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Infinite dimensional state space

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 tex2html_wrap_inline4034 , is defined by tex2html_wrap_inline4036 , tex2html_wrap_inline4000 .

An important observation is that the evolution of the state tex2html_wrap_inline4034 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 tex2html_wrap_inline4044 be a solution of


where tex2html_wrap_inline4048 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).

Fri Mar 20 16:01:06 MET 1998