Abstract
The problem of the exact bounded control of oscillations of the two-dimensional membrane is considered. Control force is applied to the boundary of the membrane, which is located in a domain on a plane. The goal of the control is to drive the system to rest in finite time.
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References
Chernousko, F.L.: Bounded control in distributed-parameter systems. J. Appl. Math. Mech. 56(5), 707–723 (1992)
Russell, D.L.: Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20(4), 639–739 (1978)
Lions, J.L.: Exact controllability. Stabilization and perturbations for distributed systems. SIAM Rev. 30(1), 1–68 (1988)
Lions, J.L.: Contrôlabilité Exacte, Perturbations et Stabilization de Systèmes Distribués. Tome 1: Contrôlabilité Exacte, Masson, Paris (1988)
Lasiecka, I., Triggiani, R.: Exact controllability of the wave equation with Neumann boundary control. Appl. Math. Optim. 1, 243–290 (1989)
Butkovskiy, A.G.: Distributed control systems. Translated from the Russian by Scripta Technica, Inc. Translation Editor: George M. Kranc. Modern Analytic and Computational Methods in Science and Mathematics, No. 11 American Elsevier Publishing Co., Inc., New York (1969)
Ivanov, S.: Control norms for large control times. ESAIM Control Optim. Calc. Var. 4, 405–418 (1999)
Romanov, I.V., Shamaev, A.S.: On a boundary controllability problem for a system governed by the two-dimensional wave equation. J. Comput. Syst. Sci. Int. 58(1), 105–112 (2019)
Quinn, J.P., Russell, D.L.: Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. R. Soc. Edinb. Sect. A 77, 97–127 (1977)
Lagnese, J.: Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50, 163–182 (1983)
Mikhlin, S.G.: Linear Partial Differential Equations. Higher School, Moscow (1977). in Russian
Agranovich, M.S.: Sobolev Spaces, Their Generalizations and Elliptic Problems in Domains with Smooth and Lipschitz Boundary. MCCME, Moscow (2013). in Russian
Lions, J.L., Madgenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, New-York (1972)
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This paper was partially supported by a grant of the Ministry of Science and Higher Education of the Russian Federation, Project No. 075-15-2019-1621.
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Communicated by Felix L. Chernousko.
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Romanov, I., Shamaev, A. Exact Bounded Boundary Controllability to Rest for the Two-Dimensional Wave Equation. J Optim Theory Appl 188, 925–938 (2021). https://doi.org/10.1007/s10957-021-01817-y
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DOI: https://doi.org/10.1007/s10957-021-01817-y