# Control and Stabilization of the Periodic Fifth Order Korteweg-de Vries Equation

## TL;DR

This paper develops control and stabilization methods for periodic solutions of the fifth order Korteweg-de Vries equation, demonstrating local exact control and exponential stability in a specific Sobolev space.

## Contribution

It introduces a dissipative control approach combined with regularity propagation to achieve stabilization of the fifth order KdV equation.

## Key findings

- Achieves local exact control of periodic solutions.
- Establishes local exponential stability in H^s space.
- Demonstrates smoothing effects via dissipative control.

## Abstract

We establish local exact control and local exponential stability of periodic solutions of fifth order Korteweg-de Vries type equations in $H^s(\mathbb{T})$, $s>2$. A dissipative term is incorporated into the control which, along with a propagation of regularity property, yields a smoothing effect permitting the application of the contraction principle.