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Uniformly exponentially stable approximations for a class of second order evolution equations

Application to LQR problems

Published online by Cambridge University Press:  05 June 2007

Karim Ramdani ,
Takéo Takahashi  and
Marius Tucsnak
Karim Ramdani
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
Takéo Takahashi
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
Marius Tucsnak
Affiliation:
Institut Elie Cartan University of Nancy-I, POB 239, Vandœuvre-les-Nancy 54506, France; [email protected] INRIA Lorraine, Projet CORIDA.
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Abstract

We consider the approximation of a class ofexponentially stable infinite dimensional linear systems modellingthe damped vibrations of one dimensional vibrating systems or ofsquare plates. It is by now well known that the approximatingsystems obtained by usual finite element or finite difference arenot, in general, uniformly stable with respect to the discretizationparameter. Our main result shows that, by adding a suitablenumerical viscosity term in the numerical scheme, our approximationsare uniformly exponentially stable. This result is then applied toobtain strongly convergent approximations of the solutions of thealgebraic Riccati equations associated to an LQR optimal controlproblem. We next give an application to a non-homogeneous stringequation. Finally we apply similar techniques for approximating theequations of a damped square plate.

Keywords

Uniform exponential stabilityLQR optimal control problemwave equationplate equationfinite elementfinite difference
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 13 , Issue 3 , July 2007 , pp. 503 - 527
Copyright
© EDP Sciences, SMAI, 2007

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References

Banks, H.T. and Kunisch, K., The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22 (1984) 684–698. CrossRef
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.
Chen, G., Fulling, S.A., Narcowich, F.J. and Sun, S., Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math. 51 (1991) 266–301. CrossRef
R.F. Curtain and H. Zwart, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics 21. Springer-Verlag, New York (1995).
Fernandez-Cara, E. and Zuazua, E., On the null controllability of the one-dimensional heat equation with BV coefficients. Comput. Appl. Math. 21 (2002) 167–190.
Gibson, J.S., An analysis of optimal modal regulation: convergence and stability. SIAM J. Control Optim. 19 (1981) 686–707. CrossRef
Gibson, J.S. and Adamian, A., Approximation theory for linear-quadratic-Gaussian optimal control of flexible structures. SIAM J. Control Optim. 29 (1991) 1–37. CrossRef
Glowinski, R., Li, C.H. and 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 (1990) 1–76. CrossRef
Huang, F.L., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equ. 1 (1985) 43–56.
Infante, J.A. and Zuazua, E., Boundary observability for the space semi-discretizations of the 1-D wave equation. ESAIM: M2AN 33 (1999) 407–438. CrossRef
Kappel, F. and Salamon, D., An approximation theorem for the algebraic Riccati equation. SIAM J. Control Optim. 28 (1990) 1136–1147. CrossRef
Z. Liu and S. Zheng, Semigroups associated with dissipative systems, Notes in Mathematics 398. Chapman & Hall/CRC Research, Chapman (1999).
M. Naimark, Linear differential operators. Ungar, New York (1967).
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, Appl. Math. Sci. 44 (1983).
Prüss, J., On the spectrum of $C\sb{0}$ -semigroups. Trans. Amer. Math. Soc. 284 (1984) 847–857. CrossRef
P.-A. Raviart and J.-M. Thomas, Introduction à l'analyse numérique des équations aux dérivées partielles. Dunod, Paris (1998).
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).
Tcheugoué Tébou, L.R. and 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 (2003) 563–598. CrossRef
Zuazua, E., 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. CrossRef
Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. CrossRef