Controllability of Infinite-dimensional Systems with Delays in Control

T his p a p e r considers th e vario u s ty p es of co n tro llab ility of lin e a r in fin ite-dim ensional d ynam ical system s defined in a B an ach space, w ith m u ltip le tim e-v ary in g delays in control. N ecessary an d su fficien t conditions fo r ap p ro x im a ­ te co n tro llab ility , ap p ro x im ate re la tiv e co n tro llab ility an d ap p ro x im ate absolute-co n tro llab ility of th ese system s a re obtained. S pecial cases of system s defined in a H ilb e rt space a re also considered. 1. Introduction and notation. C ontrollability is one of the most im ­ portant notions in m odern systems theory. Various types of controlla­ bility of linear abstract dynamical system s defined in a Banach or Hil­ bert spaces have been recently extensively explored by several authors (see e. The m ain purpose of this paper is to exam ined some fundam ental questions con­ cerning approxim ate controllability, approxim ate relative controllability and approxim ate absolute controllability of linear abstract dynam ical systems defined in Banach or H ilbert spaces, w ith m ultiple tim e-varying delays in control. From the well known consequence of the H ahn-B a-nach theorem [13], [14], necessary and sufficient conditions for various types of controllability are derived. In particular for systems defined in H ilbert spaces, stronger conditions are obtained. This paper extends the results given in the papers [3], [4], [13], [14] to the systems w ith m ultiple tim e-varying delays in control. In the sequel th e following notation and terminology, which are adopted from the papers [3], [4], [13], [14], will be used. AMS (MOS) su b je ct classification (1980). P rim a ry 93B05.