Introduction

In this lecture which is the first in a series of four lectures we will lay down the foundation for the ideas presented later. We are concerned with mathematical tools that will enable us the analysis and the construction of efficient algorithms for the solution of shape optimization problems governed by fluid dynamics models, ranging from the full potential equation to the full compressible Navier-Stokes equations. We restrict all our discussions to gradient based methods. We begin this lecture with a short review of basic ideas in optimization where we start with algebraic problems and constraints. We derive the optimality conditions for such cases as an introduction to our problems of interest which include optimal control and shape design. We demonstrate the derivation of the optimality conditions for a control problem and discuss the case of finite dimensional control as well as the infinite dimensional control case. Our last topic is the derivation of optimality conditions for shape design problem. We derive the variation of functionals with respect to the domain (shape) of integration. Examples for optimal shape design problems which minimize the deviation of the pressure from a given pressure distribution, are given for the full potential equation and the Euler equation.