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Then
In the Hilbert space with standard scalar product, the system–dynamics equations (8.5) can be written in the abstract form proposed in Section 7.6:
(8.6) |
where
and is a closed linear operator. Notice that the first equation of the system in (8.5) was encountered into the definition of while the second one defines .
Now we seek the factor control vector such that and . Elementary calculations show that
where denotes the constant function taking the value on . The knowledge of the factor control vector allows us to derive, as in Section 7.6, a final form of an abstract model of the system:
(8.7) |
where , i.e.
(8.8) |
From the theory of neutral delay systems, generates a –semigroup on (or even a –group if ). This semigroup is EXS iff [30, p. 148 - 154] or equivalently . Since and , these inequalities are always satisfied. Observe that the condition corresponds to (unloaded line), while means that (the line loaded by the wave impedance), and means (shunted line). To get the transfer function of (8.7), we rewrite the first equation of the system (8.7) in an equivalent form
Next, assuming null initial conditions and applying the Laplace transform we get
Hence,
(8.9) |
because , i.e., the compatibility assumption (7.5) holds. Now, by (8.8),
(8.10) |
where is the finite Laplace transform of . Substituting (8.10) into (8.9), we obtain
(8.11) |