Abstract
The aim of this work is to obtain optimal-order error estimates for the LQR (Linear-quadratic regulator) problem in a strongly damped 1-D wave equation. We consider a finite element discretization of the system dynamics and a control law constant in the spatial dimension, which is studied in both point and distributed case. To solve the LQR problem, we seek a feedback control which depends on the solution of an algebraic Riccati equation. Optimal error estimates are proved in the framework of the approximation theory for control of infinite-dimensional systems. Finally, numerical results are presented to illustrate that the optimal rates of convergence are achieved.
Similar content being viewed by others
References
Badra, M.: Stabilisation par feedback et approximation des equations de Navier–Stokes. Thèse Doctorat de l’Université Paul Sabatier (2006)
Banks, H.T., Ito, K.: Approximation in LQR problems for infinite-dimensional systems with unbounded input operators. J. Math. Syst. Est. Control 7(1), 1–34 (1997)
Becker, R., Vexler, B.: Optimal control of the convection-diffusion equation using stabilized finite element methods. Numer. Math. 106(3), 349–367 (2007)
Benner, P.: Solving large-scale control problems. IEEE Control Syst. Mag. (2004)
Benner, P., Gärner, S., Saak, J.: Numerical solution of optimal control problems for parabolic systems. In: Parallel Algorithms and Cluster Computing: Implementations, Algorithms and Applications. Lecture Notes in Computational Science and Engineering, vol. 52, pp. 151–169. Springer, Berlin (2006)
Bermúdez, A., Gamallo, P., Rodríguez, R.: Finite element methods in local active control of sound. SIAM J. Control Optim. 43, 437–465 (2004)
Chen, S., Triggiani, R.: Proof of extensions of two conjectures on structural damping for elastic systems, The case 1/2≤α≤1. Pac. J. Math. 136(1), 15–55 (1989)
Chen, S., Triggiani, R.: Characterization of fractional powers of certain operators arising in elastic systems, and applications. J. Differ. Equ. 88, 279–293 (1990)
Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam (1978)
Dáger, R.: Insensitizing controls for the 1-D wave equation. SIAM J. Control Optim. 45(5), 1758–1768 (2006)
Fuller, C.R., Elliot, S.J., Nelson, P.A.: Active Control of Vibration. Academic Press, New York (1997)
Gibson, J.S., Adamian, A.: Approximation theory for LQG linear-quadratic-Gaussian optimal control of flexible structures. SIAM J. Control Optim. 29(1), 1–37 (1991)
Hernández, E., Otárola, E.: A locking free FEM in active vibration control of a Timoshenko beam. Submitted; available at http://www.mat.utfsm.cl/publicaciones/preprints2008/files/2008-1.pdf
Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems. Cambridge University Press, Cambridge (2000)
Lasiecka, I.: Convergence estimates for semidiscrete approximation of nonselfadjoint parabolic equations. SIAM J. Numer. Anal. 21(5), 894–909 (1984)
Morris, K.A., Navasca, C.: Approximation of linear quadratic feedback control for partial differential equations. In: Zolesio, J.P., Cagnol, J. (eds.) Control of Distributed Parameter Systems, pp. 259–281. Marcel Dekker, New York (2004)
Nelson, P.A., Elliot, S.J.: Active Control of Sound. Academic Press, London (1999)
Tadi, M.: Computational algorithm for controlling a Timoshenko beam. Comput. Methods Appl. Mech. Eng. 153, 153–165 (1998)
Zuazua, E.: Propagation, observation, control and numerical approximation of waves approximated by finite difference method. SIAM Rev. 47(2), 197–243 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hernández, E., Kalise, D. & Otárola, E. Numerical approximation of the LQR problem in a strongly damped wave equation. Comput Optim Appl 47, 161–178 (2010). https://doi.org/10.1007/s10589-008-9213-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-008-9213-6