Halanay inequality with Hadamard derivative and application to a neural network system

In this paper, we generalize the well-known Halanay inequality from the integer-order case to the fractional case. Namely, we consider Halanay inequalities involving Hadamard fractional derivative, and discrete and distributed delays. This inequality is then applied to study the stability of neural network systems of Hopfield type. To this end, we establish some properties like the Hadamard derivative of the product of two functions and the permutation of the Hadamard derivative and the integral or the Hadamard derivative of the convolution of two functions. It is proved that the stability is of Mittag–Leffler type, in case the distributed delay kernels themselves are Mittag–Leffler decaying to zero.