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

  • * Corresponding author: José R. Quintero

    * Corresponding author: José R. Quintero 

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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  • 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.

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


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