A note on the control problem for 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. A general decomposition result is given, showing that every behavior can be expressed as the intersection of an observable and a locally undetectable behavior. Finally, the control problem is 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 achievable for the resulting controlled system, are given