A Basic Theorem on Distributed Control and Point Control

Abstract

This paper is concerned with linear distributed parameter systems whose input-output operators are representable in integral form. Two types of control are considered: (i) distributed control which is a function of both a spatial variable x (lying in a compact set Ω) and a time variable t, and (ii) “point” control which is applied at a specific point in Ω and is a function only of t. For such systems, a basic theorem is stated and proved, namely, that there exists a countable subset E of Ω with the following property: any state which can be attained by applying a distributed control can also be attained arbitrarily closely by applying a finite number of point controls applied at points in the set E. The theorem is applied to some specific systems, and further possible applications of the theorem are discussed.