Abstract
We consider a boundary stabilization problem for the plate equation in a square. The feedback law gives the bending moment on a part of the boundary as function of the velocity field of the plate. The main result of the paper asserts that the obtained closed loop system is exponentially stable if and only if the controlled part of the boundary contains a vertical and a horizontal part of non-zero length (the geometric optics condition introduced by Bardos, Lebeau and Rauch in [2] for the wave equation is thus not necessary in this case). The proof of the main result uses the methodology introduced in Ammari and Tucsnak [1], where the exponential stability for the closed loop problem is reduced to an observability estimate for the corresponding uncontrolled system combined to a boundedness property of the transfer function of the associated open loop system. The second essential ingredient of the proof is an observability inequality recently proved by Ramdani, Takahashi, Tenenbaum and Tucsnak [7]
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References
K. Ammari and M. Tucsnak, Stabilization of second order evolution equations by a class of unbounded feedbacks, ESAIM COCV, 6 (2001), 361–386.
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control. Optim., 30 (1992), 1024–1065.
W. Krabs, G. Leugering and T. Seidman, On boundary controllability of a vibrating plate, Appl. Math. Optim., 13 (1985), 205–229.
I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Euler-Bernoulli equations with boundary control only in Δw|Σ, Boll. Un. Mat. Ital. B, 7 (1991), 665–702.
G. Lebeau, Contrôle de l’équation de Schrödinger, Journal de Mathématiques Pures et Appliquées, 71 (1992), 267–291.
G. Leugering, Boundary control of a vibrating plate, in Optimal control of partial differential equations (Oberwolfach, 1982), vol. 68 of Internat. Schriftenreihe Numer. Math., Birkhäuser, Basel, 1984, 167–172.
K. Ramdani, T. Takahashi, G. Tenenbaum and M. Tucsnak, A spectral approach for the exact observability of infinite dimensional systems with skew-adjoint generator, J. Funct. Anal., 226 (2005), 193–229.
G. Weiss and M. Tucsnak, How to get a conservative well-posed linear system out of thin air. I. Well-posedness and energy balance, ESAIM Control Optim. Calc. Var., 9 (2003), pp. 247–274.
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© 2007 Birkhäuser Verlag Basel/Switzerland
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Ammari, K., Tucsnak, M., Tenenbaum, G. (2007). A Sharp Geometric Condition for the Boundary Exponential Stabilizability of a Square Plate by Moment Feedbacks only. In: Kunisch, K., Sprekels, J., Leugering, G., Tröltzsch, F. (eds) Control of Coupled Partial Differential Equations. International Series of Numerical Mathematics, vol 155. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7721-2_1
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DOI: https://doi.org/10.1007/978-3-7643-7721-2_1
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