Stability Analysis using New Variant of Halanay’s Inequality

Vector Extensions of Halanay’s Inequality

“…The usual Halanay's inequality result [8] is the following: if a > b, then v(t) exponentially converges to zero as t → +∞. In addition to our works [16] and [17] that relax the requirement that the decay rate a is strictly larger than the gain b, several other extensions of this result are available in the literature e.g., in [6], [20], [23], and [24]. Time-varying versions have been studied in [2] and [15].…”

“…Using (2) to obtain a scalar valued function v satisfying the requirements of Halanay's theory does not seem to be possible in general. These remarks motivate this paper which continues our search for generalized or relaxed versions of Halanay's inequality, which we began in [16] and [17]. While [16] provided less restrictive versions of Halanay's inequality where the gain in the overshoot term can exceed the decay rate including applications to systems with scarce arbitrarily long sample intervals, and while [17] covered sampled cases that were beyond the scope of earlier Halanay's inequality formulations such as [16], here we pursue a very different direction, where the usual scalar decaying function in Halanay's inequality is replaced by a vector valued function.…”

“…These remarks motivate this paper which continues our search for generalized or relaxed versions of Halanay's inequality, which we began in [16] and [17]. While [16] provided less restrictive versions of Halanay's inequality where the gain in the overshoot term can exceed the decay rate including applications to systems with scarce arbitrarily long sample intervals, and while [17] covered sampled cases that were beyond the scope of earlier Halanay's inequality formulations such as [16], here we pursue a very different direction, where the usual scalar decaying function in Halanay's inequality is replaced by a vector valued function. This provides an analog to vector Lyapunov function results that is applicable to stabilization problems that were beyond the scope of earlier Razumikhin function or diagonal stability based methods; see [10], [22], and [26] for vector Lyapunov functions and [5] for input-to-state stability (or ISS) for interconnected systems via combinations of Lyapunov functions, or under small gain conditions that we do not require here.…”

New Versions of Halanay’s Inequality With Multiple Gain Terms

“…. , k. By combining the preceding inequalities with (9), it follows that v(t ) − ẏ (t ) < 0, so there is t ∈ [t 0 − T 2,k , t ) such that v(t ) − y (t ) > 0, because v(t ) − y (t ) = 0. This contradicts the definition of t .…”

“…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.…”