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A Study on Stability Analysis for Complex Dynamical Networks with Time Delays

A Study on Stability Analysis for Complex Dynamical Networks with Time Delays
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This thesis considers stability conditions for networks with time-varying de- lays. In this thesis, sufficient stability conditions of complex dynamical net- works (CDN) and those of genetic regulatory networks (GRN) are derived. The derived conditions are based on linear matrix inequality (LMI) framework. This thesis mainly contributes to make the stability conditions be less conservative. For two different type of CDN, e.g., singular CDN and neutral CDN, syn- chronization problems are considered. First, delay-dependent synchronization of singular complex dynamical networks with time-varying delays is consid- ered. A modified Lyapunov-Krasovskii functional is used to derive a sufficient condition for synchronization in terms of LMIs which can be easily solved by various convex optimization algorithms. Second, the synchronization problem for a class of neutral complex dynamical networks with coupling time-varying delays is considered. A delay-dependent synchronization criterion is derived for the synchronization of neutral complex dynamical networks. By the use of a convex representation of the sector-restricted nonlinearity in system dynamics, the stability condition based on the discretized Lyapunov-Krasovskii functional is obtained via LMI formulation. On the other hand, an improved robust delay-dependent stability criteria for genetic regulatory networks with delays which varies in an interval. Based on Lyapunov-Krasovskii functional, a new delay-dependent sufficient conditions in terms of LMIs is derived, which is represented by a convex combination. The derived stability condition is without free-weighting matrices so that it reduces the computation burden. All the stability conditions are less conservative than ones in literature. Numerical simulations are provided to show the effectiveness of the proposed method.
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