Oscillating PDE in a rough domain with a curved interface: Homogenization of an Optimal Control Problem

Nandakumaran, A. K.; Sufian, Abu

doi:10.1051/cocv/2020045

Cited by 8 publications

(4 citation statements)

References 32 publications

(40 reference statements)

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“…We are under investigation of the applicability of the introduced unfolding operator in particular to optimal control problems. At this stage we would like to recall that the unfolding operator can be used to characterize the optimal control in homogenization problem (see [2,12,14]).…”

Section: Homogenization Via Periodic Unfolding Operatormentioning

confidence: 99%

“…Observe that definition of two-scale convergence reduced to weakly convergence in L 2 (Ω × Y ) and it is easy to apply as it is technicality less demanding. There are some advantages of using this method; for example, while doing optimal control problems in periodic setup, the optimal control is easily characterized by the unfolding of the adjoint state, which helps to analyze asymptotic behavior see [2,3,12,13,14]. This method reduces the definition of two-scale convergence in L p (Ω) to weak convergence of the unfolding sequence in L p (Ω × Y ) for 1 < p < ∞.…”

Section: Introductionmentioning

confidence: 99%

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Unfolding operator on Heisenberg Group and applications in Homogenization

Nandakumaran¹,

Sufian²

2021

Preprint

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The periodic unfolding method is one of the latest tool after multi-scale convergence to study multi-scale problems like homogenization problems. It provides a good understanding of various micro scales involved in the problem which can be conveniently and easily applied to get the asymptotic limit. In this article, we develop the periodic unfolding for the Heisenberg group which has a non-commutative group structure. In order to do this, the concept of greatest integer part, fractional part for the Heisenberg group have been introduced corresponding to the periodic cell. Analogous to the Euclidean unfolding operator, we prove the integral equality, L 2 -weak compactness, unfolding gradient convergence and other related properties. Moreover, we have the adjoint operator for the unfolding operator which can be recognized as an average operator. As an application of the unfolding operator, we have homogenized the standard elliptic PDE with oscillating coefficients. We have also considered an optimal control problem and characterized the interior periodic optimal control in terms of unfolding operator.

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Section: Homogenization Via Periodic Unfolding Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unfolding operator on Heisenberg Group and applications in Homogenization

Nandakumaran¹,

Sufian²

2021

Preprint

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“…Later on, in [49] the authors deal with the case of a Neumann boundary value problem in the same framework. In [2]- [5], [12]- [16], [30]- [32], [55]- [58] optimal control and exact controllability problems in domains with highly oscillating boundary are studied. Moreover we refer to [42] and [9,10,11] for the exact controllability of hyperbolic problems with oscillating coefficients in fixed and in perforated domains respectively, to [36,37] and [34,35,54] for the optimal control and the exact controllability, respectively, of hyperbolic problems in composites with imperfect interface.…”

Section: γ Interface Boundary ∂ωmentioning

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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“…Significant works are carried out for the homogenization of problems on highly oscillating boundaries; we refer to the reader (cf. other works [11][12][13][14][15][16][17][18][19] ).…”

Section: Introductionmentioning

confidence: 99%

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

Nandakumaran

Sankar

2022

Math Methods in App Sciences

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Award Number: No.F.4-2/2006 (BSR)/MA/19-20/0060In this article, we study the homogenization of heat equations in a domain with randomly oscillating boundary parts. The random oscillating boundary is time-dependent and confined by a stationary random field. Here, we follow a new homogenization technique that deals with the evolving domains, which covers many applications. We obtain the asymptotic limit as 𝜀 → 0 in the reference configuration, in which the heat equation becomes a parabolic equation with random oscillating coefficients in the reference domain. To the best of our knowledge, this is the first result of the homogenization of problems on the random evolving boundary domain. One of the major contributions is the corrector result which we establish in this article.

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Oscillating PDE in a rough domain with a curved interface: Homogenization of an Optimal Control Problem

Cited by 8 publications

References 32 publications

Unfolding operator on Heisenberg Group and applications in Homogenization

Unfolding operator on Heisenberg Group and applications in Homogenization

Homogenization and exact controllability for problems with imperfect interface

Homogenization of the heat equation in a noncylindrical domain with randomly oscillating boundary

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