A simple characterization of H-convergence for a class of nonlocal problems

Bellido, José C.; Evgrafov, Anton

doi:10.1007/s13163-020-00349-9

Cited by 4 publications

(3 citation statements)

References 12 publications

(18 reference statements)

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“…A more accurate analysis could provide the proof of the weak convergence in X 0 of (26). As far as the authors know, this result has only been proved for certain kind of kernels (see [11,9]). See also [4,Th.…”

Section: Remarkmentioning

confidence: 99%

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Optimal design problems governed by the nonlocal p -Laplacian equation

Andrés¹,

Múñoz²,

Rosado³

2021

Mathematical Control &Amp; Related Fields

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In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal p-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal p-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288-310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed.

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Section: Remarkmentioning

confidence: 99%

“…Even though there are not so many results about explicit computations of the limit, we can find some theoretical advances. In this sense we must underline [34,11,9].…”

mentioning

confidence: 99%

Optimal design problems governed by the nonlocal p -Laplacian equation

Andrés¹,

Múñoz²,

Rosado³

2021

Mathematical Control &Amp; Related Fields

View full text Add to dashboard Cite

show abstract

“…We can find some theoretical advances about the explicit computation of the limit problem. In this sense we must underline among others [35,11,10,50,25]. Much more should be commented about the influence that this type of problems has received from an outstanding list of seminal papers whose main topic, has been the analysis and characterization of Sobolev Spaces.…”

Section: 21mentioning

confidence: 99%

Local and nonlocal optimal control in the source

Múñoz¹

2020

Preprint

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The analysis of an optimal control problem of nonlocal type is analyzed. The results obtained are applied to the study the corresponding local optimal control problems. The state equations are governed by p-laplacian elliptic operators, of local and nonlocal type, and the costs belong to a wide class of integral functionals. The nonlocal problem is formulated by means of a convolution of the states with a kernel. This kernel depends on a parameter, called horizon which, is responsible for the nonlocality of the equation. The input function is the source of the elliptic equation. Existence of nonlocal controls is obtained and a G-convergence result is employed in this task. The limit of the solutions of the nonlocal optimal control, when the horizon tends to zero, is analyzed and compared to the solution of the underlying local optimal control problem.

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A simple characterization of H-convergence for a class of nonlocal problems

Cited by 4 publications

References 12 publications

Optimal design problems governed by the nonlocal p -Laplacian equation

Optimal design problems governed by the nonlocal p -Laplacian equation

Local and nonlocal optimal control in the source

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