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Exact controllability and stability of the Sixth Order Boussinesq equation (1811.05943v1)

Published 14 Nov 2018 in math.AP

Abstract: The article studies the exact controllability and the stability of the sixth order Boussinesq equation [ u_{tt}-u_{xx}+\beta u_{xxxx}-u_{xxxxxx}+(u2)_{xx}=f, \quad \beta=\pm1, ] on the interval $S:=[0,2\pi]$ with periodic boundary conditions. It is shown that the system is locally exactly controllable in the classic Sobolev space, $H{s+3}(S)\times Hs(S)$ for $s\geq 0$, for "small" initial and terminal states. It is also shown that if $f$ is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in $H{s+3}(S)\times Hs(S)$ for $s\geq 0$ with "small" initial data assumption.

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