The intention of this textbook is to provide both, the theoretical and computational tools that are necessary to investigate and to solve optimal control problems with ordinary differential equations and differential-algebraic equations. An emphasis is placed on the interplay between the continuous optimal control problem, which typically is defined and analyzed in a Banach space setting, and discrete optimal control problems, which are obtained by discretization and lead to finite dimensional optimization problems.

The book addresses primarily master and PhD students as well as researchers in applied mathematics, but also engineers or scientists with a good background in mathematics and interest in optimal control. The theoretical parts of the book require some knowledge of functional analysis, the numerically oriented parts require knowledge from linear algebra and numerical analysis. Examples are provided for illustration purposes.

Autorentext

Matthias Gerdts, Universität der Bundeswehr München, Germany.

Inhalt

1 Introduction
2 Basics from Functional Analysis
2.1 Vector Spaces
2.2 Mappings, Dual Spaces, and Properties
2.3 Function Spaces
2.4 Stieltjes Integral
2.5 Set Arithmetic
2.6 Separation Theorems
2.7 Derivatives
2.8 Variational Equalities and Inequalities
3 Infinite and Finite Dimensional Optimization Problems
3.1 Problem Classes
3.2 Existence of a Solution
3.3 Conical Approximation of Sets
3.4 First Order Necessary Conditions of Fritz-John Type
3.5 Constraint Qualifications
3.6 Necessary and Sufficient Conditions in Finte Dimensions
3.7 Perturbed Nonlinear Optimization Problems
3.8 Numerical Methods
3.9 Duality
3.10 Mixed-Integer Nonlinear Programs and Branch&Bound
4 Local Minimum Principles
4.1 Local Minimum Principles for Index-2 Problems
4.2 Local Minimum Principles for Index-1 Problems
5 Discretization Methods for ODEs and DAEs
5.1 General Discretization Theory
5.2 Backward Differentiation Formulae (BDF)
5.3 Implicit Runge-Kutta Methods
5.4 Linearized Implicit Runge-Kutta Methods
6 Discretization of Optimal Control Problems
6.1 Direct Discretization Methods
6.2 Calculation of Gradients
6.3 Numerical Example
6.4 Discrete Minimum Principle and Approximation of Adjoints
6.5 Convergence
7 Selected Applications and Extensions
7.1 Mixed-Integer Optimal Control
7.2 Open-Loop-Real-Time Control
7.3 Dynamic Parameter Identification