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Global exponential stability and periodicity of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions. (English) Zbl 1197.35144

Summary: Global exponential stability and periodicity of a class of reaction-diffusion recurrent neural networks with distributed delays and Dirichlet boundary conditions are studied by constructing suitable Lyapunov functionals and utilizing some inequality techniques. We first prove global exponential convergence to 0 of the difference between any two solutions of the original neural networks, the existence and uniqueness of equilibrium is the direct results of this procedure. This approach is different from the usually used one where the existence, uniqueness of equilibrium and stability are proved in two separate steps. Secondly, we prove periodicity. Sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the equilibrium and periodic solution are given. These conditions are easy to verify and our results play an important role in the design and application of globally exponentially stable neural circuits and periodic oscillatory neural circuits.

Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

MSC:
35K57Reaction-diffusion equations
35B10Periodic solutions of PDE
35B35Stability of solutions of PDE
35K51Second-order parabolic systems, initial bondary value problems
92B20General theory of neural networks (mathematical biology)