Lyapunov-based nonlinear boundary control design with predefined convergence for a class of 1D linear reaction-diffusion equations

Abstract

In this paper, we treat the problem of Lyapunov-based nonlinear boundary stabilization of a class of one-dimensional reaction-diffusion systems with any predefined convergence (asymptotic or non-asymptotic). As an application, we focus on the non-asymptotic notions (finite-time and fixed-time) for which we give some particular explicit control designs followed by some numerical simulations. The key idea of our approach is to use a “spatially weighted L2-norm” as a Lyapunov functional to design a nonlinear controller and to ensure stability with any desired convergence.