We demonstrate the asymptotic analysis of a semi-linear optimal control problem posed on a smooth oscillating boundary domain in the present paper. We have considered a more general oscillating domain than the usual “pillar-type” domains. Consideration of such general domains will be useful in more realistic applications like circular domain with rugose boundary. We study the asymptotic behavior of the problem under consideration using a new generalized periodic unfolding operator. Further, we are studying the homogenization of a non-linear optimal control problem and such non-linear problems are limited in the literature despite the fact that they have enormous real-life applications. Among several other technical difficulties, the absence of a sufficient criteria for the optimal control is one of the most attention-grabbing issues in the current setting. We also obtain corrector results in this paper.
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...
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
This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak L 2 convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong L 2 convergence of average preserving extensions of the solutions and their flows is also considered.
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