Synchronization Of Complex Dynamical Networks With Derivative Coupling Via Adaptive Control

Nowadays, the study of complex dynamical network with derivative couplingsbecomes a hotspot issue and has attracted more and more researchers. These networkscontain not only information of the state but also information of the derivative of thestate. In reality, the complex network systems of neutral type have been used widely indifferent fields, which have become an important research topic in complexity sciencefield. Therefore, synchronous phenomenon of neutral type complex dynamical networkis a significant researching task. Besides, many network systems’ structure containsunknown parameters. Adaptive control is an effective research method for uncertainsystems which has unknown parameters. It is a challenging problem about how to applyadaptive control method into synchronization problem for neutral type complexnetworks with unknown parameters.Adaptive control theory is used to design adaptive controllers and laws in thispaper. Combining with Lyapunov stability theory, the synchronization problem forcomplex networks with derivative couplings and unknown parameters is addressed.Details are summarized as:Firstly, we apply a new adaptive control approach for nonlinear time-varyingcomplex dynamical network with non-identical node and derivative coupling. Using thecontents of Kronecker product properties, the effect of derivative coupling can beeliminated. A new Lyapunov-Krasovskii-like composite function is constructed toensure the convergence of synchronization error in the sense of square error norm andthe boundedness of all the closed-loop signals. And the adaptive strategy we designedcan be directly applied into the complex dynamical networks with unknown derivativecoupling. Finally, examples are given to illustrate the proposed theoretical results.Secondly, on the basis of the approaches aforementioned，a new adaptive law andadaptive control technology are proposed for nonlinearly time-varying parameterizedcomplex dynamical networks model with non-identical nodes and unknown derivativecoupling. The adaptive method which contains the information of the derivative of thestate is designed to ensure the asymptotic convergence of the synchronization error inthe sense of square error norm. Finally, simulation example shows the effectiveness ofthe method.Thirdly, adaptive synchronization control problem for complex dynamicalnetworks with nonlinearly derivative coupling is proposed. By means of the parameters separation, the nonlinear functions can be transformed into linear form. Then effectivedistributed adaptive techniques are designed to eliminate the effect of time-varyingparameters. The synchronization of the considered network and the boundedness of allthe closed-loop signals are proved by constructing a new Lyapunov-Krasvoskii functionand using the Barbalat lemma. A simulation example shows the applicability andfeasibility of the approach.