Optimal State Feedback Input-Output Stabilization of Infinite-Dimensional Discrete Time-Invariant Linear Systems

We study the optimal input-output stabilization of discrete time-invariant linear systems in Hilbert spaces by state feedback. We show that a necessary and sufficient condition for this problem to be solvable is that the transfer function has a right factorization over H-infinity. A necessary and sufficient condition in terms of an (arbitrary) realization is that each state which can be reached in a finite time from the zero initial state has a finite cost. Another equivalent condition is that the control Riccati equation has a solution (in general unbounded and even non densely defined). The optimal state feedback input-output stabilization problem can then be solved explicitly in terms of the smallest solution of this control Riccati equation. We further show that after renorming the state space in terms of the solution of the control Riccati equation, the closed-loop system is not only input-output stable, but also strongly internally stable.