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Exact boundary controllability for the Boussinesq equation with variable coefficients

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  • 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 $ρ\equiv\sigma \equiv q\equiv1$.

    Mathematics Subject Classification: Primary: 93B05, 93B07, 93B12; Secondary: 93B60.


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