Characteristic cones and stability properties of two-dimensional autonomous behaviors

In the paper, the notions of characteristic set and, in particular, of char acteristic cone of a two-dimensional (2-D) behavior are introduced. Autonom ous behaviors are (linear shift-invariant) complete 2-D behaviors endowed w ith nontrivial characteristic sets. For this class of behaviors, a characte rization of all characteristic cones, based on the supports of the greatest common divisors (g.c.d.'s) of the maximal order miners of any matrix invol ved in the behavior description, is given. Stability property of an autonomous behavior, with respect to any of its ch aracteristic cones, Is defined first for finite-dimensional behaviors and t hen for autonomous behaviors which are kernels of nonsingular square matric es. For both classes, stability is related to the algebraic varieties of th e Laurent polynomial matrices appearing in the behavior representations. Fi nally, upon explicitly proving that any autonomous behavior can be expresse d as the sum of a finite dimensional behavior and of a square autonomous on e, stability of general 2-D autonomous behaviors is stated and characterize d.