Algebraic sufficient conditions for bounded-type linear input/output maps

This paper studies the problem of realization for a class of constant (stationary) linear input/output maps, which are to be called input/output maps of bounded type. A constant linear input/output map is of bounded type if its canonical realization possesses the property that the initial state determination can be done in a well-posed way based on bounded-time output data. Intuitive considerations suggest that this be the most natural (and easiest) class to consider in the theory of realization as an extension of the now classical realization theory for finite-dimensional constant linear systems. The main body of the paper is devoted to the characterization of bounded-type linear input/output maps. To this end, the module theoretic treatment of infinite-dimensional continuous-time linear systmes initiated by Kamen is fully utilized. As an application of the main result, it is shown that the input/output maps of delay-differential systems and periodic input/output maps belong to this class. In the final section, the canonical state space model is naturally derived from the input/output data for a delay-differential system, and it is also seen that the derived model exactly agrees with what is known as the M2-space model for such a retarded delay-differential system. Also, a way of constructing approximate finite-dimensional models for such systems is illustrated, and such approximate models are seen to agree with the well-known averaging approximations for retarded delay-differential systems.