Exponential Characterization in Linear Viscoelasticity Under Delay Perturbations

We present a complete characterization of the (uniform) exponential stabilization for a class of viscoelastic models under small delay perturbations. The main ingredient under consideration is the notion of admissible kernels. While in the standard literature it is mostly common to request a exponential/general kernel as a sufficient condition for the exponential/general stability of the whole viscoelastic system under study, here our objective is to employ the much more general concept of admissible kernels and prove that it is not only sufficient but also a necessary assumption for exponential stability in linear viscoelasticity under small delay perturbations.

Automated summary

We provide a comprehensive understanding of the (uniform) exponential stabilization for a specific class of viscoelastic models in the presence of small delay perturbations. The key focus lies on the concept of admissible kernels. While previous literature often emphasizes the use of exponential/general kernels as sufficient conditions for exponential/general stability of the entire viscoelastic system, our objective is to demonstrate that the more generalized concept of admissible kernels is not only sufficient but also necessary for exponential stability in linear viscoelasticity under small delay perturbations.