We study asymptotic analysis (homogenization) of second-order partial differential equations(PDEs) posed on an oscillating domain. In general, the motivation for studying problems defined on oscillating domains, come from the need to understand flow in channels with rough boundary, heat transmission in winglets, jet engins and so on. There are various methods developed to study homogenization problems namely; multi-scale expansion, oscillating test function method, compensated compactness, two-scale convergence, block-wave method, method of unfolding etc.

In this thesis, we consider a two dimensional oscillating domain (comb shape
type) `$\Omega_{\epsilon}$`

consists of a fixed bottom region `$\Omega^-$`

and an oscillatory
(rugose) upper region $\Omega_{\epsilon}^{+}$. We introduce an optimal control problems in
`$\Omega_{\epsilon}$`

for the Laplacian operator. There are mainly two types of optimal
control problems; namely distributed control andboundary control. For distributed control
problems in the oscillatingdomain, one can put control on the oscillating part or on the fixed
part and similarly for boundary control problem (control on the oscillatingboundary or on the
fixed part the boundary). Considering controls on theoscillating part is more interesting and
challenging than putting control on fixed part of the domain. Our main aim is to characterize
the controlsand study the limiting analysis (as `$\epsilon \to 0$`

) of the optimalsolution.

In the thesis, we consider all the four cases, namely distributed and boundary controls both
on the oscilalting part and away from the oscillating part. Since, controls on the oscillating
part is more exciting, in this talk, we present the details of two sections. First we consider
distributed optimal control problem, where the control is supported on the oscillating part
`$Omega_{\epsilon}^{+}$`

with periodic controls and with Neumann condition on the oscillating
boundary `$\gamma_{\epsilon}$`

. Secondly, we introduce boundary optimal control
problem, control applied through Neumann boundary condition on the oscillating boundary
`$\gamma_{\epsilon}$`

with suitable scaling parameters. We characterize the optimal control
using unfolding and boundary unfolding operators and study limiting analysis. In the limit, we
obtain two limit problems according to the scaling parameters and we observe that limit
optimal control problem has three control namely; a distributed control, a boundary control
and an interface control.

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Last updated: 11 Apr 2024