Time-Varying Heterogeneous Alternation Control for Synchronization of Delayed Neural Networks

This paper proposes a time-varying heterogeneous alternation controller for the synchronization of delayed neural networks (DNNs). Different from the mixture of traditional impulse and intermittent control, we consider a time-varying intermittent controller which is more representative than traditional periodic or aperiodic intermittent controllers. To make the system under study more applicable, we consider the impact of time-varying delay. After doing a theoretical analysis, the sufficient condition of the synchronization of DNNs is obtained. Finally, two simulations are offered to verify the validity of our results.

impulsive control. It enables the controller to work at a certain 26 discrete instant or for a certain amount of time [20]. Intermit- 27 tent control, in particular, is based on the piecewise function 28 The associate editor coordinating the review of this manuscript and approving it for publication was Wei Jiang . technique which includes work and rest time [21]. Its usual 29 benefit include cheap control costs and ease of deployment. 30 For instance, [11] put forwards a periodically intermittent 31 control with discrete-time state observations; [21] synchro-32 nized the delayed complex valued dynamic network by apply- 33 ing the aperiodic intermittent controller to some nodes of the 34 network. In aperiodic intermittent control, a positive lower 35 bound must be applied to the control width of each control 36 period. To address this issue, [22] explored a time-varying 37 intermittent control strategy. 38 Besides, impulsive control acts at certain discrete instants 39 as a sampled-data-based method [23], [24], [25]. The 40 two types of impulses-synchronous and asynchronous 41 impulses-can be easily distinguished since not all impulse 42 messages are beneficial [26], [27], [28], [29]. The 43 authors [27] viewed asynchronous impulses as a distur-44 bance and used feedback and synchronous impulse control, 45 respectively, to synchronize their considered systems. Fur-46 ther, asynchronous and synchronous delay-free impulses, 47 asynchronous and synchronous delayed impulses were all 48 taken into consideration by [29]. To measure the impact 49 of mixed impulses, the average delayed impulsive gain as 50 an unique concept was introduced. Reference [30] dealt 51 with the hybrid delay-dependent impulsive problem by 52 λ max (·)(λ min (·)) signifies the maximum(minimum) eigen-

103
The dynamics of delayed neural networks (DNNs) are given 104 by: 106 with the initial state v(t) = χ 0 (t), − τ 0 ≤ t < 0, where 107 v(t) ∈ R n denotes the neuron states of DNNs, A ∈ R n×n is 108 the state coefficient matrix, B ∈ R n×n and C ∈ R n×n mean 109 the connection weight matrices, both f (·) : R n → R n and 110 g(·) : R n → R n are nonlinear functions satisfying the fol-111 lowing Lipschitz condition, i.e., f (x) − f (y) ≤ s 1 x − y 112 and g(x) − g(y) ≤ s 2 x − y with two positive Lipschitz 113 constant s 1 and s 2 for any vectors x, y ∈ R n , the time delay 114 is bounded, i.e., 0 < τ (t) < τ 0 . Here the system (1) is set 115 as the master DNNs, which corresponds to the slave DNNs 116 listed below: with the initial state u(t) = γ 0 (t), − τ 0 ≤ t < 0, where 120 u(t) ∈ R n denotes the neuron states of DNNs, w(t) is a control 121 input to be designed.

122
Remark 1: In real-world applications, the effects of time 123 delay cannot be disregarded due to the limited speed of 124 sending and switching signals [32], [33], [34], [35], [36]. 125 The delay taken into account here may be either constant 126 or time-varying. Consequently, the delay discussed in this 127 study is more broad-based compared to [22], [35], and [36]. 128 Specifically, this work just requires the delay to be bounded, 129 whereas [22], [35], and [36] require the delay to be a constant 130 or the derivative of the delay to be bounded.

136
To achieve the synchronization of master-slave DNNs, 137 a time-varying heterogeneous alternation control is designed: 138 where m(t) is an intermittent gain with the positive lower 140 bound m 0 > 0, that is, m = , v(t)) denotes the impulsive con-142 troller given next. Specifically, the derivative of the system 143 error is re-expressed below: where κ k is the impulsive control gain, the time sequence 146 {t k |k ≥ 1} satisfies t k < t k+1 and lim k→∞ t k = ∞, z(t − k ) 147 and z(t + k ) are the left-limit and right-limit of the state z(t) 148 at impulsive instant t k , respectively, δ(·) is the Dirac delta 149 function, i.e., δ(t) = 0 for t = 0, and   (t) [37], [38]. It is generally utilized to model a tall narrow 153 spike function, i.e., an impulse [38]. 154 The following definitions and lemmas, which will be 155 applied to the main results, are introduced before moving on.

171
The key results are presented here together with rigorous 172 proof. To demonstrate the wide-ranging applications of the 173 results, additional discussions are provided.

174
The Eq. (5) is rewritten as follows to make the understand-175 ing of the following deduction easier: Integrating the two sides of the above equation 179 obtains [38], [41]: where ε > 0 is sufficiently small. As ε → 0 + , it reduces to Then the error system is further expressed as: 188 Remark 2: The strength of the impulse determines 189 whether it is harmful or beneficial to the system [28], [32].

190
For this reason, we consider the effects of two cases, that 191 is, when the condition |1 + κ k | > 1 is met, the impulse will 192 destroy the synchronization of complex systems. In this case, 193 it is usually described as internal or external noise. When the 194 condition is opposite, i.e., |1 + κ k | < 1, it will help to realize 195 system synchronization.

196
Remark 3: Different from traditional impulse intermittent 197 systems [20], [31], [32], the intermittent technique taken 198 into consideration here is time-varying, which is more broad 199 than periodic or nonperiodic intermittent methods and may 200 cover a wider range of situations. Note that the proposed 201 controller degenerates to the traditional impulse intermittent 202 control [20], [31], [32] if the control gain switches between 203 fixed constants and zero in one period. The impulse intermit-204 tent paradigm may also be applied to other systems, such as 205 the systems with intermittent communication or with impulse 206 disturbances, even though it is primarily considered from 207 control level.

259
The following formula can be obtained by taking the root 260 sign from both sides of the above inequality:

264
The following inequality can easily be obtained using a 265 straightforward deduction [27]:

279
From the sufficient condition ln 1+κ * dsyn T min + ln 1+κ * syn T max + ζ < 280 0 given in Theorem 1, we know that the master and slave 281 DNNs achieve globally exponential synchronization. This 282 completes the proof.

283
If system (1) suffers from the impulse disturbances, 284 we apply the time-varying intermittent control technique to 285 force the network to the synchronous trajectory (2) while 286 also preventing the impact of disturbances. The time-varying 287 intermittent control is represented as: These are the conclusions that we can reach.

290
Theorem 2: The globally exponential synchronization of 291 master-slave DNNs is achieved under the time-varying inter-292 mittent control (28) if the following condition is met: Proof: Similar to the proof of Theorem 1, one also has 296 the following deduction as given in Eq. (26): for any ∀t ∈ [t k−1 , t k ). We can arrive to the following 299 derivation since the considered impulses are asynchronous: 300 Based on the sufficient condition ln 1+κ * dsyn T min + ζ < 0 given 304 in Theorem 2, one has that the master-slave DNNs achieve 305 globally exponential synchronization. This completes the 306 proof.

308
Two examples in this part serve to validate the effectiveness 309 of the theoretical analysis presented above.

310
Case 1. The first example is a small-scale delayed neu-311 ral network with seven nodes with the neuron state [43] 312 described as follows:  the nonlinear terms [43].
The control gain is set as c(t) = 5 + 2 * sin(2 * pi * t/0.1).  It should be emphasized that when the system is only exposed 334 to external impulse noise, the proposed strategy can also 335 deal with this situation well. This reflects that the obtained 336 conclusion can be simply extended to other systems.

337
Based on the above system, we make a comparative sim-338 ulation experiment with the controller in [31], [32], and [20] 339 to show the benefits of our method, as shown in Figure 3.  rate of method is obviously faster than that in [31], [32], 345 and [20]. It is due to that the controller we used in this case 346 is continuous as opposed to intermittent, in addition to the 347 different setting of control parameters.

348
Remark 4: The same system under different controllers 349 will have different trajectories, as shown in Figure 3. 350 We would like to emphasize once again that the strategy 351 proposed in this paper covers two cases. In other words, 352 it can degenerate into the traditional impulse intermittent 353 control [20], [31], [32], which demonstrates the broad appli-354 cability of our strategy.   where v(t) = (v 1 (t) T , v 2 (t) T , . . . , v 200 (t) T ) T and v i (t) ∈ R 3 , 359 f (v(t)) = tanh(v(t)) and g(v(t − τ (t))) = tanh(v(t − τ (t))), 360   Theoretical analysis and discussion presented above 389 demonstrate the benefits of our proposed controller. However, 390 because the controller in this study is used by each network 391 node, it will consume a large amount of energy. So, in the 392 upcoming work, we will take into account other energy-393 saving control methods like pinning control and event-trigger 394 control [17]. 396 The authors declare that they have no known competing 397 financial interests or personal relationships that could have 398 appeared to influence the work reported in this paper.