On norm continuity, differentiability and compactness of perturbed semigroups

Abstract

The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.

Feedback theory approach to positivity and stability of evolution equations

“…We mention that the control operator B defined in (7) is positive if and only if the Dirichlet operator D λ is positive for all λ > s(A) (see Remark 2.2 of [5]).…”

“…From Example 2.6.8 [31], the operator A := A m with domain D(A) = {f ∈ Z : f ′ (0) = 0} generates a positive and exponentially stable semigroup on L 2 [0, π]. Since the triple (A, −A −1 D 0 , kδ 0 ) is a regular linear system with I : R −→ R is an admissible feedback operator (see Example 1 of [7] and Lemma 3 of [8]) then the solution of equation ( 14) is positive. In addition, it is uniformly exponentially stable if and only if k < 1 π .…”