TY - JOUR

T1 - Stabilizability properties of a linearized water waves system

AU - Su, Pei

AU - Tucsnak, Marius

AU - Weiss, George

N1 - Publisher Copyright:
© 2020 Elsevier B.V.

PY - 2020/5

Y1 - 2020/5

N2 - We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ̇. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u=Fz renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1+t)−1/6. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).

AB - We consider the strong stabilization of small amplitude gravity water waves in a two dimensional rectangular domain. The control acts on one lateral boundary, by imposing the horizontal acceleration of the water along that boundary, as a multiple of a scalar input function u, times a given function h of the height along the active boundary. The state z of the system consists of two functions: the water level ζ along the top boundary, and its time derivative ζ̇. We prove that for suitable functions h, there exists a bounded feedback functional F such that the feedback u=Fz renders the closed-loop system strongly stable. Moreover, for initial states in the domain of the semigroup generator, the norm of the solution decays like (1+t)−1/6. Our approach uses a detailed analysis of the partial Dirichlet to Neumann and Neumann to Neumann operators associated to certain edges of the rectangular domain, as well as recent abstract non-uniform stabilization results by Chill et al. (2019).

KW - Collocated actuators and sensors

KW - Dirichlet to Neumann map

KW - Linearized water waves equation

KW - Neumann to Neumann map

KW - Operator semigroup

KW - Strong stabilization

UR - http://www.scopus.com/inward/record.url?scp=85082468025&partnerID=8YFLogxK

U2 - 10.1016/j.sysconle.2020.104672

DO - 10.1016/j.sysconle.2020.104672

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AN - SCOPUS:85082468025

SN - 0167-6911

VL - 139

JO - Systems and Control Letters

JF - Systems and Control Letters

M1 - 104672

ER -