The dynamics of fractional-order neural networks (FONNs) are challenging to study, since the traditional Lyapunov theory does not apply to them. Instead, Halanay-type lemmas are used to create sufficient criteria for specific dynamic properties of FONNs. The application of these lemmas, however, typically leads to conservative criteria. The Halanay-type lemma is used in a novel way in this study to develop less conservative sufficient conditions in terms of linear matrix inequalities (LMIs) for extremely general FONNs, with different types of delays, such as neutral-type, leakage, time-varying, and distributed delays. The formulation of such a general model for the fractional-order scenario is done here for the first time. In addition, a new Lyapunov-like function is established, resulting in algebraic conditions that are less conservative. Three theorems are put forward that build sufficient criteria for the asymptotic synchronization, employing state feedback control, of the proposed networks, each based on a different Lyapunov-like function. For the first time in the context of FONNs, the free weighting matrix technique is also used to greatly decrease the conservatism of the obtained sufficient conditions. One numerical simulation illustrates each of the three theorems.

Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also successfully applied to the domain of NNs. Finally, in real implementations of NNs, time delays occur inevitably. Taking all these facts into account, this paper discusses ClVNNs defined on time scales with leakage, time-varying delays, and infinite distributed delays, a type of delays which have been relatively rarely present in the existing literature. A state feedback control scheme and a generalization of the Halanay inequality for time scales are used in order to obtain sufficient conditions expressed as algebraic inequalities and as linear matrix inequalities (LMIs), using two general Lyapunov-like functions, for the exponential synchronization of the proposed model. Two numerical examples are given in order to illustrate the theoretical results.