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 non densely 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 closedloop system is not only input-output stable, but also strongly internally ∗-stable.