Analytical Background and Optimality Theory

Abstract

This chapter provides an introduction to the analytical background and optimality theory for optimization problems with partial differential equations (PDEs). Optimization problems with PDE-constraints are posed in infinite dimensional spaces. Therefore, functional analytic techniques, function space theory, as well as existence- and uniqueness results for the underlying PDE are essential to study the existence of optimal solutions and to derive optimality conditions. These results form the foundation of efficient optimization methods in function space, their adequate numerical realization, mesh independence results and error estimators. The chapter provides first an introduction to the necessary background in functional analysis, Sobolev spaces and the theory of weak solutions for elliptic and parabolic PDEs. These ingredients are then applied to study PDE-constrained optimization problems. Existence results for optimal controls, derivative computations by the sensitivity and adjoint approaches and optimality conditions for problems with control-, state- and general constraints are considered. All concepts are illustrated by elliptic and parabolic optimal control problems. Finally, the optimal control of instationary incompressible Navier-Stokes flow is considered.