On norm continuity,differentiability and compactness of perturbed semigroups

Semigroup Forum - The main purpose of this paper is to treat semigroup properties like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular,...     

Stabilization of Infinite-Dimensional Semilinear Systems with Dissipative Drift

In this paper we study feedback stabilization for distributed semilinear control systems . Here, A is the infinitesimal generator of a linear C ₀ -semigroup of contractions on a real Hilbert space H and is a nonlinear operator on H into itself. A sufficient ad-condition is provided for strong feedback stabilization. The result is illustrated by means of partial differential systems. Accepted 31 July 1996     

Admissibility of Control Operators in UMD Spaces and the Inverse Laplace Transform

This paper investigates the admissibility of control and observation operators in UMD spaces. Necessary and/or sufficient conditions for unbounded control operators to be admissible and weakly admissible in the Salamon–Weiss sense are presented. This is illustrated by an example which shows that the UMD-property is essential. In particular, we get a direct proof of the known result of Driouich and and El-Mennaoui (Arch Math 72:56–63, 1999) on the validity of the inverse formula of the Laplace transform for C ₀-semigroups on UMD-spaces and in Hilbert spaces, as proved earlier by Yao (SIAM J Math Anal 26(5):1331–1341, 1995). We outline how these results can be used to prove a partial validity of the inverse Laplace transform for semigroups in general Banach spaces. In particular, we obtain the result on the inverse Laplace transform due to Hille and Philllips (Am Math Soc Transl Ser 2, 1957).     

A direct approach to the Weiss conjecture for bounded analytic semigroups

We give a new proof of the Weiss conjecture for analytic semigroups. Our approach does not make any recourse to the bounded H ^∞-calculus and is based on elementary analysis.     

Regular bilinear systems

This paper deals with (weakly) regular and abstract bilinearsystems which allow some degree of unboundedness in the controland observation operators for infinite-dimensional bilinearsystems. By adaption of the notion of abstract linear systemsintroduced by Salamon (1989) and Weiss (1989a), we give a bilinearversion of their approach which works in our setting. The paperinvestigates also the relation between the shift-invariant bilinearoperator and an analytic function. This latter can be consideredas a transfer function for bilinear systems. The results thusobtained are used to study the problem of observer design forsuch systems.     

Regular linear systems governed by neutral FDEs

The aim of this paper is to prove that a class of distributed parameter systems governed by neutral FDEs provides regular linear systems. Employing the well established theory of representation, transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems.     

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.