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Abstract
This paper addresses the issue of three dynamical behaviors including global Mittag-Leffler stability, robust stability and projection synchronization for fractional-order quaternion-valued neural networks (FQVNNs). Some linear matrix inequality conditions for these dynamical behaviors of FQVNNs are given by Lyapunov stability theory, quaternion matrix theory, Homeomorphic mapping theory and fractional differential equation theory. Furthermore, these obtained sufficient conditions for stability and synchronization are superior to those in existing literature. Finally, three examples are given to illustrate the effectiveness of the theoretical results.
Highlights
Fractional calculus, as a prolongation of integer calculus, was traceable in the 17th century [1, 2]
The main contributions made in this paper are as follows: (1) Different from the approaches in the existing literature, we investigate the dynamical behaviors of fractional-order quaternion-valued neural networks (FQVNNs) directly instead of converting them into complex-valued or real-valued system, which avoids the increase of system dimension
In order to discuss the synchronization problem between two FQVNNs, we introduce the response system associated with the drive system (2) as follows: Dαy(t) = –Cy(t) + Bf y(t) + I + u(t), (4)
Summary
Fractional calculus, as a prolongation of integer calculus, was traceable in the 17th century [1, 2]. Theorem 2 Under Assumptions 1 and 2, FQVNNs (2) have a unique equilibrium point and the equilibrium point is globally Mittag-Leffler robust stable, if there exist a real positive diagonal matrix Q and a Hermitian matrices P > 0 such that the following LMI holds:. Corollary 1 Under Assumption 1, FQVNNs (2) have a unique equilibrium point xwhich is globally Mittag-Leffler stable, if there exist a real positive diagonal matrix Q, a Hermitian matrix P1 ∈ Cn×n and a skew-symmetric matrix P2 ∈ Cn×n satisfying. Corollary 2 Under Assumptions 1 and 2, FQVNNs (2) have a unique equilibrium point and the equilibrium point is globally Mittag-Leffler robust stable, if there exist a real positive diagonal matrix Q, a Hermitian matrix P1 ∈ Cn×n and a skew-symmetric matrix P2 ∈ Cn×n such that the following LMI holds: H1 –H2 < 0,
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