# Local exponential stabilization for a class of Korteweg-de Vries equations by means of time-varying feedback laws

## TL;DR

This paper develops time-varying feedback laws to achieve exponential stabilization of a nonlinear Korteweg-de Vries equation on a bounded interval, even when the linearized system is not controllable.

## Contribution

It introduces a novel class of time-varying feedback laws that ensure exponential decay of solutions for small initial data in a challenging control setting.

## Key findings

- Solutions decay exponentially to zero with the proposed feedback laws.
- Well-posedness of the closed-loop system is established for general time-varying feedback.
- The control design works despite the linearized system being uncontrollable.

## Abstract

We study the exponential stabilization problem for a nonlinear Korteweg-de Vries equa- tion on bounded interval in cases where the linearized control system is not controllable. The system has Dirichlet boundary conditions at the end-points of the interval, a Neumann nonhomogeneous boundary condition at the right end-point which is the control. We build a class of time-varying feedback laws for which the solutions of the closed-loop systems with small initial data decay exponentially to 0. We present also results on the well-posedness of the closed-loop systems for general time-varying feedback laws.