Stability Results for Second-Order Evolution Equations with Switching Time-Delay

Abstract

We consider second-order evolution equations in an abstract setting with intermittently delayed/not-delayed damping. We give sufficient conditions for asymptotic and exponential stability, improving and generalising our previous results from [19]. In particular, under suitable conditions, we can consider unbounded damping operators. Some concrete examples are finally presented.2000 Mathematics Subject Classification: 35L05, 93D15

Stability of solutions to nonlinear wave equations with switching time delay

“…Here we extend the results of [22,23] to nonlinear models but also we significantly improve some results there proved. Indeed, we are able to remove an assumption (see (2.14) below) on the feedback bounds obtaining more general results also in the linear case.…”

“…f ≡ 0) we can improve previous results given in [23] by removing the assumption (2.14) on the coefficients. As in [23] we can determine more explicitely, in terms of the coefficients T 2n , m 2n , M 2n , the constantd n of Proposition 3.10, for all n ∈ IN .…”

“…B 2 ≡ 0 , as of course it happens for the standard energy E S (·). In order to have a decay estimate for E(·) in the intervals I 2n , we should assume, as in [23], inf n∈IN m 2n M 2n+1 > 0, (2.14) and define E(·) as…”

Stabilization of second-order evolution equations with time delay

“…Under the assumptions of point 1 of Remark 4.2 and the additional assumptions of this subsection, the observability estimate for the associated conservative system has been recently proved by the authors (see [22], Theorem 6.1).…”

Stability Results for Abstract Evolution Equations with Intermittent Time-Delay Feedback

“…On the contrary, in previous stability results for damped wave equation and intermittent delay feedback (see e.g. [24,9]) the set ω 2 has to be a subset of ω 1 . On the other hand, now the standard (not delayed) frictional damping is always present in time while in the quoted papers it is on-off like the delay feedback and it acts only on the complementary time intervals with respect to this one.…”