Stabilizability of linear discrete-time systems defined over a commutative normed algebra

An algebraic Riccati equation and a Riccati difference equation, each defined over a commutative normed algebra B, are used to study stabilizability of linear discrete-time systems defined over B. This framework can be applied to the problem of stabilizing linear shift-invariant half-plane two-dimensional digital filters. Conditions for the existence of a stabilizing feedback are given in terms of a solution in the limit to a Riccati difference equation over B. Results are also given on the relationship between local and global stabilizability.