Homogenization of optimal control problems governed by PDEs with oscillating coefficients in a domain with a highly oscillating boundary

Abstract

In this thesis, we explore the homogenization of the optimal control problems which involves oscillating coefficients and also involving domains with oscillating boundaries (Prototype domain are shown in Figure 1.1 ). There are six chapters in this thesis. The motivation, literature review, and preliminary are covered in Chapter 1. The following is a concise summary of chapters two through five, which present the research work’s primary findings. The last Chapter 6 provides the work’s future scope. In Chapter 2, we consider the optimal control problem governed by the wave equation with oscillating coefficients (given by matrix Aϵ) in a 2-dimensional oscillating domain Ωϵ. The domain Ωϵ consists of a fixed lower part denoted by Ω− and an oscillating upper part denoted by Ω+ϵ (see Figure 1.2). Homogeneous Neumann condition is considered on the boundary of the domain. The control is applied in the interior of the domain. We take into account the Dirichlet cost functional with an oscillating coefficient (given by matrix Bϵ), which may differ from the coefficients of the wave equation. We obtain the homogenized problem and introduce the limit optimal control problems with appropriate cost-functionals. It is found that the coefficients in the limit cost functional and the adjoint state have the contribution from both the matrices. Finally, we prove the convergence of the optimal solution, optimal state and associated adjoint solution. In Chapter 3, we explore the homogenization of an optimal control problem driven by a semi-linear parabolic equation within a two-dimensional oscillating domain, denoted as Ωϵ. The state equation and cost function in this scenario involve periodic coefficients, Aϵ and Bϵ, which exhibit significant oscillations. We analyze the limiting behavior of both the optimal control and the corresponding state as the oscillations become increasingly fine. Furthermore, we obtain the optimal control problem that encapsulates the effects of these oscillating coefficients and also, we establish a corrector result for the state variable. In Chapter 4, we investigate the boundary optimal control problem associated with the Heat equation in a 2- dimensional highly oscillating domain Ωϵ in which the control is applied periodically via Neumann condition on the oscillating part of the boundary. We characterize the optimal control in terms of unfolding operator and then study the homogenization to obtain two limit optimal control problems depending on the scalar parameter α. In the limit optimal control problem, we obtain three controls, namely interior control, an interfacial control and a boundary control. The Chapter 5 introduces boundary optimal control problem in an N-dimensional domain governed by the stationary Stokes equations. Controls are applied to the states through Neumann data on the boundary. we study the asymptotic behavior of optimal control and states variable. We obtain the limit optimal control problem in the framework of the two-scale convergence and also prove strong convergence of L2− cost functional. One of the main contribution is proving a corrector result for velocities.