## Section: New Results

### Stability of nonlinear high frequency amplifiers and stability of linear time-varying time-delay systems

Participants : Laurent Baratchart [FACTAS project-team] , Sébastien Fueyo, Jean-Baptiste Pomet, Gilles Lebeau.

These amplifiers contain on the one hand nonlinear active components and on the other hand lines, that induce some sort of delays and make the system infinite-dimensional: they are, for each choice of a periodic input, a nonlinear infinite dimensional dynamical system. The Computer Aided Design tools mentioned in Section 4.4 provide a periodic solution under this periodic forcing and may also give the frequency response of the linearized system along this trajectory with some artificial “small” excitation. The goal is to deduce stability from these data.

It is an opportunity to build theoretical basis and justification to a stability analysis through harmonic identification; the latter is one of the specialties of FACTAS, we collaborate on the infinite-dimensional non-linear stability analysis for periodic solutions and how it works with the results of harmonic identification. This is the topic of Sébastien Fueyo's PhD.

On academic examples of simple circuits, we have given full justification (with some possible obstructions) to the prediction of stability through transfer function identification. The theoretical interest is that the spectrum of the operator that gives stability is not as elementary as predicted in the literature, but stability can be predicted nonetheless. Publication in progress on this point, a preliminary version was presented in [17].

It was also the oppoprtunity to re-visit stibility of time-delay time-varying linear system. A new sufficient condition can be found in [22], and a more general result is the purpose of a publication to come. These result are important to the domain of linear time-delay systems because the time-varying case has sedom been touched.