On the Evolution Operators of a Class of Linear Time-Delay Systems

This paper studies the properties of the evolution operators of a class of time-delay systems with linear delayed dynamics. The considered delayed dynamics may, in general, be time-varying and associated with a finite set of finite constant point delays. Three evolution operators are defined and characterized. The basic evolution operator is the so-called point delay operator, which generates the solution trajectory under point initial conditions at t0=0. Furthermore, this paper also considers the whole evolution operator and the delay strip evolution operator, which define the solution trajectory, respectively, at any time instant and along a strip of time whose size is that of the maximum delay. These operators are defined for any given bounded piecewise continuous function of initial conditions on an initialization time interval of measure being identical to the maximum delay. It is seen that the semigroup property of the time-invariant undelayed dynamics, which is generated by a C0-semigroup, becomes lost by the above evolution operators in the presence of the delayed dynamics. This fact means that the point evolution operator is not a strongly and uniformly continuous one-parameter semigroup, even if its undelayed part has a time-invariant associated dynamics. The boundedness and the stability properties of the time-delay system, as well as the strong and uniform continuity properties of the evolution operators, are also discussed.