15 - Well-Posed State/Signal Systems

Summary

In this chapter, we study well-posed s/s systems. By definition, a s/s system is well-posed if it has at least one well-posed i/s/o representation (and then usually infinitely many wellposed i/s/o representations). The results presented here are analogous to the corresponding well-posed i/s/o results in Chapter 14, and most of the proofs consist of showing how to reduce a particular well-posed s/s result to the corresponding well-posed i/s/o result. In the well-posed i/s/o setting, we can interpret a well-posed i/s/o system as a realization of a continuous linear causal shift-invariant exponentially bounded operator. In the well-posed s/s setting, we can instead interpret a well-posed s/s system as a realization of a well-posed future, past, or two-sided behavior. Each one of these behaviors determine, the other two uniquely. Two well-posed s/s systems are externally equivalent if and only if they have the same well-posed behaviors. At the end of this chapter, we define the notion of a passive state/signal system and show that passive state/signal systems are well-posed. A passive state/signal system is characterized by the fact that it is regular, and its generating subspace is a maximally nonnegative subspace of the Kreĭn node space.