Polynomial stability for a system of coupled strings

Abstract

We study the long‐time behavior of two vibrating strings which are coupled at a common boundary point by a damper. We show that the classical solutions converge polynomially with a uniform rate, where the decay exponent depends on number theoretic properties of the quotient of the wave speeds of the two springs. The proof is based on a resolvent characterization of polynomial stability due to Borichev–Tomilov and Batty–Duyckaerts.

Non-uniform Stability of Damped Contraction Semigroups

“…In this section we consider the one-dimensional wave equation with pointwise damping studied in [2, Sec. 5.1] (see also [46] for a closely related problem on the stability of two serially connected strings). Given an irrational number ξ 0 ∈ (0, 1) let us consider w tt (ξ, t) − w ξξ (ξ, t) + w t (t, ξ 0 )δ ξ 0 (ξ) = 0, ξ ∈ (0, 1), t > 0, (6.3a) w(0, t) = 0, w(1, t) = 0, t > 0, (6.3b)…”

Remarks on Exponential Stability for a Coupled System of Elasticity and Thermoelasticity with Second Sound

“…While an energy functional approach, employed for example in [1,6,10,13,14,18,21,23], is beneficial in that it directly applies to some semilinear settings, there might appear disadvantages when necessary conditions for stability are desired or in cases where suitable Lyapunov type functionals are difficult to obtain. On the contrary, the linear semigroup approach offers several equivalent characterizations for exponential stability and some criteria for weaker forms such as polynomial or strong stability, which have also proven to be helpful for showing the lack of exponential stability, see for example [1,2,3,4,7,15,16,20,22,24,25].…”