Methods of Algebraic Geometry in Control Theory: Part I

Peter Falb is a Professor Emeritus of Applied Mathematics at Brown University.

• 0. Introduction.- 1. Scalar Linear Systems over the Complex Numbers.- 2. Scalar Linear Systems over a Field k.- 3. Factoring Polynomials.- 4. Affine Algebraic Geometry: Algebraic Sets.- 5. Affine Algebraic Geometry: The Hilbert Theorems.- 6. Affine Algebraic Geometry: Irreducibility.- 7. Affine Algebraic Geometry: Regular Functions and Morphisms I.- 8. The Laurent Isomorphism Theorem.- 9. Affine Algebraic Geometry: Regular Functions and Morphisms II.- 10. The State Space: Realizations.- 11. The State Space: Controllability, Observability, Equivalence.- 12. Affine Algebraic Geometry: Products, Graphs and Projections.- 13. Group Actions, Equivalence and Invariants.- 14. The Geometric Quotient Theorem: Introduction.- 15. The Geometric Quotient Theorem: Closed Orbits.- 16. Affine Algebraic Geometry: Dimension.- 17. The Geometric Quotient Theorem: Open on Invariant Sets.- 18. Affine Algebraic Geometry: Fibers of Morphisms.- 19. The Geometric Quotient Theorem: The Ring of Invariants.- 20. Affine Algebraic Geometry: Simple Points.- 21. Feedback and the Pole Placement Theorem.- 22. Affine Algebraic Geometry: Varieties.- 23. Interlude.- Appendix A: Tensor Products.- Appendix B: Actions of Reductive Groups.- Appendix C: Symmetric Functions and Symmetric Group Actions.- Appendix D: Derivations and Separability.- Problems.- References.