Stability analysis of degenerately-damped oscillations

  • Thomas Anderson
  • George Avalos
  • Elizabeth Galvin
  • Ian Kessler
  • Michelle Kleckner
  • Daniel Toundykov
  • William Tritch

Abstract

This is a study of the well-posedness and asymptotic stability of a “degenerately damped†PDE that models a vibrating elastic string. The coefficient of the damping may vanish at small amplitudes thus weakening the effect of the dissipation. It is shown that the resulting dynamical system has strictly monotonically decreasing energy and uniformly decaying lower-order norms of the solutions, however, it is not uniformly stable on the associated finite-energy space. These theoretical findings were motivated by numerical simulations of this model using an FEM scheme and successive approximations. A description of the numerical approach and sample plots of energy decay are supplied. In addition, for certain initial data the solution can be determined in closed form up to a dissipative nonlinear ordinary differential equation. These special solutions can be used to assess the accuracy of the numerical examples.

Published
2016-02-28
Section
Articles