Characterization of " low-pass filtering " for some nonlinear ODE's

Samer Riachy (1, 2)

1 NON-A - Non-Asymptotic estimation for online systems
2 QUARTZ - Laboratoire QUARTZ

The characterization of low-pass linear filters is nowadays a simple question relying on eigenvalue criteria, Bode diagram, Nyquist or Nichols plots. For nonlinear systems, the problem is more tricky. The present work tempts to develop a methodology for the characterization of " low-pass filtering " for nonlinear systems. Inspired by the DiPerna-Lions theory and relying on the theory of characteristics, the proposed method associates to the nonlinear ODE a linear transport PDE. A characterization of " low-pass filtering " is then deduced from developments on the PDE. Smooth ODE's of the form \dot{x} = F (x) are considered where the vector field F (x) is perturbed by an additive, rapidly oscillating, noise m which may have a big magnitude F (x + m). An intuitive observation is proved in this first contribution: if F has bounded derivatives, then the sensitivity of x to m decreases as the bound gets smaller or as m fluctuates faster.

Mots clés: Big magnitude noise; Characterization of " low-pass filtering " for nonlinear systems; transport equation

Domaines: Optimisation et contrôle [math.OC]

Samer Riachy. Characterization of " low-pass filtering " for some nonlinear ODE's. IFAC WC 2017, Jul 2017, Toulouse, France. ⟨hal-01586248⟩