Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization

Nandakumaran, A. K.; Prakash, Ravi; Sardar, Bidhan Chandra

doi:10.1137/140994575

Cited by 29 publications

(12 citation statements)

References 22 publications

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“…By the continuity of the unfolding operator, we have

‖ T^{ε} ū_{ε} {false‖}_{L^{2} ({(0, 1)}^{2}, H^{1} (𝒢))} ⩽ ‖ ū_{ε} ‖_{H^{1} ({normalΩ}_{ε})} .

Using the estimate , we get

false‖ {\overset{θ}{true}}_{ε} ‖_{L^{2} false(γ false)} ⩽ C

. Hence, we have

(||, \int_{S^{ε}} {\overset{\overset{θ}{true}}{true}}_{ε}^{ε} ū_{ε}) ⩽ C false‖ ū_{ε} {false‖}_{H^{1} false(Ω_{ε} false)} .

From the Lemma 5.6 in Nandakumaran et al, we have the following:

({, \begin{matrix} left left \\ array & array |\int_{F_{M_{i}}^{ε}} {\hat{\bar{θ}}}_{ε}^{ε} ū_{ε}| ⩽ C ‖ u_{ε} {false‖}_{H^{1} ({normalΩ}_{ε})}, for i = 0, 1 \\ array & array |\int_{⋃_{k = 1}^{N} F_{t_{k}}^{ε}} {\hat{\bar{θ}}}_{ε}^{ε} ū_{ε}| ⩽ C ‖ u_{ε} {false‖}_{H^{1} ({normalΩ}_{ε})} . \end{matrix})

From Equation and inequalities (, ) give, …”

Section: Some Preliminary Results and A Priori Estimatementioning

confidence: 88%

“…In other works, control problems where the controls are acting away from the oscillating part of the domain have been investigated with certain error estimates. Recently, Nandakumaran, Prakash, and Sardar considered a second‐order elliptic interior optimal control problem. In their work, they have used the method of periodic unfolding to characterize the control and studied asymptotic behavior.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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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“…By the continuity of the unfolding operator, we have

‖ T^{ε} ū_{ε} {false‖}_{L^{2} ({(0, 1)}^{2}, H^{1} (𝒢))} ⩽ ‖ ū_{ε} ‖_{H^{1} ({normalΩ}_{ε})} .

Using the estimate , we get

false‖ {\overset{θ}{true}}_{ε} ‖_{L^{2} false(γ false)} ⩽ C

. Hence, we have

(||, \int_{S^{ε}} {\overset{\overset{θ}{true}}{true}}_{ε}^{ε} ū_{ε}) ⩽ C false‖ ū_{ε} {false‖}_{H^{1} false(Ω_{ε} false)} .

From the Lemma 5.6 in Nandakumaran et al, we have the following:

({, \begin{matrix} left left \\ array & array |\int_{F_{M_{i}}^{ε}} {\hat{\bar{θ}}}_{ε}^{ε} ū_{ε}| ⩽ C ‖ u_{ε} {false‖}_{H^{1} ({normalΩ}_{ε})}, for i = 0, 1 \\ array & array |\int_{⋃_{k = 1}^{N} F_{t_{k}}^{ε}} {\hat{\bar{θ}}}_{ε}^{ε} ū_{ε}| ⩽ C ‖ u_{ε} {false‖}_{H^{1} ({normalΩ}_{ε})} . \end{matrix})

From Equation and inequalities (, ) give, …”

Section: Some Preliminary Results and A Priori Estimatementioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

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

Aiyappan

Nandakumaran

Sufian

2019

Math Methods in App Sciences

Self Cite

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“…Homogenization of oscillating boundaries with fixed amplitude is widely studied and we refer to the following main papers: [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [15], [21], [22], [23], [24], [26], [27], [28], [29], [30], [31], [32], [41] [42], [45], [46], [47], [48], and [53].…”

Section: Introductionmentioning

confidence: 99%

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

Gaudiello¹,

Lenczner²

2020

SIAM J. Appl. Math.

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A periodic homogenization model of the electrostatic equation is constructed for a comb drive with a large number of fingers and whose mode of operation is in-plane and longitudinal. The model is obtained in the case where the distance between the rotor and the stator is of an order ε α , α ≥ 2, where ε denotes the period of distribution of the fingers. The model derivation uses the two-scale convergence technique. Strong convergences are also established. This allows us to find, after a proper scaling, the limit of the electrostatic force applied to the rotor in the longitudinal direction.

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“…The domain is a standard one considered by many authors in the literature. See, for example , and so on. The domain Ω ε consists of bottom and upper parts, respectively, denoted by Ω − and

{normalΩ}_{ε}^{+}

(Figure ).…”

Section: Introductionmentioning

confidence: 99%

“…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 . For general literature in homogenization, we refer to and the reference therein.…”

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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Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization

Cited by 29 publications

References 22 publications

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

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

A Two-Dimensional Electrostatic Model of Interdigitated Comb Drive in Longitudinal Mode

Asymptotic analysis of Neumann periodic optimal boundary control problem

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