Abstract
We study the control and stabilization of the Benjamin-Ono equation in \({L^2(\mathbb{T})}\), 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 L 2–norm. Moreover, by proving a theorem of controllability in L 2, 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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Laurent, C., Linares, F. & Rosier, L. Control and Stabilization of the Benjamin-Ono Equation in \({L^2({\mathbb{T})}}\) . Arch Rational Mech Anal 218, 1531–1575 (2015). https://doi.org/10.1007/s00205-015-0887-5
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DOI: https://doi.org/10.1007/s00205-015-0887-5