Spectral conditions for admissibility and observability of wave systems: Applications to finite element schemes

Numerische Mathematik

Volumn 113, Issue 3, 2009, Pages 377-415

  Ervedoza, Sylvain ^a  

^a UNIVERSITÉ DE VERSAILLES SAINT QUENTIN EN YVELINES   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 71749120776     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-009-0235-5     Document Type: Article

References (45)

1. Banks, H.T., Ito, K., Wang, C.: Exponentially stable approximations of weakly damped wave equations. In: Estimation and control of distributed parameter systems (Vorau, 1990) Internat. Ser. Numer. Math., vol. 100, pp. 1-33. Birkhäuser, Basel (1991).
2. Bardos C., Lebeau G., Rauch J.: Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control Optim. 30(5), 1024-1065 (1992).
3. Burq N., Gérard P.: Condition nécessaire et suffisante pour la contrôlabilité exacte des ondes. C. R. Acad. Sci. Paris Sér. I Math. 325(7), 749-752 (1997).
4. Burq N., Zworski M.: Geometric control in the presence of a black box. J. Am. Math. Soc. 17(2), 443-471 (2004) (electronic).
5. Castro C., Micu S.: Boundary controllability of a linear semi-discrete 1-d wave equation derived from a mixed finite element method. Numer. Math. 102(3), 413-462 (2006).
6. Castro C., Zuazua E.: Low frequency asymptotic analysis of a string with rapidly oscillating density. SIAM J. Appl. Math. 60(4), 1205-1233 (2000) (electronic).
7. Castro C., Zuazua E.: Concentration and lack of observability of waves in highly heterogeneous media. Arch. Ration. Mech. Anal. 164(1), 39-72 (2002).
8. Dáger, R., Zuazua, E.: Wave propagation, observation and control in 1-d flexible multi-structures. Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 50. Springer, Berlin (2006).
9. Ervedoza, S.: Observability of the mixed finite element method for the 1d wave equation on nonuniform meshes. ESAIM Control Optim. Calc. Var. (2009, to appear).
10. Ervedoza, S.: Admissibility and observability for Schrödinger systems: applications to finite element approximation schemes. Asymptot. Anal. (2009, to appear).
11. Ervedoza, S.: Observability in arbitrary small time for discrete approximations of conservative systems (2009, preprint).
12. Ervedoza S., Zheng C., Zuazua E.: On the observability of time-discrete conservative linear systems. J. Funct. Anal. 254(12), 3037-3078 (2008).
13. Ervedoza, S., Zuazua, E.: Uniform exponential decay for viscous damped systems. To appear in Proc. of Siena Phase Space Analysis of PDEs 2007. Special issue in honor of Ferrucio Colombini.
14. Ervedoza S., Zuazua E.: Perfectly matched layers in 1-d: Energy decay for continuous and semi- discrete waves. Numer. Math. 109(4), 597-634 (2008).
15. Ervedoza S., Zuazua E.: Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91, 20-48 (2009).
16. Glowinski R.: Ensuring well-posedness by analogy: Stokes problem and boundary control for the wave equation. J. Comput. Phys. 103(2), 189-221 (1992).
17. 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. Jpn. J. Appl. Math. 7(1), 1-76 (1990).
18. Haraux, A.: Séries lacunaires et contrôle semi-interne des vibrations d'une plaque rectangulaire. J. Math. Pures Appl. (9), 68(4):457-465 (1990), 1989.
19. Haraux A.: Une remarque sur la stabilisation de certains systèmes du deuxième ordre en temps. Port. Math. 46(3), 245-258 (1989).
20. Imanuvilov O.Y.: On Carleman estimates for hyperbolic equations. Asymptot. Anal. 32(3-4), 185-220 (2002).
21. Infante J.A., Zuazua E.: Boundary observability for the space semi discretizations of the 1-d wave equation. Math. Model. Num. Ann. 33, 407-438 (1999).
22. Ingham A.E.: Some trigonometrical inequalities with applications to the theory of series. Math. Z. 41(1), 367-379 (1936).
23. Komornik, V.: Exact controllability and stabilization. RAM: Research in Applied Mathematics. Masson, Paris (1994). [The multiplier method].
24. Lebeau, G.: Équations des ondes amorties. Séminaire sur les Équations aux Dérivées Partielles, 1993-1994,École Polytech (1994).
25. Lions, J.-L.: Contrôlabilité exacte, Stabilisation et Perturbations de Systèmes Distribués. Tome 1. Contrôlabilité exacte, vol. RMA 8. Masson (1988).
26. Liu K.: Locally distributed control and damping for the conservative systems. SIAM J. Control Optim. 35(5), 1574-1590 (1997).
27. Macià, F.: The effect of group velocity in the numerical analysis of control problems for the wave equation. In: Mathematical and Numerical Aspects of Wave Propagation-WAVES, pp. 195-200. Springer, Berlin (2003).
28. Miller L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal. 218(2), 425-444 (2005).
29. Miller, L.: Private communication (2009).
30. Münch, A., Pazoto, A.F.: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var. 13(2), 265-293 (2007) (electronic).
31. Negreanu M., Matache A.-M., Schwab C.: Wavelet filtering for exact controllability of the wave equation. SIAM J. Sci. Comput. 28(5), 1851-1885 (2006) (electronic).
32. Negreanu M., Zuazua E.: Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Math. Acad. Sci. Paris 338(5), 413-418 (2004).
33. Osses A.: A rotated multiplier applied to the controllability of waves, elasticity, and tangential Stokes control. SIAM J. Control Optim. 40(3), 777-800 (2001) (electronic).
34. Ramdani K., Takahashi T., Tenenbaum G., Tucsnak M.: A spectral approach for the exact observability of infinite-dimensional systems with skew-adjoint generator. J. Funct. Anal. 226(1), 193-229 (2005).
35. Ramdani K., Takahashi T., Tucsnak M.: Uniformly exponentially stable approximations for a class of second order evolution equations-application to LQR problems. ESAIM Control Optim. Calc. Var. 13(3), 503-527 (2007).
36. Raviart, P.-A., Thomas, J.-M.: Introduction à l'analyse numérique des équations aux dérivées partielles. Collection Mathématiques Appliquées pour la Maitrise. [Collection of Applied Mathematics for the Master's Degree]. Masson, Paris (1983).
37. Tcheugoué Tebou L.R., Zuazua E.: Uniform boundary stabilization of the finite difference space discretization of the 1d wave equation. Adv. Comput. Math. 26(1-3), 337-365 (2007).
38. Tcheugoué Tébou L.R., Zuazua E.: Uniform exponential long time decay for the space semi- discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95(3), 563-598 (2003).
39. Trefethen L.N.: Group velocity in finite difference schemes. SIAM Rev. 24(2), 113-136 (1982).
40. Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. Advanced Texts, vol. XI. Springer, Birkhäuser (2009).
41. Weiss G.: Admissibility of unbounded control operators. SIAM J. Control Optim. 27(3), 527-545 (1989).
42. Young R.M.: An Introduction to Nonharmonic Fourier Series, 1st edn. Academic Press Inc., San Diego (2001).
43. Zhang X.: Explicit observability estimate for the wave equation with potential and its application. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456(1997), 1101-1115 (2000).
44. Zuazua E.: Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. (9) 78(5), 523-563 (1999).
45. Zuazua E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197-243 (2005) (electronic).