On the control of finite support behaviors

In the paper the main properties of finite support behaviors are introduced and characterized in terms of the polynomial matrices involved in their representation. Necessary and sufficient conditions for the existence of an input/output (possibly, non-anticipating) description of a finite support behavior are given. The control problem is also addressed, and properly restated in the behavior theory context as an interconnection problem. Conditions for the existence of controllers that make the resulting connected system observable, as well as a complete description of the class of characteristic polynomials that the resulting controlled system possibly exhibits, are given. Finally, the problem of controlling a dynamical system in input/output from by means of a controller with the same structure, is investigated and solved.