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Control and stabilization of the Benjamin-Ono equation on a periodic domain

Abstract : It was proved by Linares and Ortega that the linearized Benjamin- Ono equation posed on a periodic domain T with a distributed control sup- ported on an arbitrary subdomain is exactly controllable and exponentially stabilizable. The aim of this paper is to extend those results to the full Benjamin-Ono equation. A feedback law in the form of a localized damp- ing is incorporated into the equation. A smoothing effect established with the aid of a propagation of regularity property is used to prove the semi-global stabilization in L2(T) of weak solutions obtained by the method of vanish- ing viscosity. The local well-posedness and the local exponential stability in Hs(T) are also established for s > 1/2 by using the contraction mapping the- orem. Finally, the local exact controllability is derived in Hs(T) for s > 1/2 by combining the above feedback law with some open loop control.
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https://hal-mines-paristech.archives-ouvertes.fr/hal-01261645
Contributor : François Chaplais <>
Submitted on : Monday, January 25, 2016 - 4:26:56 PM
Last modification on : Thursday, September 24, 2020 - 5:04:18 PM

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  • HAL Id : hal-01261645, version 1

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Felipe Linares, Lionel Rosier. Control and stabilization of the Benjamin-Ono equation on a periodic domain. Transactions American Mathematical Society, American Mathematical Society, 2015, 367 (7), pp.4595-4626. ⟨hal-01261645⟩

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