Homogenization of certain PDEs and associated optimal control problems on various rough domains

Abstract

This thesis is devoted to the study of the asymptotic behavior of partial differential equations (PDEs) and associated boundary and interior optimal control problems in various oscillatory domains. The present thesis contains six chapters. Chapter 1 is about giving motivation, literature survey, and content of the thesis. The main findings of the research work are presented through chapters 2-5, which are briefly given below. The future scope of the work is given in the concluding Chapter 6.

In Chapter 2, 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. We use appropriate scaling on the controls acting on the oscillating boundary. Depending on the scaling factor, we get different limit control problems, namely, optimal boundary control and interior optimal control problem. Moreover, we visualize the domains as a branched structure, and we introduce unfolding operators to get contributions from each level at every branch.

Chapter 3 deals with an optimal control problem subject to the stationary Stokes system in a three-dimensional domain with a highly oscillating boundary. The controls are acting on the state through the Neumann data on the oscillating part of the boundary with appropriate scaling parameters $\varepsilon^\alpha$ with $\alpha \geq 1$. The periodic unfolding operator is used to characterize the optimal control. Using the unfolding operator, we analyze the asymptotic behavior of the optimal control problem under consideration. For $\alpha=1$, the limit optimal control problem has both boundary and interior controls. For $\alpha>1$, the limit optimal control problem has only boundary controls.

In Chapter 4, homogenization of an elliptic PDE with periodic oscillating coefficients and associated optimal control problems with energy type cost functional is considered. The domain is a 3-dimensional region (method applies to any n dimensional region) with oscillating boundary, where the base of the oscillation is curved, and it is given by a Lipschitz function. Further, we consider general elliptic PDE with oscillating coefficients. We also include very general type functional of Dirichlet type given with oscillating coefficients, which can be different from the coefficient matrix of the state equation. We introduce appropriate unfolding operators and approximate unfolded domains to study the limiting analysis.

The main discussion of Chapter 5 is the study of optimal control problems based on the unfolding method in rough domains. The oscillating part consists of two contrasting diffusivity materials, where one is a nearly insulating type material (low conductivity), and the other material is highly conductive (O(1)). The interesting result is the difference in the limit behavior of the optimal control problem, which depends on the control's action, whether it is on the conductivity part or insulating part. In both cases, we derive the two-scale limit controls problems which are quite similar as far as analysis is concerned. But, if the controls are acting on the conductive region, a complete-scale separation is available, whereas a complete separation is not visible in the insulating case due to the intrinsic nature of the problem. In this case, to obtain the limit optimal control problem in the macro scale, two cross-sectional cell problems are introduced. We do obtain the homogenized equation for the state, but the two-scale cost functional remains as it is.