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.
About the Author
Matthias Gerdts, Universität der Bundeswehr München, Germany.
Table of Contents
1 Introduction2 Basics from Functional Analysis2.1 Vector Spaces2.2 Mappings, Dual Spaces, and Properties2.3 Function Spaces2.4 Stieltjes Integral2.5 Set Arithmetic2.6 Separation Theorems2.7 Derivatives2.8 Variational Equalities and Inequalities3 Infinite and Finite Dimensional Optimization Problems3.1 Problem Classes3.2 Existence of a Solution3.3 Conical Approximation of Sets3.4 First Order Necessary Conditions of Fritz-John Type3.5 Constraint Qualifications3.6 Necessary and Sufficient Conditions in Finte Dimensions3.7 Perturbed Nonlinear Optimization Problems3.8 Numerical Methods3.9 Duality3.10 Mixed-Integer Nonlinear Programs and Branch&Bound4 Local Minimum Principles4.1 Local Minimum Principles for Index-2 Problems4.2 Local Minimum Principles for Index-1 Problems5 Discretization Methods for ODEs and DAEs5.1 General Discretization Theory5.2 Backward Differentiation Formulae (BDF)5.3 Implicit Runge-Kutta Methods5.4 Linearized Implicit Runge-Kutta Methods6 Discretization of Optimal Control Problems6.1 Direct Discretization Methods6.2 Calculation of Gradients6.3 Numerical Example6.4 Discrete Minimum Principle and Approximation of Adjoints6.5 Convergence7 Selected Applications and Extensions7.1 Mixed-Integer Optimal Control7.2 Open-Loop-Real-Time Control7.3 Dynamic Parameter Identification