#### PhD Thesis colloquium

##### Venue: Lecture Hall I,Dept of Mathematics

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.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 28 Nov 2023