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

Quintero José R.; Montes Alex M.

doi:10.3934/mcrf.2019039

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2020, Volume 10, Issue 2: 275-304. Doi: 10.3934/mcrf.2019039

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

Quintero José R.^1, , and
Montes Alex M.^2,

1.
Mathematics Department, Universidad del Valle, Cali, Valle del Cauca, Colombia
2.
Mathematics Department, Universidad del Cauca, Popayán, Cauca, Colombia

^* Corresponding author: José R. Quintero
^* Corresponding author: José R. Quintero

Received: November 30, 2018

Revised: April 30, 2019

Early access: August 2019

Published: April 2020

JRQ is supported by the Mathematics Department at Universidad del Valle and AMM is supported by the Mathematics Department at Universidad del Cauca

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Abstract

In this work we consider the exact controllability and the stabilization for the generalized Benney-Luke equation

$\begin{equation} u_{tt}-u_{xx}+a u_{xxxx}-bu_{xxtt}+ p u_t u_{x}^{p-1}u_{xx} + 2 u_x^{p}u_{xt} = f, ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(1)\end{equation}$

on 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.

Keywords:

Mathematics Subject Classification: Primary: 74J30, 35Q35, 93B05, 93D15; Secondary: 35Q53.

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