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Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary. (English) Zbl 0694.35102
Let \(\Omega \subseteq {\mathbb{R}}^ n\) be an open bounded domain with smooth boundary \(\Gamma =\Gamma_ 0\cup \Gamma_ 1\), where \(\Gamma_ 0\) and \(\Gamma_ 1\) are disjoint components of the boundary relatively open in \(\Gamma\). The author considers wave equation with nonlinear dissipative boundary conditions \[ y_{tt}=\Delta y,\quad x\in \Omega,\quad t>0;\quad y(x,0)=y_ 0(x),\quad y_ t(x,0)=y_ 1(x),\quad x\in \Omega, \]
\[ y(x,t)=0,\quad x\in \Gamma_ 0,\quad t>0;\quad y(x,t)+\gamma [Fy_ t(x,t)]\ni 0,\quad x\in \Gamma_ 1,\quad t>0. \] Here \(\gamma\) (u) is a monotone increasing, possibly multivalued function defined on \({\mathbb{R}}^ 1\) such that \(0\in \gamma (0)\). In the case \(\gamma (0)=0\), it is shown that a linear dissipative feedback operation F: \(L_ 2(\Omega)\to L_ 2(\Gamma)\) will “steer” the state \((y,y_ t)\) to zero as \(t\to \infty\). Energy estimates on all limit solutions are obtained for the case \(\gamma\) (0)\(\neq 0\). Similar results on plate like equations \(y_{tt}=-\Delta^ 2y\) are also given.
Reviewer: P.K.Wong

MSC:
35L05 Wave equation
35L35 Initial-boundary value problems for higher-order hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
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[1] Brezis, H, Maximal monotone operators, ()
[2] {\scJ. Bartolomeo, R. Triggiani}, Uniform energy decay rates of Euler-Bernoulli equations with feedback in the Dirichlet and Neumann boundary conditions, to appear. · Zbl 0753.35037
[3] Chen, G, A note on boundary stabilization of the wave equation, SIAM J. control optim., 19, 106-113, (1981) · Zbl 0461.93036
[4] Chen, G; Delfour, M.C; Krall, A.M; Payre, G, Modelling, stabilization and control of serially connected beams, SIAM J. control optim., 25, 526-546, (1987) · Zbl 0621.93053
[5] {\scG. Chen and H. K. Wang}, Asymptotic behavior of solutions on the one-dimensional wave equations with a nonlinear elastic dissipative boundary condition, manuscript.
[6] Dafermos, C.M; Slemrod, M, Asymptotic behavior of nonlinear contraction semigroups, J. funct. anal., 13, 97-106, (1973) · Zbl 0267.34062
[7] Grisvard, P, Une caractérisation de quelques espaces d’interpolation, Arch. rational mech. anal., 25, 40-63, (1967) · Zbl 0187.05901
[8] Kim, J.U; Renardy, Y, Boundary control of the Timoshenko beam, SIAM J. control optim., 25, 1417-1430, (1987) · Zbl 0632.93057
[9] Komornik, V; Zuazua, E, Stabilisation frontière de l’équation des ondes: une méthode directe, C. R. acad. sci. Paris. Sér. I math., 305, 605-608, (1987) · Zbl 0661.93054
[10] Lagnese, J, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. differential equations, 50, 163-182, (1983) · Zbl 0536.35043
[11] {\scJ. L. Lions}, Un résultat de régularité (paper dedicated to S. Mizohata).
[12] Lions, J.L, Exact controllability, stabilization and perturbations, () · Zbl 0644.49028
[13] Lasiecka, I, Strong stabilization of a nonlinear wave equation with dissipation on the boundary and related problems. paper dedicated to A. V. balskrishman on the occasion of his 60th birthday, ()
[14] Lagnese, J, Uniform boundary stabilization of homogeneous isotropic plates, (), 204-216
[15] Lagnese, J, Uniform stabilization of the kirchoff system by nonlinear feedback, (1988), manuscript
[16] Lasiecka, I; Lions, J.L; Triggiani, R, Nonhomogeneous boundary value problems for second-order hyperbolic generators, J. math. pures appl., 65, 149-192, (1986), (9) · Zbl 0631.35051
[17] Lasiecka, I; Triggiani, R, A cosine operator approach to modelling L2(0t;L2(γ1)) boundary input hyperbolic equations, Appl. math. optim., 7, 35-83, (1981) · Zbl 0473.35022
[18] Lasiecka, I; Triggiani, R, Regularity of hyperbolic equations under L2(0T; L2(γ)) boundary terms, Appl. math. optim., 10, 275-286, (1983) · Zbl 0526.35049
[19] Lasiecka, I; Triggiani, R, Regularity theory for a class of nonhomogeneous Euler-Bernoulli equations: A cosine operator approach, Bulletin unione Mathematica italiana, 7-2B, (1988)
[20] Lasiecka, I; Triggiani, R, Uniform exponential energy decay in a bounded region with L2(0, ∞; L2(γ))—feedback control in the Dirichlet boundary conditions, J. differential equations, 66, 340-390, (1987) · Zbl 0629.93047
[21] Lasiecka, I; Triggiani, R; Lasiecka, I; Triggiani, R, Exact controllability of the Euler-Bernoulli equation with L2(∑)-control only in the Dirichlet boundary conditions, Atti delle Academic nezionale dei lincei. rendiconti classe di sienze fisiche, metematiche e naturali, SIAM J. control, Vol. LXXXI, (March 1989), to appear
[22] {\scI. Lasiecka and R. Triggiani}, Exact controllability of the Euler-Bernoulli equations with boundary controls for displacement and moment, J. Math. Anal. Appl., to appear. · Zbl 0749.93048
[23] Lasiecka, I; Triggiani, R, Uniform decay rates for Euler-Bernoulli’s equations with feedback only in the bending moments, (), 1260-1263
[24] {\scR. Triggiani}, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl., in press
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