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Uniform stabilization of some damped second order evolution equations with vanishing short memory

  • Louis Tebou (a1)


We consider a damped abstract second order evolution equation with an additional vanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, we do not assume the operator defining the main damping to be bounded. First, using a constructive frequency domain method coupled with a decomposition of frequencies and the introduction of a new variable, we show that if the limit system is exponentially stable, then this evolutionary system is uniformly − with respect to the calibration parameter − exponentially stable. Afterwards, we prove uniform polynomial and logarithmic decay estimates of the underlying semigroup provided such decay estimates hold for the limit system. Finally, we discuss some applications of our results; in particular, the case of boundary damping mechanisms is accounted for, which was not possible in the earlier work mentioned above.



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[1] Alabau-Boussouira, F., Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006) 95112.
[2] Alabau, F. and Komornik, V., Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Control Optim. 37 (1999) 521542.
[3] Arendt, W., Batty, C. J. K., Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837852.
[4] Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Optim. 30 (1992) 10241065.
[5] Bátkai, A., Engel, K.-J., Prüss, J. and Schnaubelt, R., Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 14251440.
[6] Batty, C.J.K. and Duyckaerts, T., Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765780.
[7] Borichev, A. and Tomilov, Y., Optimal polynomial decay of functions and operator semigroups. Math. Annal. 347 (2010) 455478.
[8] H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983).
[9] Burq, N., Décroissance de l’énergie locale de l’équation des ondes pour le problème extérieur et absence de résonance au voisinage du réel. Acta Math. 180 (1998) 129.
[10] Chen, G., Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979) 6681.
[11] Chen, G., Fulling, S.A., Narcowich, F.J. and Sun, S., Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266301.
[12] Chen, G. and Russell, D.L., A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39 (1981/1982) 433-454.
[13] Conrad, F. and Rao, B., Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptotic Anal. 7 (1993) 159177.
[14] S. Ervedoza, E. Zuazua, Uniform exponential decay for viscous damped systems. Advances in phase space analysis of partial differential equations. Progr. Nonlinear Differential Equ. Appl. vol. 78. Birkhäuser Boston, Inc., Boston, MA (2009) 95–112.
[15] Ervedoza, S. and Zuazua, E., Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 2048.
[16] Fu, X., Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011) 667680.
[17] Fu, X., Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957975.
[18] Guzmán, R.B. and Tucsnak, M., Energy decay estimates for the damped plate equation with a local degenerated dissipation. Systems Control Lett. 48 (2003) 191197.
[19] Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245258.
[20] Huang, F.L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annal. Differ. Equ. 1 (1985) 4356.
[21] Komornik, V., Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29 (1991) 197208.
[22] V. Komornik, Exact controllability and stabilization. The multiplier method, RAM. Masson and John Wiley, Paris (1994).
[23] V. Komornik and V. Boundary stabilization of isotropic elasticity systems. Control of partial differential equations and applications (Laredo, 1994), vol. 174. Lect. Notes Pure and Appl. Math. Dekker, New York (1996) 135–146.
[24] Komornik, V., Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 15911613.
[25] Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 3354.
[26] Lagnese, J., Boundary stabilization of linear elastodynamic systems. SIAM J. Control Opt. 21 (1983) 968984.
[27] Lagnese, J., Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50 (1983) 163182.
[28] J. Lagnese, Boundary Stabilization of Thin Plates, vol. 10. SIAM Stud. Appl. Math. Philadelphia, PA (1989).
[29] Lasiecka, I. and Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6 (1993) 507533.
[30] Lasiecka, I. and Triggiani, R., Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25 (1992) 189224.
[31] Lebeau, G., Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht 19 (1996) 73109.
[32] Lebeau, G. and Robbiano, L., Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465491.
[33] J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 8 of RMA. Masson, Paris (1988).
[34] Liu, K., Locally distributed control and damping for the conservative systems. SIAM J. Control and Opt. 35 (1997) 15741590.
[35] Liu, Z. and Rao, B., Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56 (2005), 630644.
[36] Liu, K. and Rao, B., Exponential stability for the wave equations with local Kelvin–Voigt damping. Z. Angew. Math. Phys. 57 (2006) 419432.
[37] P. Martinez, Ph.D. Thesis, University of Strasbourg (1998).
[38] Martinez, P., Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim. 37 (1999) 673694.
[39] Nakao, M., Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 2542.
[40] Osses, A., A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim. 40 (2001) 777800.
[41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983).
[42] Phung, K. D., Polynomial decay rate for the dissipative wave equation. J. Differ. Equ. 240 (2007) 92124.
[43] Phung, K. D., Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst. 20 (2008) 10571093.
[44] Prüss, J., On the spectrum of C 0-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847857.
[45] Quinn, J.P., Russell, D.L., Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 97127.
[46] Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639739.
[47] Tcheugoué Tébou, L.R., Sur la stabilisation de l’équation des ondes en dimension 2. C. R. Acad. Sci. Paris Ser. I Math. 319 (1994) 585588.
[48] Tcheugoué Tébou, L.R., On the stabilization of the wave and linear elasticity equations in 2-D. Panamer. Math. J. 6 (1996) 4155.
[49] Tcheugoué Tébou, L.R., On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math. 55 (1998) 293306.
[50] Tcheugoué Tébou, L.R., Stabilization of the wave equation with localized nonlinear damping J.D.E. 145 (1998) 502524.
[51] Tcheugoué Tébou, L.R., Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coefficient, Commun. in P.D.E. 23 (1998) 18391855.
[52] Tcheugoué Tébou, L.R., Energy decay estimates for the damped Euler–Bernoulli equation with an unbounded localizing coefficient. Portugal. Math. 61 (2004) 375391.
[53] Tcheugoué Tébou, L.R., On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differ. Integral Equ. 19 (2006) 785798.
[54] Tebou, L., Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Syst. Control Lett. 56 (2007) 538545.
[55] Tebou, L., Well-posedness and stabilization of an Euler–Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A 32 (2012) 23152337.
[56] Tcheugoué Tébou, L.R. and Zuazua, E., Uniform exponential long time decay for the space finite differences semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numerische Mathematik 95 (2003) 563598.
[57] Tebou, L.T. and Zuazua, E., Uniform boundary stabilization of the finite differences space discretization of the 1 − d wave equation. Adv. Comput. Math. 26 (2007) 337365.
[58] Tucsnak, M., Semi-internal stabilization for a non-linear Bernoulli-Euler equation. Math. Methods Appl. Sci. 19 (1996) 897907.
[59] Wyler, A., Stability of wave equations with dissipative boundary conditions in a bounded domain. Differential Integral Equ. 7 (1994) 345366.
[60] Zuazua, E., Robustesse du feedback de stabilisation par contrôle frontière. C. R. Acad. Sci. Paris Ser. I Math. 307 (1988) 587591.
[61] Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990) 466477.
[62] Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Commun. P.D.E. 15 (1990) 205235.
[63] Zuazua, E., Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures. Appl. 70 (1991) 513529.



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