Energy decay and weak observability for some evolution systems

In an abstract functional framework, we investigate the equivalence between two fundamental properties: weak observability and energy decay associated with certain classes of dissipative operators.

This analysis is motivated by the study of evolution equations and their long-time behavior.

Our main result establishes that under appropriate assumptions, these two notions are not only closely related but in fact equivalent.

The proof relies on recent advances in the theory of weak observability, particularly those presented in \cite{AT1}, where new techniques have been developed to quantify the weak observability inequalities.

These tools allow us to bridge the gap between qualitative decay properties and spectral estimates, offering a unified perspective on stability and control in abstract dynamical systems.