Stability of the heat and of the wave equations with boundary time-varying delays

Abstract

Exponential stability analysis via Lyapunov method is extended to the one-dimensional heat and wave equations with time-varying delay in the boundary conditions. The delay function is admitted to be timevarying with an a priori given upper bound on its derivative, which is less than 1. Sufficient and explicit conditions are derived that guarantee the exponential stability. Moreover the decay rate can be explicitly computed if the data are given.

Existence and asymptotic stability of a viscoelastic wave equation with a delay

“…Using the estimate (10) in Lemma 2.2, Young's inequality and the fact that t 0 g(s)ds ≤ ∞ 0 g(s)ds = 1−l, We get for any η > 0, (see relation (20) in [14])…”

“…In other words, the decay is slower when τ becomes larger. The case of time-varying delay in the wave equation has been studied recently by Nicaise et al [20] in one space dimension. In that work, an exponential stability result was given under the condition…”

Asymptotic Behavior for a Viscoelastic Wave Equation with a Time-varying Delay Term

“…And, they also obtained the same results if both the damping and the delay act in the domain. The case of time-varying delay in the wave equation has been studied by Nicaise et al [10] in one space dimension, in which they obtained an exponential decay result subject to the condition…”

“…In fact, they proved exponential stability of the solution for the wave equation with a timevarying delay in the boundary condition in a bounded and smooth domain in R N . Recently, inspired the works of Nicaise et al [11] and M. Kirane et al [5] 10) with the same initial and boundary conditions (1.3)-(1.4), where a global existence result for γ≥0 and an exponential decay result for γ>0 were established under the assumptions…”

Stabilization of the Gas Flow in Star-Shaped Networks by Feedback Controls with Varying Delay

“…It is an extension of the Lyapunov function introduced in [4]. Delay terms of the form (21) have been presented in [12] for the time-delayed stabilization of the wave equation. Related questions of the stabilization of the wave equation are e.g.…”