Global control aspects for long waves in nonlinear dispersive media

TL;DR

This paper establishes the first global exact controllability results for long waves in nonlinear dispersive media with coupled quadratic nonlinearities, using spectral analysis and Bourgain spaces in a periodic setting.

Contribution

It introduces a novel approach combining spectral analysis and Bourgain space properties to achieve global control of coupled dispersive systems, a first in the literature.

Findings

Proves local controllability in $H^s(\mathbb{T})$, $s\geq0$.

Shows global exponential stability.

Achieves global exact controllability in large time.

Abstract

A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain $\mathbb{T}$ are studied. We used two distributed controls, supported in $\omega\subset\mathbb{T}$ and assumed to be generated by a linear feedback law conserving the "mass" (or "volume"), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in $H^s(\mathbb{T})$, for $s\geq0$. After that, by certain properties of Bourgain spaces we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg-de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.