# 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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BibTeX:

@Article{2016-01-25,

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},

}

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.

Download PDF

BibTeX:

@Article{2016-01-25,

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},

}