that the term “coupling matrix” here refers to the matrix that specifies the inner connections of each network. This is different from the network-to-network coupling scheme, which is not given by the coupling matrix but ideally has to be designed to ensure synchronization.
13.Most of the subsequent results that stem from this theorem are also expressed in terms of LMIs that need to be satisfied.
14.Note that we skip the time dependence in Eq. (1) for notational simplicity. We follow this practice (common in the literature) throughout the paper, unless we need to explicitly remark the time dependence (e.g., when introducing new variables). The top dot, , notation should be read as “differentiation with respect to the time (t) variable.”
15. S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory (Siam, Philadelphia, 1994).
16. R. B. Bapat, Graphs and Matrices (Springer, 2010).
17. A. M. Lyapunov, Stability of Motion (Academic Press, New York, 1966).
20. S. Gershgorin, in Proceedings of the Russian Academy of Sciences (1931), p. 749.
21. K. Ogata, Modern Control Engineering, 4th ed. (Prentice Hall, 2002).
25.The elements of must either be real or appear in conjugate pairs in order to be valid eigenvalues of a real matrix.
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