Stability theory for infinite-dimensional linear systems based on an extension method

The state for a homogeneous infinite-dimensional linear system can be extended to the dual of a countably Hilbertian space with graph topology. The stability theory for the extension of the system is illustrated. We prove that the exponential growth bound of the extended system is the infimum of ω such that the resolvent is a funtion of finite order in the half plane Re λ ≥ ω. Relationships between various notions of stability of the original system and of its extension are considered. For the question of exponential stability for physical problem, the original theory may lead to opposite results while the extension method will avoid the contradiction.