Control theory represents an attempt to codify, in mathematical terms, the principles and techniques used in the analysis and design of control systems. Algebraic geometry may, in an elementary way, be viewed as the study of the structure and properties of the solutions of systems of algebraic equations. The aim of these notes is to provide access to the methods of algebraic geometry for engineers and applied scientists through the motivated context of control theory. I began the development of these notes over fifteen years ago with a series of lectures given to the Control Group at the Lund Institute of Technology in Sweden. Over the following years, I presented the material in courses at Brown several times and must express my appreciation for the feedback (sic!) received from the students. I have attempted throughout to strive for clarity, often making use of constructive methods and giving several proofs of a particular result. Since algebraic geometry draws on so many branches of mathematics and can be dauntingly abstract, it is not easy to convey its beauty and utility to those interested in applications. I hope at least to have stirred the reader to seek a deeper understanding of this beauty and utility in control theory. The first volume dea1s with the simplest control systems (i. e. single input, single output linear time-invariant systems) and with the simplest algebraic geometry (i. e. affine algebraic geometry).
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.
