Integrally Small Perturbations of Semigroups and Stability of Partial Differential Equations

Abstract EN

Let A be a generator of an exponentially stable operator semigroup in a Banach space, and let Ct t≥0 be a linear bounded variable operator.

Assuming that ∫0tCsds is sufficiently small in a certain sense for the equation dx/dt=Ax+C(t)x, we derive exponential stability conditions.

Besides, we do not require that for each t0≥0, the “frozen” autonomous equation dx/dt=Ax+C(t0)x is stable.

In particular, we consider evolution equations with periodic operator coefficients.

These results are applied to partial differential equations.