Abstract
The fixed-time cluster synchronization problem for a class of directed community networks with linear couplings and discontinuous nodes is investigated in this paper. First, by means of reduction to absurdity and mathematical induction, two new and vital differential equalities are proposed and proved to study the fixed-time stability. Besides, by designing periodically switching controllers and aperiodically adaptive switching controllers, as well as using the theory of differential inclusions, some fixed-time cluster synchronization criteria are derived in which the settling time function is bounded for any initial values. Finally, the numerical simulations are performed to show the feasibility and effectiveness of the control methodology by comparing with the corresponding finite-time synchronization problem.
Highlights
Complex network is a useful modeling tool in understanding the dynamical behaviors of many natural and artificial systems
MODELING AND PRELIMINARY Consider a class of directed community networks consisting of N dynamical nodes and r communities with linearly couplings and discontinuous nodes, which can be described as follows: r xi(t) = Bμi xi(t) + fμi (xi(t)) + c aij xj(t), (1)
Let β = 0, the results for finite-time cluster synchronization of directed community networks with discontinuous nodes based on periodically switching control can be derived as: Corollary 2: Suppose that Assumptions 1-3 hold
Summary
Complex network is a useful modeling tool in understanding the dynamical behaviors of many natural and artificial systems. To the best of our knowledge, there are seldom results, or even no results concerning the fixed-time cluster synchronization of directed community networks with discontinuous nodes via switching control.
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