Stability Analysis of Quaternion-Valued Neural Networks: Decomposition and Direct Approaches

In this paper, we investigate the global stability of quaternion-valued neural networks (QVNNs) with time-varying delays. On one hand, in order to avoid the noncommutativity of quaternion multiplication, the QVNN is decomposed into four real-valued systems based on Hamilton rules: <inline-formula> <tex-math notation="LaTeX">$ij=-ji=k,~jk=-kj=i$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$ki=-ik=j$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$i^{2}=j^{2}=k^{2}=ijk=-1$ </tex-math></inline-formula>. With the Lyapunov function method, some criteria are, respectively, presented to ensure the global <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>-stability and power stability of the delayed QVNN. On the other hand, by considering the noncommutativity of quaternion multiplication and time-varying delays, the QVNN is investigated directly by the techniques of the Lyapunov-Krasovskii functional and the linear matrix inequality (LMI) where quaternion self-conjugate matrices and quaternion positive definite matrices are used. Some new sufficient conditions in the form of quaternion-valued LMI are, respectively, established for the global <inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula>-stability and exponential stability of the considered QVNN. Besides, some assumptions are presented for the two different methods, which can help to choose quaternion-valued activation functions. Finally, two numerical examples are given to show the feasibility and the effectiveness of the main results.

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