Article Contents
Article Contents

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

• * 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

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

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

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.

 Citation:

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