Asymptotic Behaviour of Solutions of the One-Dimensional Wave Equation with a Nonlinear Boundary Stabilizer

Abstract

The modeling of nonlinear passive damping devices or boundary frictions of an otherwise linear vibrating system often results in nonlinear elastic dissipative boundary conditions. Such systems occur increasingly often in engineering applications, whose control and stability analysis appear much more complex than the classical linear distributed parameter systems.This paper uses the method of characteristics and nonlinear semigroup theory to study the effect of nonlinear boundary stabilization and analyze the asymptotic behavior of solutions of such systems. The authors are able to determine the $\omega $-limit set of the dynamical system and the asymptotic rates of various solutions to the $\omega $-limit set.