Abstract
In this paper, we are interested to prove the uniform exponential decay of the energy for the finite difference fully-discretization of 1-D wave equation with an interior damping at \(\xi \). To this end, we decompose the fully discrete system into two subsystems: a conservative system, and a non conservative one. We show under a numerical hypothesis on \(\xi \) that a uniform observability inequality holds for a conservative system, when the mesh size tends to zero using Fourier series technique. Then, we use the observability inequality to prove the uniform exponential decay of the energy for the damped system. Finally, we describe some numerical experiments to illustrate the exponential decay of the energy.
Similar content being viewed by others
References
Ammari, K., Henrot, A., Tucsnak, M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal. 28, 215–240 (2001)
Banks, H.T., Kunisch, K.: The linear regulator problem for parabolic systems. SIAM J. Control Opt. 22, 684–698 (1984)
Banks, H.T., Wang, C.: Optimal feedback control of infinite-dimensional parabolic evolution systems: approximation techniques. SIAM J. Control Optim. 27, 1182–1219 (1989)
Banks, H.T., Ito, K., Wang, B.: Exponentially stable approximations of weakly damped wave equations. Int. Ser. Numer. Math. 100, 1–33 (1991)
Bouslous, H., El Boujaoui, H., Maniar, L.: Uniform boundary stabilization for the finite difference semi-discretization of 2-D wave equation. Afr. Mat. 25, 623–643 (2014)
Castro, C., Micu, S.: Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numerische Mathematik 102, 413–462 (2006)
El Boujaoui, H., Maniar, L., Bouslous, H.: Uniform boundary stabilization for the finite difference discretization of the 1-D wave equation. Afr. Mat. (2016). https://doi.org/10.1007/s13370-016-0406-3
Ervedoza, S., Zuazua, E.: Uniformly exponentially stable approximations for a class of damped systems. J. Math. Pures Appl. 91, 20–48 (2009)
Glowinski, R., Kinton, W., Wheeler, M.F.: A mixed finite element formulation for the boundary controllability of the wave equation. Int. J. Numer. Methods Eng. 27, 623–635 (1989)
Infante, J., Zuazua, E.: Boundary observability for the space semi-discretization of the 1-D wave equation. M2AN 33, 407–438 (1999)
Isaacson, E., Keller, H.B.: Analysis of numerical methods. Wiley, New York (1966)
Leon, L., Zuazua, E.: Boundary controllability of the finite-difference space semi-discretization of the beam equation. ESAIM Control Optim Calc. Var. 8, 827–862 (2002)
Loreti, P., Mehrenberger, M.: An Ingham type proof for a two-grid observability theorem. ESAIM Control Optim. Calc. Var. 14, 604–631 (2008)
Münch, A., Pazoto, A.F.: Uniform stabilization of a viscous numerical approximation for a locally damped wave equation. ESAIM Control Optim. Calc. Var. 13, 265–293 (2007)
Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Syst. Control Lett. 48, 261–280 (2003)
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, 413–418 (2004)
Nicaise, S., Valein, J.: Stabilization of the wave equation on 1-D networks with a delay term in the nodal feedbacks. Netw. Heterog. Media 2, 425–479 (2007)
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, 303–334 (2010)
Rebarber, R.: Exponential stability of coupled beams with dissipative joints: a frequency domain approach. SIAM J. Control Optim. 33, 1–28 (1995)
Tebou, L.T., 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, 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, 337–365 (2007)
Zuazua, E.: Boundary observability for the finite difference space semi-discretization of the 2-D wave equation in the square. J. Math. Pures Appl. 78, 523–563 (1999)
Zuazua, E.: Controllability of partial differential equations and its semi-discrete approximations. Discrete Contin. Dyn. Syst. 8, 469–513 (2002)
Zuazua, E.: Propagation, observation, control and numerical approximation of waves. SIAM Rev. 47, 197–243 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
El Boujaoui, H. Uniform interior stabilization for the finite difference fully-discretization of the 1-D wave equation. Afr. Mat. 29, 557–574 (2018). https://doi.org/10.1007/s13370-018-0559-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13370-018-0559-3
Keywords
- Interior stabilization
- Pointwise internal damping
- Uniform exponential decay
- Uniform interior observability
- Finite difference method
- Fully-discretization