D¹-Input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances

Abstract

D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> -Input-to-state stability (D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ISS) of a diffusive equation with Dirichlet boundary conditions is shown, in the L <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -norm, with respect to boundary disturbances. In particular, the spatially distributed diffusion coefficients are allowed to be time-varying within a given set, without imposing any constraints on their rate of variation. Based on a strict Lyapunov function for the system with homogeneous boundary conditions, D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> ISS inequalities are derived for the disturbed equation. A heuristic method used to numerically compute weighting functions is discussed. Numerical simulations are presented and discussed to illustrate the implementation of the theoretical results.