Control and stabilization for the dispersion generalized Benjamin equation on the circle

TL;DR

This paper investigates the controllability and stabilization of the dispersion generalized Benjamin equation on the circle, demonstrating global controllability and exponential stabilization using advanced functional analysis techniques.

Contribution

It establishes global controllability and stabilization results for the equation on the periodic domain, introducing dissipation-normalized Bourgain spaces for the analysis.

Findings

System is globally exactly controllable in Sobolev spaces.

The equation is globally well-posed with a damping feedback law.

Achieved local exponential stabilization in L^2 space.

Abstract

This paper is concerned with controllability and stabilization properties of the dispersion generalized Benjamin equation on the periodic domain $\mathbb{T}.$ First, by assuming the control input acts on all the domain, the system is proved to be globally exactly controllable in the Sobolev space $H^{s}_{p}(\mathbb{T}),$ with $s\geq 0.$ Second, by providing a locally-damped term added to the equation as a feedback law, it is shown that the resulting equation is globally well-posed and locally exponentially stabilizable in the space $L^{2}_{p}(\mathbb{T}).$ The main ingredient to prove the global well-posedness is the introduction of the dissipation-normalized Bourgain spaces which allows one to gain smoothing properties simultaneously from the dissipation and dispersion present in the equation. Finally, the local exponential stabilizability result is accomplished taking into account the decay of the associated semigroup combined with the fixed point argument.