A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations

Aranda, Ernesto; Pedregal, Pablo

doi:10.1016/j.jfranklin.2014.04.006

Cited by 6 publications

(7 citation statements)

References 13 publications

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“…satisfies the final time condition (2). Under a geometric control condition analogous to (A), and the above regularity assumptions on the data (u 0 , u 1 ), we introduce a finite element method that converges as…”

Section: Introductionmentioning

confidence: 99%

“…The controllability requirement is imposed via appropriate penalty terms. We also mention [46] based on the Russel principle, extended in [14] and [27,2] for least-squares based method. One the other hand, one may also employ a "control then discretize" procedure, where the optimality system (for instance associated with the control of minimal L 2 norm ) mixing the boundary condition in time and space and involving the primal and adjoint state is discretized within a priori a space-time approximation.…”

Section: Introductionmentioning

confidence: 99%

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

Burman¹,

Feizmohammadi²,

Münch³

et al. 2021

Preprint

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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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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Burman¹,

Feizmohammadi²,

Münch³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…However, at the finite dimensional level (induced by the numerical approximation in time and space), (2) can not be in general solved exactly: in other words, the constraint…”

Section: Numerical Approximation Of Exact Controls For the Wave Equationmentioning

confidence: 99%

Approximation of exact controls for semilinear wave and heat equations through space-time methods

Münch¹

2023

Handbook of Numerical Analysis

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“…Especially in linear cases, this viewpoint naturally leads to an iterative approximation scheme based on a standard descent method. It has already been tested in various scenarios and, at least numerically, it performs very well (see [1], [16], [17], [18], [19]).…”

Section: Final Commentsmentioning

confidence: 99%

A different look at controllability

Pedregal

2014

Preprint

Self Cite

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We explore further controllability problems through a standard least square approach as in [20]. By setting up a suitable error functional E, and putting m(≥ 0) for the infimum, we interpret approximate controllability by asking m = 0, while exact controllability corresponds, in addition, to demanding that m is attained. We also provide a condition, formulated entirely in terms of the error E, which turns out to be equivalent to the unique continuation property, and to approximate controllability. Though we restrict attention here to the 1D, homogeneous heat equation to explain the main ideas, they can be extended in a similar way to many other scenarios some of which have already been explored numerically, due to the flexibility of the procedure for the numerical approximation.

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A variational method for the numerical simulation of a boundary controllability problem for the linear and semilinear 1D wave equations

Cited by 6 publications

References 13 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

Approximation of exact controls for semilinear wave and heat equations through space-time methods

A different look at controllability

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