Minimax control of distributed discrete time systems through spectral factorization

We introduce a Riccati equation theory for a class of well posed (I/O-stable) discrete time linear systems Φ as presented in [6]. We shall tie together three different problems: The first problem is the general question under which conditions a minimax control problem associated to Φ can be solved by a feedback law. The second problem is the existence of certain spectral factorization of the I/O-map of Φ. The third problem is about certain solution of a Riccati equation system associated to Φ. We shall show that these three problems are in fact equivalent. This equivalence does not require any finite dimensional structure of the system Φ. The I/O-stability notion that we use throughout this paper is weaker than the conventional power stability (ρ(A) <; 1). Finally, connections to the existing power stable and finite dimensional theories are presented.