This paper establishes the global existence and uniqueness of solutions to the initial-boundary value problem for a fifth-order KdV equation posed on a finite interval 0, d. Overcoming the challenge of global solvability imposed by non-conservative boundary conditions, we introduce a nonlinear boundary feedback mechanism inspired by control theory to enforce energy dissipation. The proof hinges on deriving rigorous a priori estimates that capture both the Kato smoothing effect and boundary trace regularity, complemented by a tailored nonlinear estimate to handle the feedback term. Consequently, local solutions are extended to global ones. Furthermore, comprehensive numerical experiments validate the proposed approach and yield strong empirical evidence of exponential energy decay, a property crucial for control applications.
Zhao et al. (Sat,) studied this question.