Graphs, causality, and stabilizability: Linear, shift-invariant systems on ℒ₂[0, ∞)

This paper presents a number of basic elements for a system theory of linear, shift-invariant systems on ℒ₂[0, ∞). The framework is developed from first principles and considers a linear system to be a linear (possibly unbounded) operator on ℒ₂[0, ∞). The properties of causality and stabilizability are studied in detail, and necessary and sufficient conditions for each are obtained. The idea of causal extendibility is discussed and related to operators defined on extended spaces. Conditions for w-stabilizability and w-stability are presented. The graph of the system (operator) plays a unifying role in the definitions and results. We discuss the natural partial order on graphs (viewed as subspaces) and its relevance to systems theory.