Approximation of the controls for the wave equation with a potential

Micu, Sorin; Rovenţa, Ionel; Temereancă, Laurenţiu Emanuel

doi:10.1007/s00211-020-01106-2

“…The value of V only affects the constant. We refer to [41] for a semi-discrete (in space) approximation of exact boundary controls for a semi-discretized wave equation with potential, including experiments for small and potentials with good sign.…”

Section: Distributed Case : Initial Condition Inmentioning

confidence: 99%

Spacetime finite element methods for control problems subject to the wave equation

Burman¹,

Feizmohammadi²,

Münch³

et al. 2021

Preprint

3

2

0

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We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.

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“…Theorem 4.2. Let T > T 0 and let (w 0 M , w 1 M ) be given by (21). Suppose that the discrete initial data ((w 0 j , w 1 j )) 1≤j≤N ∈ C 2N verify ( 22) -( 23) and…”

Section: Uniform Controllability Resultsmentioning

confidence: 99%

“…We start this section by presenting the continous problem and how spectral estimates are used to obtain the moment problem. To do that we need the following variational result from [21,22].…”

Section: Preliminaries On Spectral Properties and Moment Problemsmentioning

confidence: 99%

See 1 more Smart Citation

A note on an algorithm studying the uniform controllability of a class of semidiscrete hyperbolic problems

Rovența,

Tudor

2023

AUCMCS

0

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We propose an algorithm based on the the technique introduced in [23]. The aim of the algorithm is to study, in a simple way, the approximation of the controls for a class of hyperbolic problems. It is well-known that, the finite-difference semi-discrete scheme for the approximation of controls can leads to high frequency numerical spurious oscillations which gives a loss of the uniform (with respect to the mesh-size) controllability property of the semidiscrete model. It is also known that an appropriate filtration of the high eigenfrequencies of the discrete initial data enable us to restore the uniform controllability property of the whole solution. But, the methods used to prove such results are very constructive and contains difficult and fine computations. As an example, which proves the effectiveness of our algorithm, we consider the case of the semidiscrete one dimensional wave equation. In this particular case, we are able to prove the uniform controllability, where the initial data are filtered in a range which contains as many modes as possibles, taking into account previous results obtained in literature (see [18]).

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“…The value of V only affects the constant. We refer to [40] for a semi-discrete (in space) approximation of exact boundary controls for a semi-discretized wave equation with potential, including experiments for small and potentials with good sign.…”

Section: Boundary Case: the Wave Equation With A Potentialmentioning

confidence: 99%

Spacetime finite element methods for control problems subject to the wave equation

Burman

¹

,

Feizmohammadi

²

,

Münch

³

et al. 2023

ESAIM: COCV

2

0

View full text Add to dashboard Cite

We consider the null controllability problem for the wave equation, and analyse a stabilized finite element method formulated on a global, unstructured spacetime mesh. We prove error estimates for the approximate control given by the computational method. The proofs are based on the regularity properties of the control given by the Hilbert Uniqueness Method, together with the stability properties of the numerical scheme. Numerical experiments illustrate the results.

show abstract

Approximation of the controls for the wave equation with a potential

Cited by 5 publications

References 30 publications

Spacetime finite element methods for control problems subject to the wave equation

Spacetime finite element methods for control problems subject to the wave equation

A note on an algorithm studying the uniform controllability of a class of semidiscrete hyperbolic problems

Spacetime finite element methods for control problems subject to the wave equation

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