Set-Valued Numerical Analysis and Optimal Control

R. Baier, M. Gerdts: Set-Valued Numerical Analysis and Optimal Control
in: Lecture Notes for the DAAD Intensive Course ``Optimization - Theory and Applications'' (July 05 -17, 2005), 264 pages, July 2005

Keywords: set-valued numerical analysis; optimal control; convex analysis; numerical solvers for ordinary differential equations; differential inclusions; discrete approximations; direct and indirect discretization methods for optimal control; sensitivity analysis; dynamic programming; arithmetic set operations; set-valued maps; reachable sets; necessary optimality conditions
Mathematics Subject Classification Code: 49J53 (49J15 52A20 49J21 65L05 34A60 54C60 49M25 49K40 93B03 65D30 65L06 90C46 28B20 26E25)
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This booklet covers set-valued numerical analysis and numerical methods for optimal control. Basic material in Convex and Set-Valued Analysis with a focus on arithmetical set operations, parametrizations of sets and set-valued mappings as well as numerical methods for ordinary differential equations and boundary value problems (one-step methods, shooting method) with a focus on necessary optimality conditions and sensitivity analysis are studied. As applications direct and indirect methods discretizing optimal control problems and set-valued quadrature and Runge-Kutta methods are considered.


1. Introduction
2. Preliminaries - Some Known Subdifferentials
2. Examples and Applications
3. Convex Analysis
3.1 Convex Sets
3.1.1 Basic Definitions and Properties
3.1.2 Extreme Sets
3.1.3 Separation Theorems
3.1.4 Support Function, Supporting Faces, Exposed Sets
3.1.5 Representation of Convex Sets
3.2 Arithmetic Set Operations
3.2.1 Definitions and First Properties
3.2.2 Properties of Support Functions
3.2.3 Properties of Supporting Faces
3.2.4 Metrics for Sets
4. Set-Valued Integration
4.1 Set-Valued Maps
4.2 Properties of Measurable Set-Valued Maps
4.3 Set-Valued Integrals
4.3.1 Riemann-Integral
4.3.2 Aumann's Integral
5. Numerical Solution of IVP's
5.1 Existence and Uniqueness
5.2 One-Step Methods
5.3 Convergence of One-Step Methods
5.4 Step-Size Control
5.5 Sensitivity Analysis
6. Discrete Approximation of Reachable Sets 135
6.1 Set-Valued Quadrature Methods/td>
6.2 Appropriate Smoothness of Set-Valued Mappings
6.3 Reachable Sets/Differential Inclusions
6.4 Set-Valued Combination Methods
6.5 Set-Valued Runge-Kutta Methods
6.5.1 Euler's Method
6.5.2 Modified Euler Method
7. Discrete Approximation of Optimal Control
7.1 Minimum Principles
7.2 Indirect Methods and Boundary Value Problems<<br />
7.2.1 Single Shooting
7.2.2 Multiple Shooting
7.3 Direct Discretization Methods
7.3.1 Euler Discretization
7.4 Necessary Conditions and SQP Methods
7.4.1 Necessary Optimality Conditions
7.4.2 Sequential Quadratic Programming (SQP)
7.5 Computing Gradients
7.5.1 Sensitivity Equation Approach
7.5.2 Adjoint Equation Approach
7.6 Discrete Minimum Principle
7.7 Convergence
7.8 Direct Shooting Method
7.9 Grid Refinement
7.10 Dynamic Programming
7.10.1 The Discrete Case
7.10.2 The Continuous Case
8. Examples and Applications Revisited
A. Appendix
A.1 Matrix Norms
A.2 Measurable Functions
A.3 Functions with Bounded Variation and Absolutely Continuous Functions
A.4 Additional Results
B. References

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