Velocity Stabilization of a Wave Equation with a Nonlinear Dynamic Boundary Condition

TL;DR

This paper studies the stabilization of a one-dimensional wave equation with nonlinear boundary conditions and velocity feedback control, demonstrating exponential energy decay and stability under various nonlinear boundary effects.

Contribution

It introduces a class of nonlinear velocity feedback controls for wave equations with nonlinear boundary conditions, analyzing their effectiveness in energy stabilization.

Findings

Exponential decay of energy under certain nonlinear boundary conditions.

Stability and attractivity of invariant sets are established.

Results apply to nonlinear boundary anti-damping scenarios.

Abstract

This paper deals with a one-dimensional wave equation with a nonlinear dynamic boundary condition and a Neumann-type boundary control acting on the other extremity. We consider a class of nonlinear stabilizing feedbacks that only depend on the velocity at the controlled extremity. The uncontrolled boundary is subject to a nonlinear first-order term, which may represent nonlinear boundary anti-damping. Initial data is taken in the optimal energy space associated with the problem. Exponential decay of the mechanical energy is investigated in different cases. Stability and attractivity of suitable invariant sets are established.