DELAY SYSTEMS DIFFERENTIAL STATE SPACE DESCRIPTIONS OF NONLINEAR TIME VARIANT HEREDITARY DIFFERENTIAL SYSTEMS Fritz Colonius and Diederich Hinrichsen Department of Mathematics, University of Bremen, W. Germany ABSTRACT This paper investigates the problem which trajectories of a system z, defined by a nonlinear functional differential equation in a real Banach space E, may be described by an operator differential equation in appropriate state spaces. The use of semi-group methods is avoided by a separate analysis of the differ- ential equation % = Az, where A is the first order differential operator on the state space. Necessary and sufficient conditions for the segment function to be absolutely continuous are derived. The sets of admissible initial data are de- termined. The equivalence of the functional differential equation with the op- erator differential equation in the state space (e and M~ spaces) is estab- lished for these initial data. 1. INTRODUCTION The concept of state is the central notion of modern system theory. If we want to integrate the study of hereditary systems into the context of mathema- tical system theory we first have to define the state of a hereditary system and to describe its evolution by state space equations. This may be done in many different ways, in accordance with the particular state space and the type of the state space equation, which determines the state-transition func- tion [1,p.5]. In this paper we only consider hereditary differential systems (HDS) which are described by functional differential equations of the form ; (t) = f (t,xt) . We analyze the possibilities and limits of their description by differential equa- tions in those state spaces which have been principally used in the literature as spaces of initial conditions (C and M ' spaces). There is already a rather extensive literature on the state space description of HDS, of which we can cite only a few references (see [2], [12] for more de- tailed accounts). As state spaces have first been used spaces of continuous functions ( "e -theoryn [33, c41, then Sobolev-spaces w1 C53, [6] and spaces of Lebesgue-integrable functions ( theor^" or^" t7] ,187 ,t91,[101. The bulk of the literature deals with linear HDS and studies their state space description with emphasis on the linear quadratic optimal synthesis problem [B]:, [ll], b23, spec- tra L theory C31, b33, linear structure theory [s] , C143, 0 51, and re lations to general semi-group theory [3] , [I 61 , [ l 73. Only little work has been done on the state space description of nonlinear HDS. We only know of two articles of G.F. Webb t4] resp. [101, who uses methods of nonlinear semi-group theory for the state space description of autonomous HDS within the framework of continuous resp. L'-integrable functions. The present paper investigates, under which conditions and, in particular, for which initial data the trajectories of a functional differential system can be