Optimal control theory is concerned with
finding control functions that minimize cost functions for systems
described by differential equations. The methods have found
widespread applications in aeronautics, mechanical engineering, the
life sciences, and many other disciplines.
This book focuses on optimal control problems where the state
equation is an elliptic or parabolic partial differential
equation. Included are topics such as the existence of optimal
solutions, necessary optimality conditions and adjoint equations,
second-order sufficient conditions, and main principles of selected
numerical techniques. It also contains a survey on the
Karush-Kuhn-Tucker theory of nonlinear programming in Banach
spaces.
The exposition begins with control problems with linear equations,
quadratic cost functions and control constraints. To make the book
self-contained, basic facts on weak solutions of elliptic and
parabolic equations are introduced. Principles of functional analysis
are introduced and explained as they are needed. Many simple examples
illustrate the theory and its hidden difficulties. This start to the
book makes it fairly self-contained and suitable for advanced
undergraduates or beginning graduate students.
Advanced control problems for nonlinear partial differential
equations are also discussed. As prerequisites, results on boundedness
and continuity of solutions to semilinear elliptic and parabolic
equations are addressed. These topics are not yet readily available
in books on PDEs, making the exposition also interesting for
researchers.
Alongside the main theme of the analysis of problems of optimal
control, Tröltzsch also discusses numerical techniques. The
exposition is confined to brief introductions into the basic ideas in
order to give the reader an impression of how the theory can be
realized numerically. After reading this book, the reader will be
familiar with the main principles of the numerical analysis of
PDE-constrained optimization.
Readership
Graduate students and research mathematicians interested in
optimal control theory and PDEs.
