# Rapid exponential stabilization of a 1-D transmission wave equation with in-domain anti-damping

## TL;DR

This paper develops a feedback control method using backstepping to achieve rapid exponential stabilization of a 1-D wave equation with an internal anti-damping term, ensuring desired decay rates.

## Contribution

It introduces a novel backstepping-based feedback law for pointwise stabilization of wave equations with in-domain anti-damping, guaranteeing exponential stability.

## Key findings

- Proposed feedback law achieves exponential decay
- Stability proof for closed-loop system
- Design method applicable to spatially varying anti-damping

## Abstract

We consider the problem of pointwise stabilization of a one-dimensional wave equation with an internal spatially varying anti-damping term. We design a feedback law based on the backstepping method and prove exponential stability of the closed-loop system with a desired decay rate.