Exponential stability of c0-semigroup via Lyapunov inequality in Banach space

TL;DR

This paper establishes a link between the exponential stability of $ C_0 $-semigroups in Banach spaces and solutions to a Lyapunov inequality, providing criteria for stability and invertibility based on resolvent boundedness.

Contribution

It introduces a Lyapunov inequality characterization for exponential stability of $ C_0 $-semigroups in Banach spaces, connecting stability to resolvent properties.

Findings

Lyapunov inequality solutions characterize stability

Boundedness of the resolvent relates to stability

Left invertibility of the semigroup is characterized

Abstract

We give a relation between the exponential stability of $ C_{0}- $semigroup $ \textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 0} $ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $ B^{+}(X,X^{*}) $, with $ X $ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $ R(\lambda,A) $ inside and outside of the left half-plane $ \Re\lambda\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup T.