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International audience; This paper focuses on the study of the behavior of critical roots when a dynamical system is stabilized by a PD-controller, for which the derivative action has been approximated by using two commensurate delays. The use of such an approximation leads to a characteristic quasipolynomial whose coefficients depend explicitly on the delay parameter. The aim of the paper is to address the way the delay parameter may affect the location of the roots of the corresponding characteristic function, and in particular the cases when "small" delays induce instability in the closed-loop systems. Such an analysis is performed by expressing the critical solution of the system as a delay-dependent power series. Illustrative examples complete the presentation.

International audience; This work focus on the asymptotic behavior of critical roots of linear time-delay systems with delay-dependent coefficients. Our main goal is to derive a simple methodology to characterize some ill-posed cases from the open literature. In order to accomplish such a task, our main tool will be to express the solutions as a convergent power series. More specifically, by means of an appropriate change of variable, the expansion of unbounded solutions is obtained, which allows to derive an appropriate auxiliary solution that serves to analyze the behavior of such solutions. One of the main benefits in adopting such an approach, is that stability analysis in the ill-posed cases can also be covered.