Tackling critical cases of the difference operator stability occurring
in applications described by 1 D distributed parameters.
The paper starts from the challenge of the critical cases in difference
operator stability for neutral functional differential equations (NFDE).
Such cases occur in the NFDE associated to 1 D hyperbolic partial
differential equations (PDE) dynamics in Mechanical and Hydraulic
Engineering. For some of such applications it resulted that the
aforementioned critical (non-asymptotic) stability is connected to the
character and level of the energy losses. It is shown that suitable
choice of the losses to be taken into account can remove the critical
stability properties and give the difference operator the asymptotic
stability thus ensuring asymptotic stability for the system’s dynamics
and also other asymptotic properties.