# A uniformly controllable and implicit scheme for the 1-D wave equation

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 39, Issue: 2, page 377-418
- ISSN: 0764-583X

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topMünch, Arnaud. "A uniformly controllable and implicit scheme for the 1-D wave equation." ESAIM: Mathematical Modelling and Numerical Analysis 39.2 (2010): 377-418. <http://eudml.org/doc/194266>.

@article{Münch2010,

abstract = {
This paper studies the
exact controllability of a finite dimensional system obtained by
discretizing in space and time the linear 1-D wave system with a
boundary control at one extreme. It is known that usual schemes
obtained with finite difference or finite element methods are not
uniformly controllable with respect to the discretization
parameters h and Δt. We introduce an implicit finite
difference scheme which differs from the usual centered one by
additional terms of order h2 and Δt2. Using a discrete
version of Ingham's inequality for nonharmonic Fourier series and
spectral properties of the scheme, we show that the associated
control can be chosen uniformly bounded in L2(0,T) and in such
a way that it converges to the HUM control of the continuous wave,
i.e. the minimal L2-norm control. The results are illustrated
with several numerical experiments.
},

author = {Münch, Arnaud},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Exact boundary controllability; wave system; finite
difference.},

language = {eng},

month = {3},

number = {2},

pages = {377-418},

publisher = {EDP Sciences},

title = {A uniformly controllable and implicit scheme for the 1-D wave equation},

url = {http://eudml.org/doc/194266},

volume = {39},

year = {2010},

}

TY - JOUR

AU - Münch, Arnaud

TI - A uniformly controllable and implicit scheme for the 1-D wave equation

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 39

IS - 2

SP - 377

EP - 418

AB -
This paper studies the
exact controllability of a finite dimensional system obtained by
discretizing in space and time the linear 1-D wave system with a
boundary control at one extreme. It is known that usual schemes
obtained with finite difference or finite element methods are not
uniformly controllable with respect to the discretization
parameters h and Δt. We introduce an implicit finite
difference scheme which differs from the usual centered one by
additional terms of order h2 and Δt2. Using a discrete
version of Ingham's inequality for nonharmonic Fourier series and
spectral properties of the scheme, we show that the associated
control can be chosen uniformly bounded in L2(0,T) and in such
a way that it converges to the HUM control of the continuous wave,
i.e. the minimal L2-norm control. The results are illustrated
with several numerical experiments.

LA - eng

KW - Exact boundary controllability; wave system; finite
difference.

UR - http://eudml.org/doc/194266

ER -

## References

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## Citations in EuDML Documents

top- Nicolae Cîndea, Enrique Fernández-Cara, Arnaud Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates
- Arnaud Münch, Ademir Fernando Pazoto, Uniform stabilization of a viscous numerical approximation for a locally damped wave equation
- 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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