Structure at Infinity and Impulsive Behaviors

The topic of this chapter is the structure at infinity of discrete and continuous linear time-varying systems in a unified approach. When interconnecting two subsystems, ”impulsive motions” may arise. In the continuous-time case, those impulsive motions are linear combinations of the Dirac distribution δ and its derivatives; they were first studied by Verghese [346]. In the discrete-time case, those impulsive behaviors are backward solutions with finite support; they were revealed by Lewis [216]. The space spanned by all impulsive motions of a system is called its ”impulsive behavior” and is denoted as \( \mathfrak{B}_{\infty }\). The structure of this impulsive behavior must be studied, and, for the integrity of the system resulting from the interconnection, all impulsive motions must be avoided.