Linear State-Space Systems in Infinite-Dimensional Space: The Role and Characterization of Joint Stabilizability/Detectability

In this paper several fundamenal results from the theory of linear state-space systems in finite-dimensional space are extended to encompass a class of linear state-space systems in infinite-dimensional space; specifically, we generalize those results from the finite-dimensional theory pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and more generally, the concept of joint stabilizability/detectability. In the course of doing so a complete structural characterization of jointly stabilizable/detectable systems is obtained. These results are distinguished by their generality, as we consider a large class of linear state-space systems, assuming only that (i) the evolution of the state is governed by a strongly-continuous semigroup of bounded linear operators, (ii) the state-space is a Hilbert space, (iii) the input and output spaces are finite-dimensional, and (iv) the sensing and control operators are bounded. In turn, general conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space may be drawn from these results.