Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-10T16:23:11.524Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniform stabilization of some damped second order evolutionequations with vanishing short memory

Published online by Cambridge University Press:  23 December 2013

Louis Tebou
Louis Tebou*
Affiliation:
Department of Mathematics and Statistics, Florida International University, Miami FL 33199, USA. teboul@fiu.edu
Get access

Abstract

We consider a damped abstract second order evolution equation with an additionalvanishing damping of Kelvin–Voigt type. Unlike the earlier work by Zuazua and Ervedoza, wedo not assume the operator defining the main damping to be bounded. First, using aconstructive frequency domain method coupled with a decomposition of frequencies and theintroduction 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 decayestimates of the underlying semigroup provided such decay estimates hold for the limitsystem. Finally, we discuss some applications of our results; in particular, the case ofboundary damping mechanisms is accounted for, which was not possible in the earlier workmentioned above.

Keywords

Second order evolution equationKelvin–Voigt dampinghyperbolic equationsstabilizationboundary dissipationlocalized dampingplate equationselasticity equationsfrequency domain methodresolvent estimates
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 1 , January 2014 , pp. 174 - 189
Copyright
© EDP Sciences, SMAI 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alabau-Boussouira, F., Piecewise multiplier method and nonlinear integral inequalities for Petrowsky equation with nonlinear dissipation. J. Evol. Equ. 6 (2006) 95–112. Google Scholar
Alabau, F. and Komornik, V., Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Control Optim. 37 (1999) 521–542. Google Scholar
Arendt, W., Batty, C. J. K., Tauberian theorems and stability of one-parameter semigroups. Trans. Amer. Math. Soc. 306 (1988) 837–852. Google Scholar
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) 1024–1065. Google Scholar
Bátkai, A., Engel, K.-J., Prüss, J. and Schnaubelt, R., Polynomial stability of operator semigroups. Math. Nachr. 279 (2006) 1425–1440. Google Scholar
Batty, C.J.K. and Duyckaerts, T., Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. 8 (2008) 765–780. Google Scholar
Borichev, A. and Tomilov, Y., Optimal polynomial decay of functions and operator semigroups. Math. Annal. 347 (2010) 455–478. Google Scholar
H. Brezis, Analyse fonctionnelle. Théorie et Applications. Masson, Paris (1983).
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) 1–29. Google Scholar
Chen, G., Control and stabilization for the wave equation in a bounded domain. SIAM J. Control Optim. 17 (1979) 66–81. Google Scholar
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) 266–301. Google Scholar
Chen, G. and Russell, D.L., A mathematical model for linear elastic systems with structural damping. Quart. Appl. Math. 39 (1981/1982) 433-454. Google Scholar
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) 159–177. Google Scholar
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.
Ervedoza, S. and Zuazua, E., Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91 (2009) 20–48. Google Scholar
Fu, X., Longtime behavior of the hyperbolic equations with an arbitrary internal damping. Z. Angew. Math. Phys. 62 (2011) 667–680. Google Scholar
Fu, X., Logarithmic decay of hyperbolic equations with arbitrary small boundary damping. Commun. Partial Differ. Equ. 34 (2009) 957–975. Google Scholar
Guzmán, R.B. and Tucsnak, M., Energy decay estimates for the damped plate equation with a local degenerated dissipation. Systems Control Lett. 48 (2003) 191–197. Google Scholar
Haraux, A., Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Portugal. Math. 46 (1989) 245–258. Google Scholar
Huang, F.L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Annal. Differ. Equ. 1 (1985) 43–56. Google Scholar
Komornik, V., Rapid boundary stabilization of the wave equation. SIAM J. Control Optim. 29 (1991) 197–208. Google Scholar
V. Komornik, Exact controllability and stabilization. The multiplier method, RAM. Masson and John Wiley, Paris (1994).
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.
Komornik, V., Rapid boundary stabilization of linear distributed systems. SIAM J. Control Optim. 35 (1997) 1591–1613. Google Scholar
Komornik, V. and Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1990) 33–54. Google Scholar
Lagnese, J., Boundary stabilization of linear elastodynamic systems. SIAM J. Control Opt. 21 (1983) 968–984. Google Scholar
Lagnese, J., Decay of solutions of wave equations in a bounded region with boundary dissipation. J. Differ. Equ. 50 (1983) 163–182. Google Scholar
J. Lagnese, Boundary Stabilization of Thin Plates, vol. 10. SIAM Stud. Appl. Math. Philadelphia, PA (1989).
Lasiecka, I. and Tataru, D., Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping. Differ. Integral Equ. 6 (1993) 507–533. Google Scholar
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) 189–224. Google Scholar
Lebeau, G., Equation des ondes amorties. Algebraic and geometric methods in mathematical physics (Kaciveli, 1993). Math. Phys. Stud. Kluwer Acad. Publ., Dordrecht 19 (1996) 73–109. Google Scholar
Lebeau, G. and Robbiano, L., Stabilisation de l’équation des ondes par le bord. Duke Math. J. 86 (1997) 465–491. Google Scholar
J.-L. Lions, Contrôlabilité exacte, Perturbations et Stabilisation des Systèmes Distribués, vol. 8 of RMA. Masson, Paris (1988).
Liu, K., Locally distributed control and damping for the conservative systems. SIAM J. Control and Opt. 35 (1997) 1574–1590. Google Scholar
Liu, Z. and Rao, B., Characterization of polynomial decay rate for the solution of linear evolution equation. Z. Angew. Math. Phys. 56 (2005), 630–644. Google Scholar
Liu, K. and Rao, B., Exponential stability for the wave equations with local Kelvin–Voigt damping. Z. Angew. Math. Phys. 57 (2006) 419–432. Google Scholar
P. Martinez, Ph.D. Thesis, University of Strasbourg (1998).
Martinez, P., Boundary stabilization of the wave equation in almost star-shaped domains. SIAM J. Control Optim. 37 (1999) 673–694. Google Scholar
Nakao, M., Decay of solutions of the wave equation with a local degenerate dissipation. Israel J. Math. 95 (1996) 25–42. Google Scholar
Osses, A., A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim. 40 (2001) 777–800. Google Scholar
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, vol. 44 of Appl. Math. Sci. Springer-Verlag, New York (1983).
Phung, K. D., Polynomial decay rate for the dissipative wave equation. J. Differ. Equ. 240 (2007) 92–124. Google Scholar
Phung, K. D., Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete Contin. Dyn. Syst. 20 (2008) 1057–1093. Google Scholar
Prüss, J., On the spectrum of C 0-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847–857. Google Scholar
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) 97–127. Google Scholar
Russell, D.L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978) 639–739. Google Scholar
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) 585–588. Google Scholar
Tcheugoué Tébou, L.R., On the stabilization of the wave and linear elasticity equations in 2-D. Panamer. Math. J. 6 (1996) 41–55. Google Scholar
Tcheugoué Tébou, L.R., On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation. Portugal. Math. 55 (1998) 293–306. Google Scholar
Tcheugoué Tébou, L.R., Stabilization of the wave equation with localized nonlinear damping J.D.E. 145 (1998) 502–524. Google Scholar
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) 1839–1855. Google Scholar
Tcheugoué Tébou, L.R., Energy decay estimates for the damped Euler–Bernoulli equation with an unbounded localizing coefficient. Portugal. Math. 61 (2004) 375–391. Google Scholar
Tcheugoué Tébou, L.R., On the stabilization of dynamic elasticity equations with unbounded locally distributed dissipation. Differ. Integral Equ. 19 (2006) 785–798. Google Scholar
Tebou, L., Stabilization of the elastodynamic equations with a degenerate locally distributed dissipation. Syst. Control Lett. 56 (2007) 538–545. Google Scholar
Tebou, L., Well-posedness and stabilization of an Euler–Bernoulli equation with a localized nonlinear dissipation involving the p-Laplacian. DCDS A 32 (2012) 2315–2337. Google Scholar
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) 563–598. Google Scholar
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) 337–365. Google Scholar
Tucsnak, M., Semi-internal stabilization for a non-linear Bernoulli-Euler equation. Math. Methods Appl. Sci. 19 (1996) 897–907. Google Scholar
Wyler, A., Stability of wave equations with dissipative boundary conditions in a bounded domain. Differential Integral Equ. 7 (1994) 345–366. Google Scholar
Zuazua, E., Robustesse du feedback de stabilisation par contrôle frontière. C. R. Acad. Sci. Paris Ser. I Math. 307 (1988) 587–591. Google Scholar
Zuazua, E., Uniform stabilization of the wave equation by nonlinear boundary feedback. SIAM J. Control Optim. 28 (1990) 466–477. Google Scholar
Zuazua, E., Exponential decay for the semilinear wave equation with locally distributed damping. Commun. P.D.E. 15 (1990) 205–235. Google Scholar
Zuazua, E., Exponential decay for the semilinear wave equation with localized damping in unbounded domains. J. Math. Pures. Appl. 70 (1991) 513–529. Google Scholar