Periodically intermittent control for finite-time synchronization of delayed quaternion-valued neural networks
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ORIGINAL ARTICLE
Periodically intermittent control for finite-time synchronization of delayed quaternion-valued neural networks Chaouki Aouiti1
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Mayssa Bessifi1
Received: 28 March 2020 / Accepted: 5 October 2020 Ó Springer-Verlag London Ltd., part of Springer Nature 2020
Abstract In this paper, the finite-time synchronization between two delayed quaternion-valued neural networks (QVNNs) via the periodically intermittent feedback control is studied. Firstly, the finite-time synchronization problem is presented for the first time via the periodically intermittent control approach. Secondly, the upper bounds of the settling time for finite-time synchronization are estimated. Thirdly, a kind of novel controller, state feedback controller, which contains an integral term and delayed term, is proposed. Through these, the problem of finite-time synchronization has been solved very well. Finally, several new conditions ensuring finite-time synchronization of two delayed QVNNs are derived by establishing a new differential inequality and constructing a Lyapunov function. In the end, two numerical examples with simulations show the effectiveness of the derived results and the developed method. Keywords Finite-time synchronization Quaternion-valued neural networks Periodically intermittent control Settling time
1 Introduction
quaternion multiplication does not meet the commutative law. For instance, let h1 ¼ 1 þ 2i 3k; h2 ¼ 3j 5k then h1 h2 ¼ 15 þ 9i þ 7j þ 11k 6¼ h2 h1 ¼ 15 9i 13j The quaternion-valued neural networks, which were invented by Hamilton in 1843 [46], consists of a real and k: Hence, the analysis on quaternion is much harder than three imaginary parts. The skew field of quaternion is that on plurality. As we all comprehend, complex-valued labeled by neural networks (CVNNs) can be seen as an extension of real-valued neural networks (RVNNs). By nature, CVNNs ðRÞ ðIÞ ðJÞ ðKÞ Q :¼ fh ¼ h þ ih þ jh þ kh g can be also generalized to quaternion-valued neural netðRÞ ðIÞ ðJÞ ðKÞ works (QVNNs). It is well noted that the dynamics of are real numbers and the three where h ; h ; h ; h neural networks represent a very important role, in the imaginary units i, j and k respect the Hamilton’s multiplidesign and execution of neural networks. Lately, the study cation rules: of QVNNs has attracted much attention of many ij ¼ ji ¼ k; jk ¼ kj ¼ i; ki ¼ ik ¼ j; i2 ¼ j2 ¼ k2 ¼ 1: researchers and some results about dynamical behaviors of QVNNs have been derived (see [2–43]). The conjugate quaternion of h is defined by h ¼ hðRÞ Furthermore, in the real systems, the synchronization is pffiffiffiffiffi ihðIÞ jhðJÞ khðKÞ : The norm of h is jhj ¼ hh 2 R. The a vast phenomenon, which means that two systems or more align one another to reach a common dynamical behavior. According to the synchronization, we can comprehend & Chaouki Aouiti unknown systems through the well-known systems. [email protected] sequently, it meets an important role in network control Ma
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