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

Linares, Felipe; Rosier, Lionel

doi:10.1090/S0002-9947-2015-06086-3

Control and stabilization of the Benjamin-Ono equation on a periodic domain
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by Felipe Linares and Lionel Rosier PDF

Trans. Amer. Math. Soc. 367 (2015), 4595-4626 Request permission

Abstract:

It was proved by Linares and Ortega that the linearized Benjamin-Ono equation posed on a periodic domain $\mathbb {T}$ with a distributed control supported 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 damping 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 $L^2(\mathbb {T})$ of weak solutions obtained by the method of vanishing viscosity. The local well-posedness and the local exponential stability in $H^s(\mathbb {T})$ are also established for $s>1/2$ by using the contraction mapping theorem. Finally, the local exact controllability is derived in $H^s(\mathbb {T})$ for $s>1/2$ by combining the above feedback law with some open loop control.

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