On the control issues for higher-order nonlinear dispersive equations on the circle

TL;DR

This paper establishes local and global control and stabilization results for higher-order KdV-type equations on the circle, utilizing spectral analysis, Bourgain spaces, and Strichartz estimates to extend previous findings.

Contribution

It introduces new propagation of singularities and extends control results to higher-order KdV-type equations, broadening the scope of prior work.

Findings

Local controllability and exponential stability for higher-order KdV equations.

Global controllability and stabilization in Sobolev spaces for all s ≥ 0.

Extension of Strichartz estimates and propagation results to general higher-order operators.

Abstract

The local and global control results for a general higher-order KdV-type operator posed on the unit circle are presented. Using spectral analysis, we are able to prove local results, that is, the equation is locally controllable and exponentially stable. To extend the local results to the global one we captured the smoothing properties of the Bourgain spaces, the so-called propagation of singularities, which are proved with a new perspective. These propagation, together with the Strichartz estimates, are the key to extending the local control properties to the global one, precisely, higher-order KdV-type equations are globally controllable and exponentially stabilizable in the Sobolev space $H^{s}(\mathbb{T})$ for any $s \geq 0$. Our results recover previous results in the literature for the KdV and Kawahara equations and extend, for a general higher-order operator of KdV-type, the Strichartz estimates as well as the propagation results, which are the main novelties of this work.