On some special features which are peculiar to discrete-time behaviors with trajectories on Z+

In this paper, discrete time systems defined on Z+ are analyzed within the behavioral approach. Once the main internal properties of such behaviors have been defined and characterized, our attention is focused on those aspects that make behaviors defined on Z+ dramatically different from behaviors defined on Z. Specifically, the general lack of permanence and the role of finite support trajectories. Several consequences of these two facts are analyzed. In particular, it is shown that while controllable behaviors with trajectories on Z can always be identified by means of a finite set of finite support trajectories, for controllable behaviors with trajectories on Z+ this is no more possible.