Optimal input-output stabilization of infinite-dimensional discrete time-invariant linear systems by output injection

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by output injection. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a left factorization over H-infinity. Another equivalent condition is that the filter Riccati equation (of an arbitrary realization) has a solution (in general, unbounded and even nondensely defined). We further show that after renorming the state space in terms of the inverse of the smallest solution of the filter Riccati equation, the closed-loop system is not only input-output stable but also strongly internally *-stable

Well-Posed Linear Systems - a Survey With Emphasis on Conservative Systems

We survey the literature on well-posed linear systems, which has been an area of rapid development in recent years. We examine the particular subclass of conservative systems and its connections to scattering theory. We study some transformations of well-posed systems, namely duality and time-flow inversion, and their effect on the transfer function and the generating operators. We describe a simple way to generate conservative systems via a second-order differential equation in a Hilbert space. We give results about the stability, controllability and observability of such conservative systems and illustrate these with a system modeling a controlled beam