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.