Abstract
We consider a numerical scheme associated with the iterative method developed in Ramdani et al. (ESAIM Control Optim. Calc. Var. 13(3):503–527, 2007) to recover initial conditions of conservative systems. In this method, the initial conditions are reconstructed by using observers. Here we use a finite-difference discretization in space of these observers and our aim is to prove estimates of the errors with respect to the mesh size and to the number of steps in the iterative method. This is done in the particular example of the 1d wave equation. In order to avoid restrictions of the number of steps with respect to the mesh size, we add a numerical viscosity in the numerical observers. A generalization for other equations is also given.
Similar content being viewed by others
References
Auroux, D., Blum, J.: Back and forth nudging algorithm for data assimilation problems. C. R. Math. Acad. Sci. Paris 340(12), 873–878 (2005)
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) of International Series of Numerical Mathematics, vol. 100, pp. 1–33. Birkhäuser, Basel (1991)
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)
Bensoussan, A.: Filtrage Optimal des Systèmes Linéaires. Dunod (1971)
Brezis, H.: Analyse fonctionnelle. Collection Mathématiques Appliquées pour la Maîtrise, [Collection of Applied Mathematics for the Master’s Degree]. Masson, Paris. Théorie et applications, [Theory and applications] (1983)
Chapelle, D., Cîndea, N., Moireau, P.: Improving convergence in numerical analysis using observers—the wave-like equation case. Math. Models Methods Appl. Sci. 22(12), 1250040, 35 (2012)
Chapelle, D., Cîndea, N., De Buhan, M., Moireau, P.: Exponential convergence of an observer based on partial field measurements for the wave equation. Math. Probl. Eng. , 12 (2012). Art. ID 581053
Cîndea, N., Micu, S., Tucsnak, M.: An approximation method for exact controls of vibrating systems. SIAM J. Control Optim. 49(3), 1283–1305 (2011)
Curtain, R.F., Zwart, H.: An Introduction to Infinite-Dimensional Linear Systems Theory, of Texts in Applied Mathematics, vol. 21. Springer, New York (1995)
Deguenon, J., Sallet, G., Xu, C.-Z.: Infinite dimensional observers for vibrating systems. In: Proceedings of IEEE Conference on Decision and Control, pp. 3979–3983 (2006)
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), 1–76 (1990)
Haine, G.: Recovering the initial data of an evolution equation. Application to thermoacoustic tomography. Submitted (2012)
Haine, G.: Recovering the observable part of the initial data of an infinite-dimensional linear system. Submitted (2012)
Haine, G., Ramdani, K.: Observateurs itératifs en horizon fini. Application à la reconstruction de données initiales pour des edp d’évolution. J. Eur. Syst. Autom. (JESA) 45(7–10), 715–724 (2011)
Haine, G., Ramdani, K.: Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations. Numer. Math. 120(2), 307–343 (2012)
Infante, J.-A., Zuazua, E.: Boundary observability for the space-discretizations of the 1-d wave equation. C. R. Acad. Sci. Paris Sér. I Math. 326(6), 713–718 (1998)
Infante, J.A., Zuazua, E.: Boundary observability for the space semi-discretizations of the 1-D wave equation. M2AN Math. Model. Numer. Anal. 33(2), 407–438 (1999)
Ito, K., Ramdani, K., Tucsnak, M.: A time reversal based algorithm for solving initial data inverse problems. Discrete Contin. Dyn. Syst. Ser. S 4(3), 641–652 (2011)
Kailath, T., Sayed, A.H., Babak, H.: Linear Estimation. Prentice Hall (2000)
Komornik, V.: Exact controllability and stabilization. The multiplier method. In: RAM: Research in Applied Mathematics. Masson, Paris (1994)
Li, X.-D., Xu, C.-Z., Peng, Y.-J., Tucsnak, M.: On the numerical investigation of a Luenberger type observer for infinite-dimensional vibrating systems. In: Proceedings of the 17th World Congress The International Federation of Automatic Control, pp. 7624–7629. Seoul, Corée, République De (2008)
Lions, J.-L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] vol. 9. Masson, Paris (1988). Perturbations. [Perturbations]
Micu, S.: Uniform boundary controllability of a semidiscrete 1-D wave equation with vanishing viscosity. SIAM J. Control Optim. 47(6), 2857–2885 (2008)
Moireau, P., Chapelle, D., Le Tallec, P.: Joint state and parameter estimation for distributed mechanical systems. Comput. Methods Appl. Mech. Eng. 197(6–8), 659–677 (2008)
Münch, A.: A uniformly controllable and implicit scheme for the 1-D wave equation. M2AN Math. Model. Numer. Anal. 39(2), 377–418 (2005)
Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Optimization and control of distributed systems. Syst. Control Lett. 48(3–4), 261–279 (2003)
Negreanu, M., Zuazua, E.: Discrete Ingham inequalities and applications. SIAM J. Numer. Anal. 44(1), 412–448 (2006). (electronic)
Nicaise, S., Valein, J.: Quasi exponential decay of a finite difference space discretization of the 1-d wave equation by pointwise interior stabilization. Adv. Comput. Math. 32(3), 303–334 (2010)
Phung, K.D., Zhang, X.: Time reversal focusing of the initial state for Kirchhoff plate. SIAM J. Appl. Math. 68(6), 1535–1556 (2008)
Ramdani, K., Takahashi, T., Tucsnak, M.: Internal stabilization of the plate equation in a square: the continuous and the semi-discretized problems. J. Math. Pures Appl. (9) 85(1), 17–37 (2006)
Ramdani, K., Takahashi, T., Tucsnak, M.: Semi-discrétisation en espace du problème de la stabilisation interne de l’équation des poutres. In: Paris-Sud Working Group on Modelling and Scientific Computing 2006–2007, of ESAIM Proceedings, vol. 18, pp. 48–56. EDP Sci., Les Ulis (2007)
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)
Ramdani, K., Tucsnak, M., Weiss, G.: Recovering the initial state of an infinite-dimensional system using observers. Automatica 46, 1616–1625 (2010)
Tébou, L.RT., 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)
Tebou, L.T., Zuazua, E.: Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26(1–3), 337–365 (2007)
Tucsnak, M., Weiss, G.: Observation and control for operator semigroups. In: Birkhäuser Advanced Texts: Basler Lehrbücher [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Verlag, Basel (2009)
Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47(2), 197–243 (2005) (electronic)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by: Robert Schaback
Rights and permissions
About this article
Cite this article
García, G.C., Takahashi, T. Numerical observers with vanishing viscosity for the 1d wave equation. Adv Comput Math 40, 711–745 (2014). https://doi.org/10.1007/s10444-013-9320-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10444-013-9320-5