The Wellposedness and controllability of the generalized symmetric regularized long wave system

The symmetric regularized long wave system (SRLW) is a model for weakly nonlinear ion acoustic and space-charge waves, introduced by C. Seyler D. Fenstermacher. In this paper, we investigate the Wellposedness controllability properties of generalized (g-SRLW) in different structures, such as periodic bounded domains. Firstly, obtain exact results both linear g-SRLW posed on one-dimensional torus, considering effect distributed moving control. Secondly, consider interval with certain Dirichlet-Neumann conditions. We demonstrate that not spectrally controllable, meaning no finite combination eigenfunctions associated state equations, other than zero, can be steered to zero. Despite lack spectral controllability, it shown approximately controllable.