Extensions of linear systems to duals of countably Hilbertian spaces with graph topology

The state space for an infinite-dimensional system of the form = Ax + Bu can be extended to the dual of a countably Hilbertainspace with graph toplogy. The Study of countably Hilbertianspaces with graph topology and their duals is made. The structuresof the extensions of a C₀ semigroup and the stability theoryof the restrictions and extensions of the semigroup are illustrated.An existence and uniqueness theorem of the solution of the extendedsystem is given.