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Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus

Abstract : We consider the Benjamin-Bona-Mahony (BBM) equation on the one dimensional torus T = R/(2πZ). We prove a Unique Continuation Property (UCP) for small data in H^1(T) with nonnegative zero means. Next we extend the UCP to certain BBM-like equations, including the equal width wave equation and the KdV-BBM equation. Applications to the stabilization of the above equations are given. In particular, we show that when an internal control acting on a moving interval is applied in BBM equation, then a semiglobal exponential stabilization can be derived in H^s(T) for any s ≥ 1. Furthermore, we prove that the BBM equation with a moving control is also locally exactly controllable in H^s(T) for any s ≥ 0 and globally exactly controllable in H s (T) for any s ≥ 1.
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https://hal.archives-ouvertes.fr/hal-00669334
Contributor : Lionel Rosier Connect in order to contact the contributor
Submitted on : Monday, February 13, 2012 - 8:02:32 AM
Last modification on : Tuesday, October 19, 2021 - 11:56:24 PM
Long-term archiving on: : Monday, May 14, 2012 - 2:21:31 AM

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Lionel Rosier, Bing-Yu Zhang. Unique continuation property and control for the Benjamin-Bona-Mahony equation on the torus. Journal of Differential Equations, Elsevier, 2013, 254 (1), pp.141-178. ⟨10.1016/j.jde.2012.08.014⟩. ⟨hal-00669334⟩

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