On the exact controllability and the stabilization for the Benney-Luke equation

Abstract

In this work we consider the exact controllability and the stabilization for the generalized Benney-Luke equationon a periodic domain S (the unit circle on the plane) with internal control f supported on an arbitrary sub-domain of S. We establish that the model is exactly controllable in a Sobolev type space when the whole S is the support of f , without any assumption on the size of the initial and final states, and that the model is local exactly controllable when the support of f is a proper subdomain of S, assuming that initial and terminal states are small. Moreover, in the case that the initial data is small and f is a special internal linear feedback, the solution of the model must have uniform exponential decay to a constant state.

Global control aspects for long waves in nonlinear dispersive media

“…Also, more recently, Yang and Zhang [40] proved new estimates related to this kind of system (see Section 3). • It is important to mention that the propagation results have been successfully applied in control theory in several systems represented by single equations, such as wave equation [14], after that for the Schrödinger equation [23], for the Benjamin-Ono equation [27] , KdV equation [25], the Kawahara equation [41], biharmonic Schrödinger equation [8], for the Benney-Luke equation [17] and, finally, for the Benjamin equation [32].…”