  A causal input-output system may be described by a function space for inputs,
a function space for outputs, and a causal operator mapping the input space
into the output space. A particular representation of the state of such a
system at any instant has been defined as an operator from the space of
possible future inputs to that of future outputs. This representation is called
the natural state. The purpose of this report is to investigate additional
properties of the natural state in two areas. The first area has to do with the
possibility of determining the input-output system from its natural state set.
A counterexample where this is not possible is given. Sufficient conditions for
identifying the system from its natural state set are given. The results in
this area are mostly for time-invariant systems. There are also some
preliminary observations on reachability. The second area deals with
differentiability properties involving the natural state inherited from the
input-output system, including differentiability of the natural state and
natural state trajectories. A differential equation representation is given.
The results presented in this report may be considered as aids in modeling
physical systems because system identification from state set holds in many
models and is only tacitly assumed; also, differentiability is a useful
property for many systems.
