Linear state-space systems in infinite-dimensional space: the role and characterization of joint stabilizability/detectability

Several fundamental 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. The results treated are those pertaining to the relationship between input-output and internal stability, the problem of dynamic output feedback stabilization, and the concept of joint stabilizability/detectability. A complete structural characterization of jointly stabilizable/detectable systems is obtained. The generalized theory applies to 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 Hilbert space; (iii) the input and output spaces are finite-dimensional; and (iv) the sensing and control operators are bounded. General conclusions regarding the fundamental structure of control-theoretic problems in infinite-dimensional space can be drawn from these results