Stability and Realization of Difference Equations Over \(\mathbb {Z}\) and \(\mathbb {R}\)

While the Jury and Routh-Hurwitz tests provide necessary and sufficient conditions (NASC) for stability of linear difference equations (DE), for a class of “sparse systems” sufficient conditions of much lower complexity for stability are derived based on a Riccati-equation type condition and its function-theoretic equivalent. Next, we use the Jury test to present an algorithm to get NASC for stability dependent on the delay. The second part deals with difference equations defined over \(\mathbb {R}\) . Time-varying and state-dependent delay present new problems, including a potential loss of linearity, and require an appropriate state space to define the notions of trajectory and stability. We show with a simple toy example that a discrete event space-time structure is appropriate, and that iterated functional equations characterize solutions.