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.

Keywords: uniform exponential stability, LQR optimal control problem, wave equation, plate equation, finite element, finite difference

@article{COCV_2007__13_3_503_0, 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}, pages = {503--527}, publisher = {EDP-Sciences}, volume = {13}, number = {3}, year = {2007}, doi = {10.1051/cocv:2007020}, zbl = {1126.93050}, mrnumber = {2329173}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2007020/} }

TY - JOUR AU - Ramdani, Karim AU - Takahashi, Takéo AU - Tucsnak, Marius TI - Uniformly exponentially stable approximations for a class of second order evolution equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2007 DA - 2007/// SP - 503 EP - 527 VL - 13 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007020/ UR - https://zbmath.org/?q=an%3A1126.93050 UR - https://www.ams.org/mathscinet-getitem?mr=2329173 UR - https://doi.org/10.1051/cocv:2007020 DO - 10.1051/cocv:2007020 LA - en ID - COCV_2007__13_3_503_0 ER -

%0 Journal Article %A Ramdani, Karim %A Takahashi, Takéo %A Tucsnak, Marius %T Uniformly exponentially stable approximations for a class of second order evolution equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2007 %P 503-527 %V 13 %N 3 %I EDP-Sciences %U https://doi.org/10.1051/cocv:2007020 %R 10.1051/cocv:2007020 %G en %F 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://archive.numdam.org/articles/10.1051/cocv:2007020/

[1] The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22 (1984) 684-698. | Zbl

and ,[2] 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

, and ,[3] Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266-301. | Zbl

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

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

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

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

and ,[8] 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

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

,[10] Boundary observability for the space semi-discretizations of the $1$-D wave equation. ESAIM: M2AN 33 (1999) 407-438. | Numdam | Zbl

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

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

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

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

,[15] On the spectrum of ${C}_{0}$-semigroups. Trans. Amer. Math. Soc. 284 (1984) 847-857. | Zbl

,[16] Introduction à l'analyse numérique des équations aux dérivées partielles. Dunod, Paris (1998). | Zbl

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

and ,[18] 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

and ,[19] 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

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

,*Cited by Sources: *