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The conditions ensuring EXS reduce to (3.18) only, and in the expanded form they are
(3.35) |
To get (3.13) from (3.16) we take
Elementary calculations yield
where
It is more convenient to minimize the normalized performance index
with
Simultaneously, (3.35) reduces to
The first inequality jointly with yields while the second one determines the minimization interval for . Since
we conclude that is a unimodal function of for . If (the case of transmission line loaded by the wave impedance) then the minimum of is achieved at . For the minimum is located in the interval , while for in .
Remark 3.3.1. The system discussed above is a representative of a particular case of the neutral system (3.5) where . The characteristic quasipolynomial of the latter system can be factorized as follows,
Therefore the necessary and sufficient conditions for EXS simplify to the requirements that the spectra of the matrices and are located in the left open half–plane and in the open unit circle, respectively, i.e.,
Since solves the first equation in (3.25), the remaining equations of (3.25) take the form
(3.36) |
where is a unique solution of the discrete Lyapunov matrix equation (3.24). From [57, Sections 8.38.5] we conclude that the second equation in (3.36), being a "nonsymmetric" discrete Lyapunov matrix equation, has a unique solution . This holds because for satisfying .
Finally, by (3.26) the matrix identically equals zero and we conclude that in the discussed case the operator (3.21) is fully characterized by three matrix Lyapunov equations (3.36) and (3.24).