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INTRODUCTION TO SEMIGROUP THEORY

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Abstract

In this paper some basic sytem theoretic concepts will be introduced for abstract systems of the form x(t) = Ax(t) + Bu(t), x(0) = x0, y(t) = Cx(t). (1) Here A is the infinitesimal generator of a strongly continuous semigroup S(t) on a Banach space Z and necessary and sufficient conditions for this to be the case axe given by the Hille-Yosida theorem. For U another Banach space B is-an-element-of C(U, Z) and x0 is-an-element-of Z, u(.) is-an-element-of L2(0, infinity; U) a mild solution is defined to be x(t) = S(t)x0 + integral-t/0 S(t - s)Bu(s)ds (2) and x(.) is-an-element-of C(0; infinity; Z). Various definitions of controllablity, observability, stabilizability, detectability, identifiability and realizability will be given and theorems which characterize them will be stated. Throughout the paper examples will be given (albeit trivial ones) which illustrate the way the abstract definitions and results can be applied to concrete problems defined via partial differential equations and delay equations. In preparing this introduction I have made considerable use of the following book by Ruth Curtain and Hans Zwart An Introduction to Infinite Dimensional Linear Systems Theory which is to be published soon

Topics: QA76, TL
Publisher: SPRINGER VERLAG
OAI identifier: oai:wrap.warwick.ac.uk:21273
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