In this paper, these parametric matrices of NCDN (3) and (4) are known constant matrices in certain mode r(t).
In this section, sufficient conditions are presented to ensure [H.sub.[infinity]] cluster synchronization for the neutral complex dynamical network (NCDN) (3) and 4).
Given the transition rate matrix [UPSILON], the initial positive definite matrix Y = [Y.sup.T] > 0, constant scalars [d.sub.1i], [d.sub.2i], and [d.sub.mi] satisfying [d.sub.1i] < [d.sub.mi] < [d.sub.2i], the NCDN systems (53) and (4) with sector-bounded condition (7) are [H.sub.[infinity]] cluster synchronization with a disturbance attenuation lever S if there exist symmetric positive matrices [P.sub.i] > 0, (i [member of] S), [Q.sub.j] > 0, (j = 4, 5, 6), [R.sub.k] > 0, (k = 5,6,7), [T.sub.l] > 0, [U.sub.m] > 0, and [V.sub.n] > 0, (l, m, n = 4, 5,6) for any scalars [[epsilon].sub.1],[[epsilon].sub.2] > 0 such that the following linear matrix inequalities hold:
A four-node NCDN 3) and 4) with Markovian switching between two modes is taken into consideration; that is, N = 4 and M = 2.
[H.sub.[infinity]] cluster synchronization of this NCDN based on the above criterion is tested.
It can be concluded that this neutral complex dynamical network (NCDN) has achieved [H.sub.[infinity]] cluster synchronization, which illustrates the effectiveness of Theorem 12.
Motivated by the above analysis, the [H.sub.[infinity]] cluster synchronization problem for a class of NCDNs with Markovian switching and mode-dependent time-varying delays is investigated in this paper.
In this paper, [H.sub.[infinity]] cluster synchronization of the NCDNs with Markovian jump parameters is studied for the first time, which is first introduced to quantify the attenuation level of synchronization error dynamics against the exogenous disturbance of NCDNs with Markovian switching.
Define the stochastic Lyapunov-Krasovskii function of the NCDNs (3) and (4) as V(x(t),r(t) = i, t > 0) = V(x(t),i,t) where its infinitesimal generator is defined as