LQ Dynamic Optimization and Differential Games

Notation and symbols.

1 Introduction.

1.1 Historical perspective.

1.2 How to use this book.

1.3 Outline of this book.

1.4 Notes and references.

2 Linear algebra.

2.1 Basic concepts in linear algebra.

2.2 Eigenvalues and eigenvectors.

2.3 Complex eigenvalues.

2.4 Cayley–Hamilton theorem.

2.5 Invariant subspaces and Jordan canonical form.

2.6 Semi-definite matrices.

2.7 Algebraic Riccati equations.

2.8 Notes and references.

2.9 Exercises.

2.10 Appendix.

3 Dynamical systems.

3.1 Description of linear dynamical systems.

3.2 Existence–uniqueness results for differential equations.

3.2.1 General case.

3.2.2 Control theoretic extensions.

3.3 Stability theory: general case.

3.4 Stability theory of planar systems.

3.5 Geometric concepts.

3.6 Performance specifications.

3.7 Examples of differential games.

3.8 Information, commitment and strategies.

3.9 Notes and references.

3.10 Exercises.

3.11 Appendix.

4 Optimization techniques.

4.1 Optimization of functions.

4.2 The Euler–Lagrange equation.

4.3 Pontryagin’s maximum principle.

4.4 Dynamic programming principle.

4.5 Solving optimal control problems.

4.6 Notes and references.

4.7 Exercises.

4.8 Appendix.

5 Regular linear quadratic optimal control.

5.1 Problem statement.

5.2 Finite-planning horizon.

5.3 Riccati differential equations.

5.4 Infinite-planning horizon.

5.5 Convergence results.

5.6 Notes and references.

5.7 Exercises.

5.8 Appendix.

6 Cooperative games.

6.1 Pareto solutions.

6.2 Bargaining concepts.

6.3 Nash bargaining solution.

6.4 Numerical solution.

6.5 Notes and references.

6.6 Exercises.

6.7 Appendix.

7 Non-cooperative open-loop information games.

7.1 Introduction.

7.2 Finite-planning horizon.

7.3 Open-loop Nash algebraic Riccati equations.

7.4 Infinite-planning horizon.

7.5 Computational aspects and illustrative examples.

7.6 Convergence results.

7.7 Scalar case.

7.8 Economics examples.

7.8.1 A simple government debt stabilization game.

7.8.2 A game on dynamic duopolistic competition.

7.9 Notes and references.

7.10 Exercises.

7.11 Appendix.

8 Non-cooperative feedback information games.

8.1 Introduction.

8.2 Finite-planning horizon.

8.3 Infinite-planning horizon.

8.4 Two-player scalar case.

8.5 Computational aspects.

8.5.1 Preliminaries.

8.5.2 A scalar numerical algorithm: the two-player case.

8.5.3 The N-player scalar case.

8.6 Convergence results for the two-player scalar case.

8.7 Notes and references.

8.8 Exercises.

8.9 Appendix.

9 Uncertain non-cooperative feedback information games.

9.1 Stochastic approach.

9.2 Deterministic approach: introduction.

9.3 The one-player case.

9.4 The one-player scalar case.

9.5 The two-player case.

9.6 A fishery management game.

9.7 A scalar numerical algorithm.

9.8 Stochastic interpretation.

9.9 Notes and references.

9.10 Exercises.

9.11 Appendix.

References.

Index.