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

Aiyappan, S.; Nandakumaran, A. K.; Sufian, Abu

doi:10.1002/mma.5748

Cited by 9 publications

(7 citation statements)

References 34 publications

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“…Since, ξ 0 and ∇ H u are in (L 2 (Ω × Y )) 2 , we see that u 1 ∈ L 2 (Ω; H 1 #,H (Y )/R). Hence, we have the second convergence.…”

Section: 2mentioning

confidence: 87%

“…For the second part, we will use the test function of the form 2 with div H,y ψ = 0. Let us consider the following,…”

Section: 2mentioning

confidence: 99%

“…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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“…Since, ξ 0 and ∇ H u are in (L 2 (Ω × Y )) 2 , we see that u 1 ∈ L 2 (Ω; H 1 #,H (Y )/R). Hence, we have the second convergence.…”

Section: 2mentioning

confidence: 87%

“…For the second part, we will use the test function of the form 2 with div H,y ψ = 0. Let us consider the following,…”

Section: 2mentioning

confidence: 99%

Section: Homogenization Via Periodic Unfolding Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Unfolding operator on Heisenberg Group and applications in Homogenization

Nandakumaran¹,

Sufian²

2021

Preprint

Self Cite

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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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Asymptotic analysis of an interior optimal control problem governed by Stokes equations

Garg

Sardar

2022

Math Methods in App Sciences

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This article introduces an interior optimal control problem (OCP) in a two-dimensional domain with a highly oscillatory boundary governed by the stationary Stokes equations. We consider the periodic controls in the oscillating region of the domain and use the unfolding operators to characterize the optimal controls. We establish the convergences of optimal control, state, and pressure in a suitable space to the ones of the limit system in a fixed domain.

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

Cited by 9 publications

References 34 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

Asymptotic analysis of an interior optimal control problem governed by Stokes equations

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