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

In this paper, we investigate the uniform exponential stability of the system dx(t)dt=Ax(t)−ρBx(t),(ρ>0),$$ \frac{dx(t)}{dt}= Ax(t)-\rho Bx(t),\kern3.0235pt \left(\rho >0\right), $$ where the unbounded operator A$$ A $$ is the infinitesimal generator of a linear C0−$$ {C}_0- $$semigroup of contractions S(t)$$ S(t) $$ in a Hilbert space X$$ X $$ and B$$ B $$ 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 prove the uniform exponential stabilization of some bilinear partial differential equations. [ABSTRACT FROM AUTHOR]