On estimating neuronal states of delayed neural networks based on canonical Bessel-Legendre inequalities

Abstract

In this paper, the dissipativity state estimation problem is addressed for neural networks with time-varying delays. Applying the N-order Bessel-Legendre inequality, we formulate a hierarchical criterion on the existence of state estimators for the delayed neural networks. First, an augmented Lyaponov–Krasovskii functional is introduced to incorporate the vector information of the N-order Bessel-Legendre inequality. Second, a novel bounded real lemma (BRL) is established for the estimation error system. In particular, the quadratic delay coefficients are absorbed by introducing some new vectors, which, together with smaller upper bounds for certain integral items, leads to the BRL less conservative than some existing results. As such, the state estimators are designed via the linear matrix inequality approach. Third, the obtained criterion is proven to form a hierarchy, which means that the conservatism reduces with the increasing of the order N of the Bessel-Legendre inequality. Finally, the validity of our results is demonstrated through three examples.