Novel Stability Results for Halanay Inequality and Applications to Delay Neural Networks

Abstract

This work investigates the stability of Halanay inequality. Some novel results are obtained by means of constructing an auxiliary differential equation. Some previous works are improved and extended. After that, the obtained results are applied to investigate the stability of neural networks with time-varying and distributed delays. At last, some examples along with numerical simulations are presented to illustrate the validity of the theoretical results. INDEX TERMS Stability, Halanay inequality, delay neural networks.

New Versions of Halanay’s Inequality With Multiple Gain Terms

“…The key idea which guides us consists of taking advantage of the knowledge of the constants T 1,i and T 2,i ≥ T 1,i , which can be derived from information on the delays of a system with poorly known time-varying delays. As in [13], our assumptions are satisfied by functions a which take both positive and negative values (which contrasts with our earlier results on generalized Halanay's inequalities from [8]- [11], which required a to be nonnegative valued) and the largest value of b i can be arbitrarily large over arbitrarily large intervals, provided that the values a(t) of the function a are positive and large on sufficiently long time intervals. Also, our Halanay inequality generalizations [8]- [11] did not use the comparison function approach that we use here, and they use only one gain term, instead of the multiple gain terms that we use here.…”

“…Also, our Halanay inequality generalizations [8]- [11] did not use the comparison function approach that we use here, and they use only one gain term, instead of the multiple gain terms that we use here. By contrast with the main result of [13], we establish ISS like inequalities. By the definition of ISS, this ensures that lim t+∞ v(t) = 0 when δ is the zero function.…”

“…In particular, the paper [13] presents notable results for time-varying Halanay inequalities of the type…”

“…where T ≥ 0 is a constant, v is nonnegative valued, and the term containing b is called the gain term. In [13], sufficient conditions for lim t→+∞ v(t) = 0 to hold are given for cases where (1) holds and where a(t) can take both positive and negative values and where b(t) is larger than a(t) on some arbitrarily large time intervals. This contrasts with the standard Halanay's inequality (e.g., [1, Lemma 4.2, p. 138]) having the form (1) where a > 0 and b ∈ [0, a) are constants.…”

“…This contrasts with the standard Halanay's inequality (e.g., [1, Lemma 4.2, p. 138]) having the form (1) where a > 0 and b ∈ [0, a) are constants. Motivated by the fact that inequalities of the type (1) can be used to study time-varying systems with time-varying delays, we revisit the main result of [13]. To obtain less restrictive conditions and establish input-to-state stability (or ISS) like inequalities, we consider generalized Halanay's inequalities with multiple gain terms of the type…”