Abstract
In this paper, the existence and stability analysis of the quaternion-valued neural networks (QVNNs) with time delay are considered. Firstly, the QVNNs are equivalently transformed into four real-valued systems. Then, based on the Lyapunov theory, nonlinear measure approach, and inequality technique, some sufficient criteria are derived to ensure the existence and uniqueness of the equilibrium point as well as global stability of delayed QVNNs. In addition, the provided criteria are presented in the form of linear matrix inequality (LMI), which can be easily checked by LMI toolbox in MATLAB. Finally, two simulation examples are demonstrated to verify the effectiveness of obtained results. Moreover, the less conservatism of the obtained results is also showed by two comparison examples.
Highlights
Quaternion, as a supercomplex number, was discovered by W.R
Gong et al have discussed the asymptotic stability of complex-valued neural networks (CVNNs), and sufficient criteria were obtained in the form of linear matrix inequality (LMI) by using the nonlinear measure approach [17]
To the best of our knowledge, only few results if not none have discussed the stability of delayed quaternion-valued neural networks (QVNNs) with nonlinear measure approach, which is one of our motivations to carry out this research
Summary
Quaternion, as a supercomplex number, was discovered by W.R. Hamilton in 1943, and it has been shown that the quaternion is with expansive potential for future development in three-dimensional and four-dimensional data processing. Based on matrix measure and Halanay inequality, Liu et al considered the exponential stability of delayed QVNNs, and several criteria were presented in [32, 48]. Gong et al have discussed the asymptotic stability of CVNNs, and sufficient criteria were obtained in the form of LMIs by using the nonlinear measure approach [17]. Based on the nonlinear measure approach, some delay-independent conditions were established to ascertain the stability of delayed neural networks [26]. The robust stability of inertial Cohen–Grossberg neural networks was discussed by using the nonlinear measure approach and Halanay inequality [28]. To the best of our knowledge, only few results if not none have discussed the stability of delayed QVNNs with nonlinear measure approach, which is one of our motivations to carry out this research.
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