Optimal control for a parabolic problem in a domain with highly oscillating boundary

Maio, Umberto De; Gaudiello, Antonio; Lefter, Cătălin

doi:10.1080/00036810410001724670

Cited by 38 publications

(14 citation statements)

References 11 publications

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“…The exact controllability of hyperbolic problems with oscillating coefficients in fixed domains is treated in [44] and, in the case of perforated domains, in [8,11]. In [14]÷ [18], [31]÷ [33] and [53], the authors study the optimal control and exact controllability problems in domains with highly oscillating boundary. We refer the reader to [38,39] for the optimal control of hyperbolic problems in composites with imperfect interface and to [42] for the optimal control of rigidity parameters of thin inclusions in composite materials.…”

Section: Introductionmentioning

confidence: 99%

Homogenization and exact controllability for problems with imperfect interface

Monsurrò¹,

Perugia²

2019

Networks &Amp; Heterogeneous Media

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In this paper we study the internal exact controllability for a second order linear evolution equation defined in a two-component domain. On the interface we prescribe a jump of the solution proportional to the conormal derivatives, meanwhile a homogeneous Dirichlet condition is imposed on the exterior boundary. Due to the geometry of the domain, we apply controls through two regions which are neighborhoods of a part of the external boundary and of the whole interface, respectively. Our approach to internal exact controllability consists in proving an observability inequality by using the Lagrange multipliers method. Eventually we apply the Hilbert Uniqueness Method, introduced by J. -L. Lions, which leads to the construction of the exact control through the solution of an adjoint problem. Finally we find a lower bound for the control time depending not only on the geometry of our domain and on the matrix of coefficients of our problem but also on the coefficient of proportionality of the jump with respect to the conormal derivatives.

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Section: Introductionmentioning

confidence: 99%

Homogenization and exact controllability for problems with imperfect interface

Monsurrò¹,

Perugia²

2019

Networks &Amp; Heterogeneous Media

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“…In , he has derived H 1 norm estimates for the homogenized solutions of elliptic and parabolic type PDEs. In , asymptotic analysis of optimal control problems posed on various PDEs were investigated using oscillating test functions method. In , the authors have used the Buttazzo–Dal Maso abstract scheme for variational convergence of constrained minimization problems to study the asymptotic analysis of optimal control problem in thick multi‐level junctions.…”

Section: Introductionmentioning

confidence: 99%

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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“…There is also a large amount of literature on the homogenization with oscillating boundaries, which has tremendous applications as well (for example, , , and ). For some recent work on oscillating boundaries, see and .…”

Section: Introductionmentioning

confidence: 99%

Asymptotic analysis of Neumann periodic optimal boundary control problem

Nandakumaran

Prakash

Sardar

2016

Math Methods in App Sciences

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An optimal boundary control problem in a domain with oscillating boundary has been investigated in this paper. The controls are acting periodically on the oscillating boundary. The controls are applied with suitable scaling parameters. One of the major contribution is the representation of the optimal control using the unfolding operator. We then study the limiting analysis (homogenization) and obtain two limit problems according to the scaling parameters. Another notable observation is that the limit optimal control problem has three controls, namely, a distributed control, a boundary control, and an interface control.

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Optimal control for a parabolic problem in a domain with highly oscillating boundary

Cited by 38 publications

References 11 publications

Homogenization and exact controllability for problems with imperfect interface

Homogenization and exact controllability for problems with imperfect interface

Optimal control problem in a domain with branched structure and homogenization

Asymptotic analysis of Neumann periodic optimal boundary control problem

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