Exponential stability of linear systems under a class of Desch-Schappacher perturbations

TL;DR

This paper studies the exponential stability of linear systems with unbounded operators and Desch-Schappacher perturbations, providing conditions for stability and applying results to stabilize bilinear PDEs.

Contribution

It introduces new sufficient conditions for exponential stability of systems with unbounded operators under Desch-Schappacher perturbations.

Findings

Established stability criteria for perturbed systems.

Applied results to stabilize bilinear PDEs.

Extended stability analysis to a class of unbounded operators.

Abstract

In this paper we investigate the uniform exponential stability of the system $\frac{dx(t)}{dt}=Ax(t)-\rho Bx(t), \; (\rho >0), $ where the unbounded operator $A$ is the infinitesimal generator of a linear $C_0-$semigroup of contractions $S(t)$ in a Hilbert space $X$ and $B$ is a Desch-Schappacher operator. Then we give sufficient conditions for exponential stability of the above system. The obtained stability result is then applied to show the uniform exponential stabilization of bilinear partial differential equations.