Set-Valued Numerical Analysis and Optimal Control

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