UCA Complex Days (Nice, January 11 2018)

Keywords. stability analysis, delay systems, nonlinear circuits, perturbation of linear operators, functional differential equations of neutral type , harmonic balance

Nonlinear hyperfrequency amplifiers contain nonlinear active components and lines, that can be seen as linear infinite dimensional systems inducing delays that cannot be neglected at high frequencies. Computer assisted design tools are extensively used. They provide fre- quency responses but fail to provide a reliable estimation of their stability, and this stability is crucial because an unstable response will not be observed in practice and the engineer needs to have this information between building the actual device. We shall present the models of such devices, and the current methods to compute the response to a given periodic signal to be amplified (this is a periodic solution of a periodically forced infinite dimensional dynamical system) as well as the frequency response of an input-output system associated to the linearization around this periodic solution. The goal of the talk is to present the ideas and preliminary results that on the one hand allow to deduce stability of this time-varying linear system from that frequency response and on the other hand provide a relationship between this stability and the internal stability of the actual nonlinear circuit. The first point resorts from harmonic analysis and perturba- tion of linear operators. The second one from nonlinear infinite dimensional dynamics and ad hoc linearization.

Jean-Baptiste Caillau

Keywords. slow-fast dynamical systems, minimum time control, averaging of hamiltonian dynamics, Finsler geometry, space mechanics

We consider the minimum time control of dynamical systems with slow and fast state vari- ables. Such models are ubiquitous for complex mechanical systems that exhibit behaviours driven by different time scales. With applica- tions to perturbations of integrable systems in mind, we focus on the case of problems with one or more fast angles, together with a small drift on the slow part modelling a so-called secular evolution of the slow variables. According to Pontrjagin maximum principle, time minimizing  trajectories are projections on the state space of Hamiltonian curves. In the case of a single fast angle, the slow and fast parts of these curves are identified thanks to an appropriate symplectic reduction. Then, an approximation of the Hamiltonian flow is obtained using an averaging procedure. It turns out that, pro- vided the drift on the slow part of the original system is small enough, this approximation is of metric nature: Time minimizing trajectories can be approximated by geodesics of a suitable Finsler metric. Moreover, because of the secular evolution of the slow variables, this metric is asymmetric. We report results on asymptotic controllability, existence and convergence for the original control system. As an application to space mechanics, the effect of the J2 term in the Earth potential on the control of a spacecraft is considered. The J2 perturbation accounts for the oblateness of the Earth, and we provide a qualitative analysis of its influence when minimizing time. In an ongoing work, we address the more involved question of systems having two or more fast angles. In such situa- tions, resonances come into play and complicate significantly the analysis.