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

Authors: Felipe Linares and Lionel Rosier, Trans. Amer. Math. Soc. , Vol. 367 No 7, pp. 4595-4626, 2015
It was proved by Linares and Ortega that the linearized Benjamin- Ono equation posed on a periodic domain 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 L2(T) of weak solutions obtained by the method of vanishing 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 theorem. 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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author = {Felipe Linares and Lionel Rosier},
title = {Control and stabilization of the Benjamin-Ono equation
on a periodic domain},
journal = {Trans. Amer. Math. Soc.},
volume = {367},
number = {7},
pages = {4595-4626},
year = {2015},