Characteristic cones and stability properties of two-dimensional autonomous behaviors

Abstract—In the paper, the notions of characteristic set and, in particular, of characteristic cone of a two-dimensional (2-D) behavior are introduced. Autonomous behaviors are (linear shift-invariant) complete 2-D behaviors endowed with nontrivial characteristic sets. For this class of behaviors, a characterization of all characteristic cones, based on the supports of the greatest common divisors (g.c.d.'s) of the maximal order minors of any matrix involved in the behavior description, is given. Stability property of an autonomous behavior, with respect to any of its characteristic cones, is defined first for finite-dimensional behaviors and then for autonomous behaviors which are kernels of nonsingular square matrices. For both classes, stability is related to the algebraic varieties of the Laurent polynomial matrices appearing in the behavior representations. Finally, upon explicitly proving that any autonomous behavior can be expressed as the sum of a finite-dimensional behavior and of a square autonomous one, stability of general 2-D autonomous behaviors is stated and characterized.