The Well-posedness and Controllability of the Generalized Symmetric Regularized Long Wave System

TL;DR

This paper studies the mathematical properties of the generalized symmetric regularized long wave system, demonstrating well-posedness and controllability in various domain structures, with specific results on exact and approximate controllability.

Contribution

It establishes well-posedness and controllability results for the g-SRLW system in both periodic and bounded domains, including linear and nonlinear cases, and clarifies spectral and approximate controllability limitations.

Findings

Well-posedness and exact controllability on the torus for linear and nonlinear systems.

Non-spectral controllability in bounded intervals with Dirichlet-Neumann conditions.

Approximate controllability despite lack of spectral controllability.

Abstract

The symmetric regularized long wave system (SRLW) is a model for the weakly nonlinear ion acoustic and space-charge waves, which was introduced by C. Seyler and D. Fenstermacher. In this paper, we investigated the wellposedness and controllability properties of the generalized symmetric regularized long wave system (g-SRLW) in different structures (periodic and bounded domains). Firstly, the wellposedness and the exact controllability results for both linear and nonlinear g-SRLW system posed on the one-dimensional torus are obtained under the effect of a distributed moving control. Second, we consider the g-SRLW system in a bounded interval with some Dirichlet-Neumann conditions and we show that the system is not spectrally controllable (No finite linear combination of eigenfunctions associated with the state equations, other than zero, can be steered to zero). Although the system is not spectrally controllable, it can be shown that it is approximately controllable.