Minimax Control of Distributed Discrete Time Systems through Spectral Factorization

In this paper we introduce a Riccati equation theory for (a class of) well posed (I/O-stable) discrete time linear systems \Phi as presented in [9]. We tie together three different notions: The first notion is the general question under which conditions it is possible to solve a minimax control problem associated to \Phi by static state feedback. The second notion concerns the existence of a certain spectral factorization of the I/O-map of \Phi. The third notion is about a particular (stabilizing) solution of a Riccati equation system associated with \Phi. We show that these three notions are in fact equivalent under fairly mild stability assumptions of \Phi, namely input-output stability. Furthermore, this equivalence does not require any finite dimensional structure in any of the operators of the system