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Control and Stabilization of the Benjamin-Ono Equation in L2(T)

Authors: Camille Laurent, Felipe Linares and Lionel Rosier, Arch. Mech. Anal. , Vol. 218 No 3, pp. 1531-1575, May 20 2015 DOI : 10.1007/s00205-015-0887-5
We study the control and stabilization of the Benjamin-Ono equation in L2(T), the lowest regularity where the initial value problem is well-posed. This problem was already initiated in Linares and Rosier (Trans Am Math Soc 367:4595–4626, 2015) where a stronger stabilization term was used (that makes the equation of parabolic type in the control zone). Here we employ a more natural stabilization term related to the L2–norm. Moreover, by proving a theorem of controllability in L2, we manage to prove the global controllability in large time. Our analysis relies strongly on the bilinear estimates proved in Molinet and Pilod (Anal PDE 5:365–395, 2012) and some new extension of these estimates established here.
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author = {Felipe Linares and Lionel Rosier Camille Laurent},
title = {Control and Stabilization of the Benjamin-Ono Equation in L2(T)},
journal = {Arch. Mech. Anal.},
volume = {218},
number = {3},
pages = {1531-1575},
year = {2015},