# Remark on the pointwise stabilization of an elastic string equation

## TL;DR

This paper studies the stabilization of a one-dimensional wave equation with interior damping, proving logarithmic energy decay under specific actuator placement assumptions using resolvent and Carleman estimates.

## Contribution

It introduces a new algebraic condition for actuator placement that guarantees logarithmic decay, improving upon previous diophantine approximation-based conditions.

## Key findings

- Logarithmic decay of energy under new actuator placement condition
- Use of resolvent estimate and Carleman estimate techniques
- Comparison showing less restrictive assumptions than prior work

## Abstract

We consider an initial and boundary value problem the one dimensional wave equation with damping concentrated at an interior point. We prove a result of a logarithmic decay of the energy of a system with homogeneous Dirichlet boundary conditions. The method used is based on the resolvent estimate approach which derives from the Carleman estimate technique. Under an algebraic assumption describing the right location of the actuator, we prove a logarithmic decay of the energy of solution. We show that this assumption is lower than the one given by [Tuc] and [AHT] which depends on the diophantine approximations properties of the actuator's location.