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This paper starts from several applications described by initial/boundary value problems for 1D (time and one space variable) hyperbolic partial differential equations whose basic properties and stability of equilibria are studied throughout the same properties for certain associated neutral functional differential equations. It is a common fact that asymptotic stability for neutral functional differential equations is normally obtained under the assumption of asymptotic stability of the difference operator associated to the aforementioned neutral functional differential equations. However the physically meaningful applications presented in the paper have the associated difference operator(s) in critical cases (their stability is, generally speaking, non-asymptotic). Consequently the stability of the considered application models is either non-asymptotic or fragile (in a sense introduced in the paper). The models represent an overview gathered from various fields, processed here in order to emphasize the associated neutral functional differential equations which, consequently, are a challenge to the usual approaches. In the concluding part there are suggested possible ways to overcome these difficulties. [ABSTRACT FROM AUTHOR]

We consider a system of two sets of partial differential equations describing the water hammer in a hydroelectric power plant containing the dynamics of the tunnel, turbine penstock, surge tank and hydraulic turbine. Under standard simplifying assumptions (negligible Darcy-Weisbach losses and dynamic head variations), a system of functional differential equations of neutral type, with two delays, can be associated to the aforementioned partial differential equations and existence, uniqueness and continuous data dependence can be established. Stability is then discussed using a Lyapunov functional deduced from the energy identity. The Lyapunov functional is "weak" i.e. its derivative function is only non-positive definite. Therefore only Lyapunov stability is obtained while for asymptotic stability application of the Barbashin-Krasovskii-LaSalle invariance principle is required. A necessary condition for its validity is the asymptotic stability of the difference operator associated to the neutral system. However, its properties in the given case make the asymptotic stability non-robust (fragile) in function of some arithmetic properties of the delay ratio. [ABSTRACT FROM AUTHOR]

The present paper starts from a model with distributed parameters (i.e. described by hyperbolic partial differential equations with non-standard (derivative) boundary conditions) of a hydroelectric power plant with tunnel, surge tank and penstock. The association of a system of functional differential equations of neutral type and the one-to-one correspondence between the solutions of the two mathematical objects is given. Further, it is given the deduction - via singular pertur-bations - of the nonlinear ordinary differential equations for modeling the surge tank in order to discuss its stability under constant power delivery of the hydraulic turbine. Some other unsolved problems are pointed out. [ABSTRACT FROM AUTHOR]

The paper is concerned with a class of time-delay systems that are linearly equivalent to higher-order scalar equations. If the retarded case was considered some four decades ago in Halanay (1969), the non-trivial neutral case is considered here. The specific structure of these systems endows them with remarkable properties that are useful in various applications in dynamics and control. As an auxiliary necessary result, the variation of parameters formula is revisited what opens new perspectives for research progress in such directions as the Perron condition and forced linear oscillations. [ABSTRACT FROM PUBLISHER]