Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1

Durante, Tiziana; Mel'nyk, Taras A.

doi:10.1051/cocv/2011107

Cited by 31 publications

(18 citation statements)

References 15 publications

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“…Such successful applications of thick-junction constructions have stimulated active learning BVP's in thick junctions with more complex structures: thick junctions with the thin junction's body [2,3,4], thick multi-level junctions [8,9,18], thick cascade junctions [5,17], where new qualitative results were obtained. Specifically, it was shown that processes in thick multi-level junctions behave as a many-phase system and thick cascade junctions have new kind of eigenvibrations.…”

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confidence: 99%

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Mel'nyk¹

2015

Journal of Mathematical Analysis and Applications

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We consider a semi-linear parabolic problem in a model plane thick fractal junction Ω ε , which is the union of a domain Ω 0 and a lot of joined thin trees situated ε-periodically along some interval on the boundary of Ω 0 . The trees have finite number of branching levels. The following nonlinear Robin boundary conditionε is given on the boundaries of the branches from the i-th branching layer; α i and β i are real parameters. The asymptotic analysis of this problem is made as ε → 0, i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the corresponding homogenized problem is found and the existence and uniqueness of its solution in an anizotropic Sobolev space of multi-sheeted functions is proved. We construct the asymptotic approximation for the solution v ε and prove the corresponding asymptotic estimate in the space C [0, T ]; L 2 (Ω ε ) ∩ L 2 0, T ; H 1 (Ω ε ) , which shows the influence of the parameters {α i } and {β i } on the asymptotic behavior of the solution.

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confidence: 99%

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Mel'nyk¹

2015

Journal of Mathematical Analysis and Applications

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“…On the one hand, the model is of interest in many branches of science and technology: we refer to [13][14][15][16] and [17] for other specific models, as well as for further references on these models.…”

Section: State Of the Art In The Literaturementioning

confidence: 99%

“…Forests of cylinders are also considered when dealing with the homogenization of the so-called thick junction in multistructures. In this framework, let us mention, for example, [14,33] and [34] for different problems and the bibliography therein. In contrast with the problem here considered, in these papers ( [14,33,34]) the thin tubes are part of the domain (which has a strongly oscillating boundary) where the differential equation is posed, and they have a diameter of the same size of the periodicity O.…”

Section: State Of the Art In The Literaturementioning

confidence: 99%

On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems

Gómez

Лобо

Pérez

et al. 2014

Math Methods in App Sciences

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Let uϵ be the solution of the Poisson equation in a domain normalΩ⊂double-struckR3 perforated by thin tubes with a nonlinear Robin‐type boundary condition on the boundary of the tubes (the flux here being β(ϵ)σ(x,uϵ)), and with a Dirichlet condition on the rest of the boundary of Ω. ϵ is a small parameter that we shall make to go to zero; it denotes the period of a grid on a plane where the tubes/cylinders have their bases; the size of the transversal section of the tubes is O(aϵ) with aϵ≪ϵ. A certain nonperiodicity is allowed for the distribution of the thin tubes, although the perimeter is a fixed number a. Here, σ∈C1(falsenormalΩ¯×double-struckR) is a strictly monotonic function of the second argument, and the adsorption parameter β(ϵ) > 0 can converge toward infinity. Depending on the relations between the three parameters ϵ, aϵ, and β(ϵ), the effective equations in volume are obtained. Among the multiple possible relations, we provide critical relations, which imply different averages of the process ranging from linear to nonlinear. All this allows us to derive spectral convergence as ϵ→0 for the associated spectral problems in the case of σ a linear function of uϵ. Copyright © 2014 John Wiley & Sons, Ltd.

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“…Homogenization of oscillating boundaries with fixed amplitude is widely studied and we refer to the following main papers: [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [15], [21], [22], [23], [24], [26], [27], [28], [29], [30], [31], [32], [41] [42], [45], [46], [47], [48], and [53].…”

Section: Introductionmentioning

confidence: 99%

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

Gaudiello¹,

Lenczner²

2020

SIAM J. Appl. Math.

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A periodic homogenization model of the electrostatic equation is constructed for a comb drive with a large number of fingers and whose mode of operation is in-plane and longitudinal. The model is obtained in the case where the distance between the rotor and the stator is of an order ε α , α ≥ 2, where ε denotes the period of distribution of the fingers. The model derivation uses the two-scale convergence technique. Strong convergences are also established. This allows us to find, after a proper scaling, the limit of the electrostatic force applied to the rotor in the longitudinal direction.

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Homogenization of quasilinear optimal control problems involving a thick multilevel junction of type 3 : 2 : 1

Cited by 31 publications

References 15 publications

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

Asymptotic approximation for the solution to a semi-linear parabolic problem in a thick junction with the branched structure

On critical parameters in homogenization of perforated domains by thin tubes with nonlinear flux and related spectral problems

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

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