State Space Realization for Systems with State Dependent Delay

This paper analyzes simple toy systems, consisting of a difference equation in continuous time, but where the delay depends on the current state, and gives some preliminary results for the higher dimensional case. These difference equations in continuous time are conceptually simpler than the corresponding differential systems. 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. The state space is rigorously constructed for some typical state dependent difference equations in continuous time. Dynamics are represented by the associated infinitesimal generator, which is subsequently derived. In passing, we discuss a class of scalar and vector iterated functional differential equations, and present some new characterizations of their solution. Finally, we present some preliminary results for a scalar differential delay system with state dependent delay. This contribution is purely theoretical.