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