Uniform stabilization of some damped second order evolution equations with vanishing short memory
ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, p. 174-189

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.

DOI : https://doi.org/10.1051/cocv/2013060
Classification:  93D15,  35L10,  35Q74,  37L15,  74K20
Keywords: second order evolution equation, Kelvin-Voigt damping, hyperbolic equations, stabilization, boundary dissipation, localized damping, plate equations, elasticity equations, frequency domain method, resolvent estimates
@article{COCV_2014__20_1_174_0,
author = {Tebou, Louis},
title = {Uniform stabilization of some damped second order evolution equations with vanishing short memory},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
pages = {174-189},
doi = {10.1051/cocv/2013060},
zbl = {1282.93210},
mrnumber = {3182696},
language = {en},
url = {http://www.numdam.org/item/COCV_2014__20_1_174_0}
}

Tebou, Louis. Uniform stabilization of some damped second order evolution equations with vanishing short memory. ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, pp. 174-189. doi : 10.1051/cocv/2013060. http://www.numdam.org/item/COCV_2014__20_1_174_0/

[1] F. Alabau-Boussouira, Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006) 95-112. | MR 2215634 | Zbl 1117.34054

[2] F. Alabau and V. Komornik, Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Control Optim. 37 (1999) 521-542. | MR 1665070 | Zbl 0935.93037

[3] W. Arendt, C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837-852. | MR 933321 | Zbl 0652.47022

[4] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | MR 1178650 | Zbl 0786.93009

[5] A. Bátkai, K.-J. Engel, J. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425-1440. | MR 2269247 | Zbl 1118.47034

[6] C.J.K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765-780. | MR 2460938 | Zbl 1185.47043

[7] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups. Math. Annal. 347 (2010) 455-478. | MR 2606945 | Zbl 1185.47044

[8] H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983). | MR 697382 | Zbl 1147.46300

[9] N. Burq, 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) 1-29. | MR 1618254 | Zbl 0918.35081

[10] G. Chen, Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979) 66-81. | MR 516857 | Zbl 0402.93016

[11] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266-301. | MR 1089141 | Zbl 0734.35009

[12] G. Chen and D.L. Russell, A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39 (1981/1982) 433-454. | MR 644099 | Zbl 0515.73033

[13] F. Conrad and B. Rao, Decay of solutions of the wave equation in a star-shaped domain with nonlinear boundary feedback. Asymptotic Anal. 7 (1993) 159-177. | MR 1226972 | Zbl 0791.35011

[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. | MR 2664618 | Zbl 1201.35046

[15] S. Ervedoza and E. Zuazua, Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20-48. | MR 2487899 | Zbl 1163.74019

[16] X. Fu, Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011) 667-680. | MR 2822347 | Zbl 1264.35041

[17] X. Fu, Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957-975. | MR 2560307 | Zbl 1180.35104

[18] R.B. Guzmán and M. Tucsnak, Energy decay estimates for the damped plate equation with a local degenerated dissipation. Systems Control Lett. 48 (2003) 191-197. | MR 2020636 | Zbl 1134.93397

[19] A. Haraux, Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245-258. | MR 1021188 | Zbl 0679.93063

[20] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annal. Differ. Equ. 1 (1985) 43-56. | MR 834231 | Zbl 0593.34048

[21] V. Komornik, Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29 (1991) 197-208. | MR 1088227 | Zbl 0749.35018

[22] V. Komornik, Exact controllability and stabilization. The multiplier method, RAM. Masson and John Wiley, Paris (1994). | MR 1359765 | Zbl 0937.93003

[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. | MR 1364644 | Zbl 0849.93029

[24] V. Komornik, Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591-1613. | MR 1466918 | Zbl 0889.35013

[25] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33-54. | MR 1054123 | Zbl 0636.93064

[26] J. Lagnese, Boundary stabilization of linear elastodynamic systems. SIAM J. Control Opt. 21 (1983) 968-984. | MR 719524 | Zbl 0531.93044

[27] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50 (1983) 163-182. | MR 719445 | Zbl 0536.35043

[28] J. Lagnese, Boundary Stabilization of Thin Plates, vol. 10. SIAM Stud. Appl. Math. Philadelphia, PA (1989). | MR 1061153 | Zbl 0696.73034

[29] I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6 (1993) 507-533. | MR 1202555 | Zbl 0803.35088

[30] I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Appl. Math. Optim. 25 (1992) 189-224. | MR 1142681 | Zbl 0780.93082

[31] G. Lebeau, Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht 19 (1996) 73-109. | MR 1385677 | Zbl 0863.58068

[32] G. Lebeau and L. Robbiano, Stabilisation de l'équation des ondes par le bord. Duke Math. J. 86 (1997) 465-491. | MR 1432305 | Zbl 0884.58093

[33] J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 8 of RMA. Masson, Paris (1988). | MR 963060 | Zbl 0653.93002

[34] K. Liu, Locally distributed control and damping for the conservative systems. SIAM J. Control and Opt. 35 (1997) 1574-1590. | MR 1466917 | Zbl 0891.93016

[35] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56 (2005), 630-644. | MR 2185299 | Zbl 1100.47036

[36] K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping. Z. Angew. Math. Phys. 57 (2006) 419-432. | MR 2228174 | Zbl 1114.35023

[37] P. Martinez, Ph.D. Thesis, University of Strasbourg (1998).

[38] P. Martinez, Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim. 37 (1999) 673-694. | MR 1675149 | Zbl 0946.93025

[39] M. Nakao, Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25-42. | MR 1418287 | Zbl 0860.35072

[40] A. Osses, A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim. 40 (2001) 777-800. | MR 1871454 | Zbl 0997.93013

[41] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983). | MR 710486 | Zbl 0516.47023

[42] K. D. Phung, Polynomial decay rate for the dissipative wave equation. J. Differ. Equ. 240 (2007) 92-124. | MR 2349166 | Zbl 1130.35017

[43] K. D. Phung, Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst. 20 (2008) 1057-1093. | MR 2379488 | Zbl 1156.35417

[44] J. Prüss, On the spectrum of C0-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847-857. | MR 743749 | Zbl 0572.47030

[45] J.P. Quinn, D.L. Russell, Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping. Proc. Roy. Soc. Edinburgh Sect. A 77 (1977) 97-127. | MR 473539 | Zbl 0357.35006

[46] D.L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639-739. | MR 508380 | Zbl 0397.93001

[47] L.R. Tcheugoué Tébou, Sur la stabilisation de l'équation des ondes en dimension 2. C. R. Acad. Sci. Paris Ser. I Math. 319 (1994) 585-588. | MR 1298287 | Zbl 0809.35050

[48] L.R. Tcheugoué Tébou, On the stabilization of the wave and linear elasticity equations in 2-D. Panamer. Math. J. 6 (1996) 41-55. | MR 1366607 | Zbl 0853.35013

[49] L.R. Tcheugoué Tébou, On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math. 55 (1998) 293-306. | MR 1647163 | Zbl 0918.35079

[50] L.R. Tcheugoué Tébou, Stabilization of the wave equation with localized nonlinear damping J.D.E. 145 (1998) 502-524. | MR 1620983 | Zbl 0916.35069

[51] L.R. Tcheugoué Tébou, Well-posedness and energy decay estimates for the damped wave equation with Lr localizing coefficient, Commun. in P.D.E. 23 (1998) 1839-1855. | MR 1641733 | Zbl 0918.35078

[52] L.R. Tcheugoué Tébou, Energy decay estimates for the damped Euler-Bernoulli equation with an unbounded localizing coefficient. Portugal. Math. 61 (2004) 375-391. | MR 2113555 | Zbl 1072.35188

[53] L.R. Tcheugoué Tébou, On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differ. Integral Equ. 19 (2006) 785-798. | MR 2235895 | Zbl 1212.93260

[54] L. Tebou, Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Syst. Control Lett. 56 (2007) 538-545. | MR 2332006 | Zbl 1121.35016

[55] L. Tebou, Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A 32 (2012) 2315-2337. | MR 2885813 | Zbl 1242.93118

[56] L.R. Tcheugoué Tébou and E. Zuazua, 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) 563-598. | MR 2012934 | Zbl 1033.65080

[57] L.T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite differences space discretization of the 1 − d wave equation. Adv. Comput. Math. 26 (2007) 337-365. | MR 2350359 | Zbl 1119.65086

[58] M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation. Math. Methods Appl. Sci. 19 (1996) 897-907. | MR 1399056 | Zbl 0863.93071

[59] A. Wyler, Stability of wave equations with dissipative boundary conditions in a bounded domain. Differential Integral Equ. 7 (1994) 345-366. | MR 1255893 | Zbl 0816.35078

[60] E. Zuazua, Robustesse du feedback de stabilisation par contrôle frontière. C. R. Acad. Sci. Paris Ser. I Math. 307 (1988) 587-591. | MR 967367 | Zbl 0659.93055

[61] E. Zuazua, Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990) 466-477. | MR 1040470 | Zbl 0695.93090

[62] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping. Commun. P.D.E. 15 (1990) 205-235. | MR 1032629 | Zbl 0716.35010

[63] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures. Appl. 70 (1991) 513-529. | MR 1146833 | Zbl 0765.35010