Controllability for one-dimensional nonlinear wave equations with degenerate damping

Abstract

Internal exact controllability for the nonlinear wave equation in one space dimension ytt−yxx+g(y)yt=his studied, where g(⋅) is a nonnegative function. For the case, lim sup|s|→∞g(s)ln|s|<γ, we obtain the global exact controllability for the equation with Dirichlet boundary condition. The proof is based on the combination of fixed-point arguments and explicit observability estimates for the linearized wave equation with a potential that depends on both x and t. For the case, g(s)=|s|, we only get a local exact controllability by means of Banach fixed-point theorem.