Realizing generalized outer synchronization of complex dynamical networks with stochastically adaptive coupling

Asymptotical outer synchronization control of the complex dynamical networks via unknown interaction and links dynamics

“…37 This is different from the models in literature. [16][17][18][28][29][30] (ii) The models ( 2) and ( 4) are composed of two parts. One is linear part (PX + XP T in (2), P X + X PT in ( 4)), which represents the dynamic state of LS.…”

“…(b) If the matrices M(t) and M(t) are both non-zero bounded time-varying matrices, the connections between all nodes of each of the two networks will eventually change with the same as the dynamics of LS, which is different from some existing results in the literature. [16][17][18][28][29][30] Remark 7. For the applications of Theorem 1, the following steps can be proposed.…”

“…At present, there are many research results for the outer synchronization of CDN such as the complete outer synchronization, 16 the hybrid outer synchronization, 17,18 the projective outer synchronization, 28 and the generalized outer synchronization. 29,30 Specially, the outer synchronization can also occur in two subnetworks with the same number nodes, which are clustered from a network. This phenomenon often occurs in the real networks, for example, the two aircraft groups flying synchronously, 31,32 the synchronization of an ensemble of two groups of oscillators, 33,34 and the synchronization in the multiple pairs of matching subnetworks with a leaders' subnetwork and a followers' subnetwork.…”

“…Compared with the existing literature, the key advantage of this paper is that (i) in each CDN, the dynamic model of NS considers the unknown interaction term on another CDN, which is not considered in the literature of outer synchronization. [16][17][18][28][29][30] The synthesis of adaptive control scheme uses the boundedness information of unknown interactions. (ii) In each CDN, the dynamics of LS is modelled by the Riccati matrix differential equation, in which the coupling term related to the state of two CDNs is synthesized to assist achieving the outer synchronization, which is rare in the existing literature of outer synchronization.…”

“…That is, two LSs can track, respectively, the proposed auxiliary dynamic targets when the two NSs realize the outer synchronization, which is different from some existing results in the literature. [16][17][18][28][29][30] The rest of this paper is organized as follows. In Section 2, we propose two CDN models with unknown interactions in which the NS and the LS are mutually coupled, and the corresponding mathematical assumptions are required.…”