Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity

Numerische Mathematik

Volumn 95, Issue 3, 2003, Pages 563-598

  Tcheugoué Tébou, Louis Roder ^a,c   Zuazua, Enrique ^b  

^a YORK UNIVERSITY   (Canada)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

^c FLORIDA INTERNATIONAL UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0345491605     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-002-0442-9     Document Type: Article

References (26)

1. [BLR] Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt. 30, 1024-1065 (1992)
2. [G] Glowinski, R.: Ensuring well-posedness by analogy; Stokes problem and boundary control for the wave equation, J. Compt. Physics, 103 (2), 189-221 (1992)
3. [GLL] Glowinski, R., Li, C.H., Lions, J.-L.: A numerical approach to the exact boundary controllability of the wave equation (I). Dirichlet controls: Description of the numerical methods, Japan J. Appl. Math. 7, 1-76 (1990)
4. [H1] Haraux, A.: Semi-groupes linéaires et équations d'évolution linéaires périodiques, Publications du Laboratoire d' Analyse Numérique, Université Pierre et Marie Curie, Paris (1978), No. 78011.
5. [H2] Haraux, A.: Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps, Portugal. Math. 46, 245-258 (1989)
6. [I] Ingham, A.E.: Some trigonometric inequalities with applications to the theory of series, Math. Zeits. 41, 367-379 (1936)
7. [IK] Isaacson, E., Keller, H.B.: Analysis of numerical methods. John Wiley & Sons (1966)
8. [IZ1] Infante, J.A., Zuazua, E.: Boundary observability for the space-discretizations of the 1-D wave equation, C. R. Acad. Paris, Série I, 326, 713-718 (1997)
9. [IZ2] Infante, J.A., Zuazua, E.: Boundary observability for the space semi-discretizations of the 1-D wave equation, Math. Model. Num. An. 33, 407-438 (1999)
10. [K] Komornik, V.: Exact controllability and stabilization. The multiplier method, RAM, Masson & John Wiley, Paris, 1994
11. [La] Lagnese, J.: Control of wave processes with distributed control supported on a subregion, S.I.A.M J. Control and Opt. 21, 68-85 (1983)
12. [Lio1] Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod-Gauthier-Villars, Paris. 1969
13. [Lio2] Lions, J.L.: Contrôlabilité exacte, perturbations et stabilisation des systèmes distribués, vol. 1, RMA 8, Masson, Paris, 1988
14. [LM] Lions, J.L., Magenes, E.: Problèmes aux limites non homogènes et applications, vol. 1 Dunod, Paris, 1968
15. [Liu] Liu, K.: Locally distributed control and damping for the conservative systems, S.I.A.M J. Control and Opt. 35, 1574-1590 (1997)
16. [Ma] Macià, F.: PhD Thesis, Universidad Complutense de Madrid, 2001
17. [Mz] Martinez, P.: PhD Thesis, University of Strasbourg, 1998
18. [N1] Nakao, M.: Energy decay for the wave equation with a degenerate dissipative term, Proc. Royal Soc. Edinburgh, A100, 19-27 (1985)
19. [N2] Nakao, M.: Decay of solutions of the wave equation with a local degenerate dissipation, Israel J. Math. 95, 25-42 (1996)
20. [R] Raviart, P.A.: Sur l'approximation de certaines équations d'évolution linéaires et non linéaires, J.M.P.A. 46, 11-107 (1967)
21. [TT1] Tcheugoué Tébou, L.R.: Estimations d'énergie pour l'équation des ondes avec un amortissement nonlinéaire localisé, C. R. Acad. Paris, Série I, 325, 1175-1179 (1997)
22. [TT2] Tcheugoué Tébou, L.R.: On the decay estimates for the wave equation with a local degenerate or nondegenerate dissipation, Portugal. Math. 55, 293-306 (1998)
23. [TT3] Tcheugoué Tébou, L.R.: Stabilization of the wave equation with localized nonlinear damping, J.D.E. 145, 502-524 (1998)
24. r localizing coefficient, Comm. in P.D.E., 23, 1839-1855 (1998)
25. [Z1] Zuazua, E.: Exponential decay for the semilinear wave equation with locally distributed damping, Commun, in P.D.E., 15, 205-235 (1990)
26. [Z2] Zuazua, E.: Boundary observability for finite-difference space semi-discretizations of the 2-D wave equation in the square, J.M.P.A. 78, 523-563 (1999)