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Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects. (English) Zbl 1204.34072

For a class of complex networks of interacting identical dynamical systems with impulsive effects, a problem of synchronization is studied. The coupling in the networks is considered to be with a time delay such that each element of the ensemble receives retarded signals from other oscillators. The coupling matrix belongs to a class of irreducible matrices with zero row sums and non-negative elements except for the main diagonal. Under further assumptions on the vector fields of the interacting oscillators and the coupling matrix, exponential stability of the synchronized manifold is demonstrated. The latter means that the synchronization errors decay to zero exponentially fast with time. Sufficient conditions for the exponential synchronization are derived by the geometrical decomposition of the network states over the eigenvectors of the coupling matrix and linear matrix inequality methods. Two numerical examples are also presented to illustrate the applicability of the obtained analytical results.

MSC:

34D06 Synchronization of solutions to ordinary differential equations
34A37 Ordinary differential equations with impulses
34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
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