Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

Durante, Tiziana; Mel'nyk, Taras A.

doi:10.1007/s10957-009-9604-6

Cited by 24 publications

(11 citation statements)

References 15 publications

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“…In this paper we continue our investigation of optimal control problems in thick multilevel junctions, which we have begun in [10]. Here we improve and generalize our results in the case of the perturbed nonlinear boundary multi-phase interactions and more complicated structure of a thick multilevel junction.…”

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confidence: 53%

“…• Find an optimal control θ * ε ∈ K (1) ε , which with the corresponding state u * ε , minimize the following cost functional 10) where u ε is the unique weak solution to problem (0.4) with the following boundary conditions on the bases of the thin cylinders:…”

Section: Reformulation Of the Problemmentioning

confidence: 99%

“…Therefore boundary-value problems in thick junctions of different types are very extensively investigated at present (see [1][2][3]6,8,10], [19,20,24,25] and references therein). The aim of these researches is to develop rigorous methods to study the asymptotic behavior of solutions when the number of the attached thin domains of a thick junction infinitely increases and their thickness vanishes.…”

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confidence: 99%

“…In [8,10,24] it was shown that processes in thick multi-level junctions behave as a "many-phase system" in the region which is filled up by the thin domains from each level in the limit passage.…”

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confidence: 99%

“…To homogenize problem CP ε in the first case we will use the direct method and approach of our previous paper [10], where we studied a linear optimal control problem involving a plane thick two-level junction of type 2 : 1 : 1. Nevertheless, here we essentially simplify this approach, namely, we prove the convergence results without any special constructions of multilevel extension operators and without any additional assumptions of smoothness for the right-hand sides and controls.…”

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

Durante

Mel'nyk

2011

ESAIM: COCV

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Abstract.We consider quasilinear optimal control problems involving a thick two-level junction Ωε which consists of the junction body Ω0 and a large number of thin cylinders with the cross-section of order O(ε 2 ). The thin cylinders are divided into two levels depending on the geometrical characteristics, the quasilinear boundary conditions and controls given on their lateral surfaces and bases respectively. In addition, the quasilinear boundary conditions depend on parameters ε, α, β and the thin cylinders from each level are ε-periodically alternated. Using the Buttazzo-Dal Maso abstract scheme for variational convergence of constrained minimization problems, the asymptotic analysis (as ε → 0) of these problems are made for different values of α and β and different kinds of controls. We have showed that there are three qualitatively different cases. Application for an optimal control problem involving a thick one-level junction with cascade controls is presented as well.

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Section: Reformulation Of the Problemmentioning

confidence: 99%

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confidence: 99%

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

Durante

Mel'nyk

2011

ESAIM: COCV

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Optimal control problem in a domain with branched structure and homogenization

Aiyappan

Nandakumaran

2016

Math Methods in App Sciences

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We consider a linear parabolic problem in a thick junction domain which is the union of a fixed domain and a collection of periodic branched trees of height of order 1 and small width connected on a part of the boundary. We consider a threebranched structure, but the analysis can be extended to n-branched structures. We use unfolding operator to study the asymptotic behavior of the solution of the problem. In the limit problem, we get a multi-sheeted function in which each sheet is the limit of restriction of the solution to various branches of the domain. Homogenization of an optimal control problem posed on the above setting is also investigated. One of the novelty of the paper is the characterization of the optimal control via the appropriately defined unfolding operators. Finally, we obtain the limit of the optimal control problem. IntroductionIn this article, we consider a parabolic problem in a thick junction domain , > 0, a small parameter, and also the corresponding optimal control problem. Various materials with complex structures including multi-layer thick junctions are widely used in many fields of science. Such structures are usually known as complex structures because of its complexity both in construction and analysis. Other complex structures are perforated domains, composite materials, grid domains, and domains with oscillating boundaries to name a few.As mentioned earlier, constructions with thick junction (also multi-level) are used in many technologies, like microstrip radiator, nano technologies ([1, 2]), biological systems, fractal-type constructions, etc. Studying PDE problems in such complex structures has paramount importance. We refer to the work in [3][4][5][6] and the references therein for the study in multi-branched structures. Although the importance of optimal control may be at the junctions, we consider the controls on the entire oscillating part from which we can also understand the contribution from each branch at each level. One can apply need based controls at the appropriate junctions.The domain under consideration consists of multiple layer thick junctions known as branched structure (Figure 1). Such a domain has a fixed part and lot of thin periodically distributed parts (or handle trees) attached along certain part of the boundary of the domain at different levels. The trees have finite number of branching levels and in this paper, we take three branching levels, but one can consider any finite number of branches. The height of each branch is of O.1/, whereas the thickness is of O. /. We consider the domain in two-dimensional space. Such a domain has already been considered by Mel'nyk ([6]). Indeed the results can be extended to three dimensional problem and higher dimensions as well. Asymptotic analysis for a Robin problem in a thick junction has been investigated in [5]. In fact, our work is motivated from the work of Mel'nyk, where he considers a semi-linear parabolic problem with the source term vanishes on the oscillating interior part. He has studied the problem usin...

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Asymptotic analysis of a boundary optimal control problem on a general branched structure

Aiyappan

Nandakumaran

Sufian

2019

Math Methods in App Sciences

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We consider an optimal control problem posed on a domain with a highly oscillating smooth boundary where the controls are applied on the oscillating part of the boundary. There are many results on domains with oscillating boundaries where the oscillations are pillar-type (non-smooth) while the literature on smooth oscillating boundary is very few. In this article, we use appropriate scaling on the controls acting on the oscillating boundary leading to different limit control problems, namely, boundary optimal control and interior optimal control problem. In the last part of the article, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch. KEYWORDS asymptotic analysis, homogenization, optimal control, oscillating boundary domain, unfolding operator MSC CLASSIFICATION 80M35; 80M40; 49J20; 35B27 Math Meth Appl Sci. 2019;42:6407-6434.wileyonlinelibrary.com/journal/mma

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Asymptotic Analysis of an Optimal Control Problem Involving a Thick Two-Level Junction with Alternate Type of Controls

Cited by 24 publications

References 15 publications

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

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

Optimal control problem in a domain with branched structure and homogenization

Asymptotic analysis of a boundary optimal control problem on a general branched structure

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