Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays

Octonion-valued neural networks (OVNNs) are a type of neural networks for which the states and weights are octonions. The octonion algebra is the only normed division algebra that can be defined over the reals, besides the complex and quaternion algebras. Being nonassociative, it clearly does not belong to the Clifford algebras category, which are all associative. In this paper, sufficient conditions for the global exponential stability of neutral-type OVNNs with time-varying delays are formulated, by considering two types of Lipschitz conditions that must be satisfied by the octonion-valued activation functions. To avoid the nonassociativity of the octonions and the noncommutativity of the quaternions, the OVNNs model is decomposed into 4 complex-valued systems, using the Cayley-Dickson construction. By using Lyapunov-Krasovskii functionals with double, triple, and quadruple integral terms, the free weighting matrix method, and simple, double, and triple Jensen inequalities, the stability criteria are formulated in terms of complex-valued linear matrix inequalities. Two numerical examples are provided in order to demonstrate the effectiveness and feasibility of the theoretical results.