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

Karim Ramdani; Takéo Takahashi; Marius Tucsnak

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 3, page 503-527
- ISSN: 1292-8119

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topRamdani, Karim, Takahashi, Takéo, and Tucsnak, Marius. "Uniformly exponentially stable approximations for a class of second order evolution equations." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 503-527. <http://eudml.org/doc/250005>.

@article{Ramdani2007,

abstract = {
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.
},

author = {Ramdani, Karim, Takahashi, Takéo, Tucsnak, Marius},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Uniform exponential stability; LQR
optimal control problem; wave equation; plate equation; finite
element; finite difference; uniform exponential stability; LQR optimal control problem; finite element},

language = {eng},

month = {6},

number = {3},

pages = {503-527},

publisher = {EDP Sciences},

title = {Uniformly exponentially stable approximations for a class of second order evolution equations},

url = {http://eudml.org/doc/250005},

volume = {13},

year = {2007},

}

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

DA - 2007/6//

PB - EDP Sciences

VL - 13

IS - 3

SP - 503

EP - 527

AB -
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.

LA - eng

KW - Uniform exponential stability; LQR
optimal control problem; wave equation; plate equation; finite
element; finite difference; uniform exponential stability; LQR optimal control problem; finite element

UR - http://eudml.org/doc/250005

ER -

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## Citations in EuDML Documents

top- Farah Abdallah, Serge Nicaise, Julie Valein, Ali Wehbe, Uniformly exponentially or polynomially stable approximations for second order evolution equations and some applications
- Sylvain Ervedoza, Observability properties of a semi-discrete 1d wave equation derived from a mixed finite element method on nonuniform meshes
- Sylvain Ervedoza, Resolvent estimates in controllability theory and applications to the discrete wave equation

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