Vector Extensions of Halanay’s Inequality

Abstract

We provide two extensions of Halanay's inequality, where the scalar function in the usual Halanay's inequality is replaced by a vector valued function, under a Metzler condition. We provide an easily checked necessary and sufficient condition for asymptotic convergence of the function to the zero vector in the time invariant case. For time-varying cases, we provide a sufficient condition for this convergence, which can be easily checked when the systems are periodic. We illustrate our results in cases that are beyond the scope of prior asymptotic stability results.

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.…”

“…Since s i ∈ (t 0 , t ], we deduce from the definition of t that v(s i ) ≤ y (s i ), which we can combine with (11) to obtain…”

“…By contrast, Halanay's inequality technique (which was initiated in [3]) is a useful stability analysis tool for the stability analysis of systems with poorly known time-varying delays. This inequality and variants of it, which complement Razumikhin's theorem (which was used in contributions such as [4] and [16]), have been studied in several papers, such as [2], [5], [8], [9], [11], [12], and [15], to extend the domain of application of Halanay's stability analysis strategy.…”

ISS inequalities for vector versions of Halanay’s inequality and of the trajectory-based approach

“…In [16] and [19], ISS inequalities are established, but these results do not directly extend to vector inequalities. In [15], we provided necessary and sufficient stability conditions for functions satisfying vector inequalities of Halanay's type. However, the proofs in [15] do not lead to ISS inequalities when additive disturbances are present.…”

“…In [15], we provided necessary and sufficient stability conditions for functions satisfying vector inequalities of Halanay's type. However, the proofs in [15] do not lead to ISS inequalities when additive disturbances are present. In [1], a vector version of the trajectory-based approach is developed, but no ISS inequality is given under additive disturbances.…”