Baier R., Gerdts M. Set-Valued Numerical Analysis and Optimal Control

Sofia: Robert Baier and Matthias Gerdts, 2005. — 264 p.Introduction
Examples and Applications
Convex Analysis
Convex Sets
Basic Definitions and Properties
Extreme Sets
Separation Theorems
Support Function, Supporting Faces, Exposed Sets
Representation of Convex Sets
Arithmetic Set Operations
Definitions and First Properties
Properties of Support Functions
Properties of Supporting Faces
Metrics for Sets
Set-Valued Integration
Set-Valued Maps
Properties of Measurable Set-Valued Maps
Set-Valued Integrals
Riemann-Integral
Aumann's Integral
Numerical Solution of IVP's
Existence and Uniqueness
One-Step Methods
Convergence of One-Step Methods
Step-Size Control
Sensitivity Analysis
Discrete Approximation of Reachable Sets
Set-Valued Quadrature Methods
Appropriate Smoothness of Set-Valued Mappings
Reachable Sets/Differential Inclusions
Set-Valued Combination Methods
Set-Valued Runge-Kutta Methods
Euler's Method
Modified Euler Method
Discrete Approximation of Optimal Control
Minimum Principles
Indirect Methods and Boundary Value Problems
Single Shooting
Multiple Shooting
Direct Discretization Methods
Euler Discretization
Necessary Conditions and SQP Methods
Necessary Optimality Conditions
Sequential Quadratic Programming (SQP)
Computing Gradients
Sensitivity Equation Approach
Adjoint Equation Approach
Discrete Minimum Principle
Convergence
Direct Shooting Method
Grid Refinement
Dynamic Programming
The Discrete Case
The Continuous Case
Examples and Applications Revisited
Appendix
Matrix Norms
Measurable Functions
Functions with Bounded Variation and Absolutely Continuous Functions
Additional Results
References