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.