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Table of Contents
1 Cover
4 Preface
5 1 Algebraic Background for Numerical Methods, Control Theory and Renormalization 1.1. Introduction 1.2. Hopf algebras: general properties 1.3. Connected Hopf algebras 1.4. Pre-Lie algebras 1.5. Algebraic operads 1.6. Pre-Lie algebras (continued) 1.7. Other related algebraic structures 1.8. References
6 2 From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras 2.1. Introduction 2.2. Generalized iterated integrals 2.3. Advances in chronological calculus 2.4. Rota–Baxter algebras 2.5. References
7 3 Noncommutative Symmetric Functions, Lie Series and Descent Algebras 3.1. Introduction 3.2. Classical symmetric functions 3.3. Noncommutative symmetric functions 3.4. Lie series and Lie idempotents 3.5. Lie idempotents as noncommutative symmetric functions 3.6. Decompositions of the descent algebras 3.7. Decompositions of the tensor algebra 3.8. General deformations 3.9. Lie quasi-idempotents as Lie polynomials 3.10. Permutations and free quasi-symmetric functions 3.11. Packed words and word quasi-symmetric functions 3.12. References
8 4 From Runge–Kutta Methods to Hopf Algebras of Rooted Trees 4.1. Numerical integration methods for ordinary differential equations 4.2. Algebraic theory of Runge–Kutta methods 4.3. B-series and related formal expansions 4.4. Hopf algebras of rooted trees 4.5. References
9 5 Combinatorial Algebra in Controllability and Optimal Control 5.1. Introduction 5.2. Analytic foundations 5.3. Controllability and optimality 5.4. Product expansions and realizations 5.5. References
10 6 Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application 6.1. The Butcher group and the Connes–Kreimer algebra 6.2. Character groups of graded and connected Hopf algebras 6.3. Controlled groups of characters 6.4. Appendix: Calculus in locally convex spaces 6.5. References
12 Index
List of Illustrations
1 Chapter 5Figure 5.1. Parallel parking a car (bicycle). For a color version of this figure...Figure 5.2. The states of the bicycleFigure 5.3. An inverted pendulumFigure 5.4. Open-loop and closed-loop controls with a feedback controller KFigure 5.5. Parallel parking a car (bicycle). For a color version of this figure...Figure 5.6. Pontryagin maximum principle. For a color version of this figure, se...Figure 5.7. Needle variations also scaled by amplitudeFigure 5.8. More complex family of control variations
Tables
1 Chapter 4Table 4.1. Functions
2 Chapter 6Table 6.1. Standard examples for growth families (Dahmen and Schmeding 2018, Pro...
