Some control results for the KdV-type equation

Abstract

This work deals with the controllability and stabilization of fifth-order dispersive equations in bounded and unbounded domains. In the first result, we prove a new type of controllability for a fifth-order dispersive equation that models water waves, which we call overdetermination control problem. Precisely, we can find a control acting on the boundary that provides us that the solutions of the considered problem satisfy an overdetermined integral condition. Additionally, when the control acts internally in the system rather than at the boundary, we are also able to prove a controllability result. In the second result, we extend the overdetermined control property to unbounded domains. This condition is satisfied when the domain of the Kawahara equation is the real line, left half-line, and right half-line. Furthermore, we show a type of exact control associated with the ''mass'' of the Kawahara equation over the right half-line. The third, and last, work deals with the exponential decay of the energy associated with the solutions of the Kawahara equation. Precisely, we prove that the fifth-order dispersive system, with a damping mechanism and delay terms on the boundary, is exponentially stable. We do this using two different procedures: The first result is obtained using the Lyapunov method, which ensures exponential decay. The second result is obtained through the compactness-uniqueness argument, which reduces our study to proving an observability inequality.