State Space Realization for a Continuous Difference Equation with Varying Delay

The concept of state space is rigorously formulated for some systems with delay. The central tenet is that a state space should encode the minimal sufficient information that is needed to solve the Cauchy problem associated with the system. We focus on the conceptually simpler difference equations rather than differential equations. For a simple toy example the state space is constructed and the associated infinitesimal generators derived. Along the way we briefly discuss a class of iterated functional equations. This contribution is purely theoretical.