Employing the Friedrichs’ inequality to ensure global exponential stability of delayed reaction-diffusion neural networks with nonlinear boundary conditions

Abstract

By employing the Friedrichs’ inequality and M-matrices, we have obtained two sets of sufficient conditions to ensure global exponential stability of the reaction-diffusion Hopfield networks with S-type distributed delays and nonlinear boundary conditions. Our work demonstrates that both diffusion effects, boundary conditions and shapes of spatial regions have played critical roles in determining the global exponential stability of the network system. Comparing with the previous work, our method can further provide how to find the more accurate and larger convergence rate. We also present several examples and simulations of the network models defined respectively on the cylinder, unit ball, cube and line segment so as to show the significance and application of our theory. The theoretical result and the idea here can be applied in considering system control and synchronization with or without random terms.