REALIZATION THEORY OF INFINITE-DIMENSIONAL LINEAR SYSTEMS By YUTAKA YAMAMOTO A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA

of Dissertation Presented to the Graduate Council of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy REALIZATION THEORY OF INFINITEDIMENSIONAL LINEAR SYSTEMS By YUTAKA YAMAMOTO August, 1978 Chairman: Dr. R. E. Kalman Major Department: Mathematics This work studies the problem of realization of constant linear input/ output maps, which do not necessarily possess a finite-dimensional realization. A class of constant linear input/output maps is introduced. This class is then characterized as the family of continuous linear maps whose weighting patterns are measures. The natural state-space representation (realization) of such constant linear input/output maps is studied, and a new notion of observability, topological observability , is introduced. It is then seen that topological observability enables us to prove the existence and uniqueness of canonical (quasi-reachable and topologically observable) realizations. It is also shown that a certain subclass of realizations admits a functional-differential equation description. Necessary and sufficient conditions that the state space of a canonical realization be a Banach (or Hilbert) space are obtained. A new notion, topological observability in bounded time , plays a central role in deriving such conditions. A thorough study of a concrete example of such systems Is given. CHAPTER