Stabilization and lag synchronization of proportional delayed impulsive complex-valued inertial neural networks☆
Introduction
In 1986, Babcock and Westervelt [1] proposed a NN model based on Hopfield network with second-order derivative of the state, which was named INNs. Hereafter, Wheeler and Schieve [2] first published a result on the stability of INNs, and found that INNs are more complex than various NNs. For inertia item is a determinant element for chaos and bifurcation, many scholars have obtained many outcomes on INNs, see such as [3], [4], [5]. Meanwhile, INNs with different kinds of time delays have been extensively and in-depth probed in convergence [6], dissipativity [7], synchronization [8], passivity [9] and anti-synchronization [10]. Consequently, the previous results incorporated distinct kinds of time delays, comprising constant delays (CDs) [11], [12], time-varying delays (TDs) [8], [9], mixed-delays [13], [14], [15], [16], discrete and distributed time delays (DDTDs) [17], [18], periodic time delays [19], probabilistic TDs [20], [21]. Lately, different from previous types, the proportional delays (PDs), as , initially posed by Zhou [22], is a kind of unbounded delay, less conservative and increasing monotonically. In various fields, such as biological networks, physics, and control, the PDs’ system can play a weighty role, see [7], [22], [23], [24], [25], [26], [27], [28], [29]. PDs were introduced into INNs in [23], including finite-time and fixed-time synchronization were investigated.
If the NNs itself could not achieve stability, the stabilization of INNs could be implemented from applying conventional control schemes, as to linear feedback control [10], [12], [16], [30], [31], nonlinear feedback control [23], [26], [32], [33], [34], impulsive control [35], [36], adaptive control [12], [31], [37], [38], sampled-data control [39], fuzzy control [40], intermittent control [15], [41], [42] and so forth. The gains of adaptive control vary with the change of adaptive rule, which is better than the control, whose control gain is constant. Therefore, adaptive control has better flexibility and robustness. Two different feedback control laws guaranteed the fixed-time stabilization of fuzzy neutral INNs in [43]. Under the two types of delayed feedback controllers, it is ensured that the memristor-based INNs achieved finite time stabilization in [33]. Through drawing a state-feedback controller, Sheng [44] discussed the GES of the memristive INNs with DDTDs. Results as mentioned above were based on the method of variable alteration, INNs is transformed into two first-order systems, and controllers are attached to the two subsystems, respectively, and then synchronization or stabilization of INNs were acquired. In practical applications, the realization of two controllers is much more difficult than having one controller. Therefore, in this paper, without using the variable transformation method, only an adaptive controller is designed to obtain the GES and ELS of INNs.
On the other hand, lag synchronization signifies that the status of the drive system is synchronized with the past status of the response system, its time lag is a constant, and it has been broadly used in information security [37]. In the context of mixing infinite TDs and state-dependent switching, literature [45] obtained the finite-time LS of INNs. A kind of neutral NN with CDs was explored in [46], and its finite time generalized projection LS was achieved. The neural activation function was used to study the global ELS problem of a switching NN with TDs and its application in image encryption in [47]. Nevertheless, there are few outcomes in the current kinds of literature for ELS of INNs with PDs. Therefore, filling this gap is also very urgent and essential.
In addition, complex variables are often used in many applications, including nonlinear filtering, pattern recognition, image processing, classification, and so forth [5], [6], [10], [12], [27]. Via applying order-reduction and separation methods, a kind of p-norm fixed-time synchronization of CVINNs with TDs is revealed in [5]. Similarly, by reduced-order substitution and applying a controller to each first-order system, Wei [10] obtained synchronization and anti-synchronization of CVINNs with TDs, M. Iswarya [27] analyzed exponential input-to-state stability of memristor-based CVINNs with distributed delays and PDs, further transferred the PDs into CDs by making a nonlinear substitution, which increasing the dimensionality and complexity of the system. Recently, Yu [12] proposed a non-reduction, non-separation method, and studied exponential and adaptive synchronization of CVINNs with CDs. As far as we know, the research outputs of CVINNs with PDs are very few, and the outcomes of the corresponding LS problem are rarely addressed.
Moreover, because of the inside formation or the outside circumstances, the state of CVINNs changes suddenly or drastically from time to time. It could be thought of as an impulse effect [8], [9], [24], [28], [35], [39]. Chen [8] applied pinning impulsive control to study the synchronization of coupled INNs with TDs. Wan [9] analyzed the passivity of memristive-based impulsive INNs with TDs. Li [39] considered asymptotical synchronization for INNs with unbounded delay and saturation via impulsive control. Works mentioned above first adopted the method of order reduction, and then in the obtained first-order system, the impulse effect or impulse control was considered for each first-order equation, but the impulse was not directly considered in the original INNs. It is necessary to adopt a direct method and think about the influence of impulse effect and delay effect on CVINNs’ stability at the same time.
In view of the above discussion, this paper intends to solve the GES and ELS problems of ICVINNs with PDs. The main contributions of our work are as below:
It is the first time to build the GES and ELS of ICVINNs with PDs, which unlike from the former works [7], [22], [24], [27], [28], here the PDs is not transformed into a CDs’ system with nonlinear substitution , which makes the original system more complex and more difficult to handle.
Unlike the previous methods of order-reduction and separation [5], [6], this paper directly applies a novel complex-valued adaptive controller based on ICVINNs. The controller designed here contains the power exponent factor to achieve better convergence.
Different from the matrix form that is difficult to verify [6], [12], [33], [36], the sufficient criteria in this paper is in algebraic form, which is more concise and easier to verify, for the GES and ELS results are presented under some inequality technique. Finally, we give two numerical simulation examples.
The remainder of this paper is as below. Section 2 presents some preliminaries and lemmas. The primary outcomes are acquired in Section 3 Global exponential stabilization, 4 Exponential lag synchronization. Section 5 shows numerical examples. Lastly, we conclude in Section 6.
Section snippets
Preliminary
Let represent the space of n-dimensional complex vectors. About any , and denote its real component and imaginary component, respectively. The module is delimited as , where is the conjugate representation of . The norm for any . Denote .
Consider the following ICVINNs with PDswhere the impulsive moment
Global exponential stabilization
A type of controlled ICVINNs is investigated in this section and described aswhere is the following adaptive control strategywhere constants and are all positive. Theorem 1 If Assumption 1, , , and , are satisfied, then ICVINNs
Exponential lag synchronization
In practice, lag exists, when the synchronization happens between the drive and response systems, which could be characterized as for some constant lag time . LS error among the drive and response systems can be presented aswhere is the jth neuron state in the response systems, which can be depicted as follows:the corresponding
Numerical example
Two examples will demonstrate the validity of the main outcomes. Example 1 [Global exponential stabilization] Establish a two-dimensional ICVINNs (1) with PDs:and . A novel activation function is proposed in [49], its expression iswhere andwith and . The activation functions can be selected as
Conclusion
This paper proposes novel algebraic criteria to guarantee the GES and ELS of ICVINNs with PDs. In order to achieve the GES and the ELS, adaptive control strategies are designed to lower control costs. Since the underlying system includes proportional delays, which is unbounded, the obtained results are more general and extend some existing ones. The simulation results apparently support the obtained outcomes.
CRediT authorship contribution statement
Yongkang Zhang: Conceptualization, Methodology, Visualization, Writing - original draft. Liqun Zhou: Data curation, Writing - review & editing, Investigation, Supervision.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Yongkang Zhang received the Ph.D. degree from BeiHang University, Beijing, China, in 2013, the M.S. degree and B.S. degree in Tianjin Normal University in 2009 and 2006, respectively. He is currently a lecturer of Tianjin Normal University. His research interests include the stability of neural networks, differential equations.
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Yongkang Zhang received the Ph.D. degree from BeiHang University, Beijing, China, in 2013, the M.S. degree and B.S. degree in Tianjin Normal University in 2009 and 2006, respectively. He is currently a lecturer of Tianjin Normal University. His research interests include the stability of neural networks, differential equations.
Liqun Zhou received her Ph.D. degree from Harbin Institute of Technology, Harbin, China, in control science and engineering in 2007, M.S. degree in Mathematics Department of Harbin Institute of Technology in 2004, B.S. degree in Mathematics Department of Qiqihaer University in 1998. She is currently a professor of Tianjin Normal University. Her research interests include theory and applications of neural networks, differential equations, nonlinear systems.
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This work is supported by the NNSF of China (No.11901433, No.11902221, No.12101452).