Uniformly exponentially stable approximations for a class of second order evolution equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, p. 503-527

We consider the approximation of a class of exponentially stable infinite dimensional linear systems modelling the damped vibrations of one dimensional vibrating systems or of square plates. It is by now well known that the approximating systems obtained by usual finite element or finite difference are not, in general, uniformly stable with respect to the discretization parameter. Our main result shows that, by adding a suitable numerical viscosity term in the numerical scheme, our approximations are uniformly exponentially stable. This result is then applied to obtain strongly convergent approximations of the solutions of the algebraic Riccati equations associated to an LQR optimal control problem. We next give an application to a non-homogeneous string equation. Finally we apply similar techniques for approximating the equations of a damped square plate.

DOI : https://doi.org/10.1051/cocv:2007020
Classification:  93D15,  65M60,  65M12
Keywords: uniform exponential stability, LQR optimal control problem, wave equation, plate equation, finite element, finite difference
     author = {Ramdani, Karim and Takahashi, Tak\'eo and Tucsnak, Marius},
     title = {Uniformly exponentially stable approximations for a class of second order evolution equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {13},
     number = {3},
     year = {2007},
     pages = {503-527},
     doi = {10.1051/cocv:2007020},
     zbl = {1126.93050},
     mrnumber = {2329173},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2007__13_3_503_0}
Ramdani, Karim; Takahashi, Takéo; Tucsnak, Marius. Uniformly exponentially stable approximations for a class of second order evolution equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 13 (2007) no. 3, pp. 503-527. doi : 10.1051/cocv:2007020. http://www.numdam.org/item/COCV_2007__13_3_503_0/

[1] H.T. Banks and K. Kunisch, The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22 (1984) 684-698. | Zbl 0548.49017

[2] H.T. Banks, K. Ito and C. Wang, Exponentially stable approximations of weakly damped wave equations, in Estimation and control of distributed parameter systems (Vorau, 1990), Birkhäuser, Basel, Internat. Ser. Numer. Math. 100 (1991) 1-33. | Zbl 0850.93719

[3] G. Chen, S.A. Fulling, F.J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266-301. | Zbl 0734.35009

[4] R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics 21. Springer-Verlag, New York (1995). | MR 1351248 | Zbl 0839.93001

[5] E. Fernandez-Cara and E. Zuazua, On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21 (2002) 167-190. | Zbl 1119.93311

[6] J.S. Gibson, An analysis of optimal modal regulation: convergence and stability. SIAM J. Control Optim. 19 (1981) 686-707. | Zbl 0477.49019

[7] J.S. Gibson and A. Adamian, Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures. SIAM J. Control Optim. 29 (1991) 1-37. | Zbl 0788.93027

[8] R. Glowinski, C.H. Li and J.-L. Lions, 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 (1990) 1-76. | Zbl 0699.65055

[9] F.L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43-56. | Zbl 0593.34048

[10] J.A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-D wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | Zbl 0947.65101

[11] F. Kappel and D. Salamon, An approximation theorem for the algebraic Riccati equation. SIAM J. Control Optim. 28 (1990) 1136-1147. | Zbl 0717.49030

[12] Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Notes in Mathematics 398. Chapman & Hall/CRC Research, Chapman (1999). | MR 1681343 | Zbl 0924.73003

[13] M. Naimark, Linear differential operators. Ungar, New York (1967).

[14] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, Appl. Math. Sci. 44 (1983). | MR 710486 | Zbl 0516.47023

[15] J. Prüss, On the spectrum of C 0 -semigroups. Trans. Amer. Math. Soc. 284 (1984) 847-857. | Zbl 0572.47030

[16] P.-A. Raviart and J.-M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Dunod, Paris (1998). | Zbl 0561.65069

[17] G. Strang and G.J. Fix, An analysis of the finite element method. Prentice-Hall Inc., Englewood Cliffs, N.J. Prentice-Hall Series in Automatic Computation (1973). | MR 443377 | Zbl 0356.65096

[18] L.R. Tcheugoué Tébou and E. Zuazua, Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95 (2003) 563-598. | Zbl 1033.65080

[19] E. Zuazua, Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. 78 (1999) 523-563. | Zbl 0939.93016

[20] E. Zuazua, Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197-243. | Zbl 1077.65095