Semigroup Theory for Control on Sun-reflexive Banach Spaces

Abstract

We use the theory of dual C0-semigroups, as developed by Phillips, to define and study a new class of control systems on nonrefiexive Banach spaces. Our main results concern the (approximate) controllability and observability of such systems. We illustrate our abstract results with an application to a delay system.

A Semigroup Approach to Boundary Feedback Systems

“…Many authors have used semigroup theory to study linear initial boundary value or boundary control problems (see, e.g., the monographs by Lasiecka-Triggiani [LT00], the papers by Desch et al [DLS85], [DMS01], the - * approach in Heijmans [Hei87], or the approach via characteristic matrices by Kaashoek and Verduyn Lunel [KVL92].) In this paper we propose an approach in which we convert the given boundary value problem on some domain Ω ⊂ R n into an (inhomogeneous) Abstract Cauchy Problem Ẋ(t) = AX(t) + F (t), t ≥ 0, X(0) = X 0 (iACP)…”

A class of abstract delay differential equations in the light of suns and stars. II

“…This will be explored further in collaboration with S. M. Verduyn Lunel. We note that [21] contains an early controltheoretic application of dual perturbation theory, in a -reflexive setting. Meanwhile, O. Diekmann has suggested to me a class of abstract renewal equations inspired by structured population dynamics.…”

Strong ergodicity for perturbed dual semigroups and application to age-dependent population dynamics

“…So it is clear that the perturbation theory for dual semigroups in sun-reflexive Banach spaces could give a suitable framework to generalize the classical control theory. Heijmans [14] first gave controllability and observability results for the canonical control system in sun-reflexive Banach spaces.…”