On a Class of Applications for Difference Equations in Continuous Time

There are considered certain engineering applications described by non-standard boundary value problems for 1D hyperbolic partial differential equations. Their qualitative analysis is performed by associating a class of functional differential equations with deviated argument, in most cases of neutral type. At their turn, to these neutral equations it is associated a system of continuous time difference equations. The difference equations are required to be asymptotically stable to ensure the same behavior for the basic system via a “weak” Lyapunov functional of energy type. The explanation is as follows: the derivative of a “weak” Lyapunov function(al) being only negative semi-definite, asymptotic stability is obtained from the application of the Barbashin-Krasovskii-LaSalle invariance principle. In the case of the neutral functional differential equations the invariance principle holds under the aforementioned property of asymptotic stability for the difference equation associated to it. It is then shown how the introduction of additional dissipation in the model can lead to asymptotic stability for both the difference system and the basic one, described by the boundary value problems for 1D hyperbolic partial differential equations. There are also pointed out connections to other mathematical problems such as dissipative/conservative boundary conditions.