LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM FOR ELLIPTIC PARTIAL DIFFERENTIAL EQUATION

Abstract

This project centers on the optimal control of partial differential equations. It starts by examining optimal control problems in finite-dimensional settings, with examples that encompass both convex and non-convex cases. The project delves into various approaches to demonstrate the existence and uniqueness of control problems, such as the Lagrange multiplier method and the Karush- Kuhn-Tucker framework. Later on, the project develops into optimal control problems in infinitedimensional cases, where the control problems are in Lipschitz domains. Here, we focus only on linear elliptic partial differential equations, since a satisfactory regularity theory is available for the solutions to such equations. Elliptic equations may describe an electrical potential, a stationary distribution of temperature, a scattered field, or a velocity potential. The investigation encompasses the analysis of the existence and uniqueness of weak solutions to elliptic equations, along with an exploration of normed spaces, Sobolev spaces, and adjoint operators. The study also examines trace operators for continuous and compact embeddings of spaces, as well as the concept of differentiability in Banach spaces, which is crucial for determining the necessary condition for function optimization. Additionally, we employ the Karush-Kuhn-Tucker system to create an optimal solution for the quadratic problem in an elliptic partial differential equation.