Duality theory for discrete-time linear systems

摘要

Previous duality theories for discrete-time linear systems over a field K have been restricted to cases in which the input, state, and output spaces are finite dimensional. Direct attempts to extend such a theory to infinite-dimensional systems fail, because the category K-LS of linear spaces over the field K is not self-dual and hence does not, by itself, provide an adequate framework for a general duality theory of discrete linear systems. Instead, it is necessary to consider categories of linearly topologized spaces over K, and to use topological rather than algebraic duals. With this approach, the dimensionality of the system is of no consequence, and so finite- and infinite-dimensional systems are handled with equal ease. A general categorical duality of discrete-time linear systems is first developed within the framework of the self-dual category K-DP of dual pairs over K, so that the essential character of the theory is algebraic rather than topological. K-DP is equivalent to sK-LTS, the category of weak linearly topologized spaces, and also to kK-LTS, the category of Mackey linearly topologized spaces. This provides a linearlytopologized-space framework for discrete-time linear systems, with topological dualization the underlying duality functor. An alternative theory of duality in which the system is modeled directly in K-LS is also presented. Using the duality of maximal and minimal dual pairs, the category cK-LTS of linearly compact, linearly topologized spaces, and not K-LS, is seen to be the proper framework for studying the duals of machines in K-LS. Again, topological dualization is the underlying duality functor.