Adaptive Finite Time Stability of Delayed Systems With Applications to Network Synchronization

Abstract

The finite time stability (FnTSta) theory of delayed systems has not been set up until now. In this paper, we propose a two-phases-method (2PM), to achieve this object. In the first phase, we prove that the time for norm of system error evolving from initial values to 1 is finite; then in the second phase, we prove that the time for norm of the error evolving from 1 to 0 is also finite, thus FnTSta is obtained. Considering the cost and complexity of controller, we use only two simple terms to realize this aim. For the proposed 2PM, time delays can be the same or asynchronous, bounded or unbounded, etc; the norm can be 2-norm, 1-norm, ∞-norm, which show that 2PM is powerful and has a wide scope of applications. Furthermore, we also prove the adaptive finite time stability (AFnTSta) theory of delayed systems using 2PM. As an application of the obtained FnTSta theory, we consider the finite time outer synchronization (FnTOSyn) for complex networks with asynchronous unbounded time delays, corresponding criteria are also obtained. Finally, two numerical examples are given to demonstrate the validity of our theories.

Finite-time bipartite lag synchronization of coupled memristive neural networks with uncertain disturbances via hybrid control

“…Remark 6. Compared with the control strategies designed in section ''Bipartite synchronization with the same lag,''u m (t) in equation (38) has been added another part u m1 (t) to deal with the different lags in different clusters. Adding u m1 (t) to equation (37), one has…”

“…Then, the dynamical networks (32) and the objective network (2) with control strategies (38) can realize the finite-time bipartite synchronization with the different lags d m, m = 1, 2 in the settling time T. T as follows…”

“…28–34 Impulsive control is a vital and sensible choice from the perspective of convenience and economy. 30,31 Further, in Wang and Chen, 35,36 Liu and Li 37 and Liu, 38 some hybrid control methods were proposed to research the synchronization problem of MNN system in a finite/fixed time, namely, two-phase method. Recently, in Yu et al, 39 based on MNNs with delays that are bounded and time-varying, the stabilization process was divided into two procedures, and the time it takes in this whole process is finite and calculable.…”

“…The initial values are the same as the first four of Example 1, coupling strength s 1 = 0:3, s 2 = 0:2, and for the transmission lags d 1 = 1, d 2 = 2 for neurons in different clusters, there exists e 2 = 28:4847 and we can obtain T ave \ 0:0157 by simple calculation. Set T ave = 0:01, under the designed control strategies (38); the FTBLS has been achieved for the dynamical networks (32) and objective network (2). The settling time is T = 0:0619lnV(0) + 0:2276 when V 1 (0) .…”

“…Figure 6. State trajectories of l m (t)(m 2 F) of the MNNs (46) and l 0 (t) of the MNN (2) on the interval ½0, 10 without control(38).…”