Synchronization and Adaptive Control for Coupled Fractional-Order Reaction-Diffusion Neural Networks with Multiple Couplings

Over recent decades, the dynamical behavior for neural networks (NNs) has been intensively studied as a hot topic across multiple fields [1–6]. In [2], several exponential stability conditions were presented for a type of complex-valued memristive recurrent NNs by using the M-matrix. By artfully constructing Lyapunov functionals based on delay-product type functionals, two delay-dependent sufficient conditions for stability of NNs were presented in [3]. In [5], by applying the multiple Lyapunov functionals and the average dwell-time methods, several switching laws were presented to ensure the passivity for delayed recurrent NNs with stochastic disturbances. In [6], the authors utilized several weighted integral inequalities to cope with the exponential passivity problem for uncertain delayed NNs. It is worth pointing out that integral-order NNs are discussed in [1–6]. Since fractional-order neural networks (FONNs) can more effectively and accurately describe human brain neurons, plenty of work has been devoted to the dynamical behavior for FONNs [7–12]. The stability for fractional-order delayed Hopfield NNs was addressed in [7], and the cases that the networks consist of two neurons and three neurons were discussed. Rakkiyappan, Cao and Velmurugan [8] not only considered the uniform stability for fractional-order complex-valued delayed recurrent neural networks but also proved the uniqueness and existence of equilibrium point. Sau et al. [10] put forward a delay-dependent passivity condition for FONNs by means of Lyapunov functional method. In [12], Ding et al. gave several robust passivity conditions for FONNs with norm bounded uncertain parameters, and further investigated the robust passivity of fractional-order interval neural networks.