This paper concerns some spectral properties of the scalar dynamical system defined by a linear delay-differential equation with two positive delays. More precisely, the existing links between the delays and the maximal multiplicity of the characteristic roots are explored, as well as the dominance of such roots compared with the spectrum localization. As a by-product of the analysis, the pole placement issue is revisited with more emphasis on the role of the delays as control parameters in defining a partial pole placement guaranteeing the closed-loop stability with an appropriate decay rate of the corresponding dynamical system.

We give a sufficient condition for exponential stability of a network of lossless telegrapher's equations, coupled by linear time-varying boundary conditions. The sufficient conditions is in terms of dissipativity of the couplings, which is natural for instance in the context of microwave circuits. Exponential stability is with respect to any L p -norm, 1 ≤ p ≤ ∞. This also yields a sufficient condition for exponential stability to a special class of systems of linear time-varying difference-delay equations which is quite explicit and tractable. One ingredient of the proof is that L p exponential stability for such difference-delay systems is independent of p, thereby reproving in a simpler way some results from [4].

No abstract