# American Institute of Mathematical Sciences

September  2018, 7(3): 403-415. doi: 10.3934/eect.2018020

## Exact boundary controllability for the Boussinesq equation with variable coefficients

 Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia, Mathematical Engineering Laboratory, (LR01ES13), Tunisia Polytechnic School, Tunisia

Received  October 2017 Revised  April 2018 Published  July 2018

In this paper we study the exact boundary controllability for the following Boussinesq equation with variable physical parameters:
 \left\{ \begin{align} & \rho (x){{y}_{tt}}=-{{(\sigma (x){{y}_{xx}})}_{xx}}+{{(q(x){{y}_{x}})}_{x}}-{{({{y}^{2}})}_{xx}},\ \ \ \ \ \ \ t>0,~x\in (0,l), \\ & y(t,0)={{y}_{xx}}(t,0)=y(t,l)=0,~~\sigma (l){{y}_{xx}}(t,l)=u(t)\ \ \ \ \ t>0, \\ \end{align} \right.
where
 $l>0$
, the coefficients
 $ρ(x)>0, \sigma (x)>0$
,
 $q(x)≥0$
in
 $\left[ {0,l} \right]$
and
 $u$
is the control acting at the end
 $x=l$
. We prove that the linearized problem is exactly controllable in any time
 $T>0$
. Our approach is essentially based on a detailed spectral analysis together with the moment method. Furthermore, we establish the local exact controllability for the nonlinear problem by fixed point argument. This problem has been studied by Crépeau [Diff. Integ. Equat., 2002] in the case of constant coefficients
 $ρ\equiv\sigma \equiv q\equiv1$
.
Citation: Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
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