In general, the state of a solution x at time t is all information that
is needed to continue the solution uniquely. From equation (2) it
is clear that the state at time t is exactly given by the solution
on the interval [t-1,t]. If we translate this piece of the solution
back to the interval [-1,0], we obtain an element of C. This leads us to
the following definition. The state at time t, denoted
by 
, is defined by 
, 
.
An important observation is that the evolution of the state is given
by an abstract differential equation in the infinite dimensional state
space C. Given equation (2) this differential equation can be computed
explicitly. Let 
be a solution of
where 
is an unbounded operator defined by
So we meet the surprising fact that the solutions of the finite dimensional delayed equation (2) are in one-one correspondence with the solutions of the infinite dimensional ordinary differential equation (3) and this correspondence is given by
This observation originated with Krasovskii and has been crucial in the development of the qualitative theory of functional differential equations, see [1] and [2].
The state space approach makes it possible to use methods from operator theory. The abstract differential equation (3) has mainly been studied in the Hilbert space setting for operators G that are selfadjoint (for example, the Laplacian operator) or close to selfadjoint. However, the operator defined by (4) is nonselfadjoint and this is the source of some of the interesting behaviour that is observed for solutions to equation (2).