Topics in the geometric theory of linear systems

Title page
PREFACE
MATHEMATICAL ENGINEERING: PROBLEMS AND OPPORTUNITIES
PREFACE TO CHAPTER
GEOMETRIC SYSTEM THEORY: From Cartan to Grothendieck
1. Introduction
2. Some General Principles of System Theory
3. Exterior Differential Systems
4. Input-Output Systems as Exterior Differential Systems
5. Notions of Linear System Theory
6. State Feedback and Luenberqer Observers of Linear Systems
7. The Kronecker Theory of Pencils of Linear Maps
8. The Vector Bund1e Associated with Pairs of Linear Maps
9. The Holomorphic Vector Bundle Invariants of Linear, Time-Invariant Input Systems
10. Feedback Equivalence and the Kronecker Theory of Pairs of Linear Maps
11. Transfer Functions and Holomorphic Vector Bundles for 1-D and n-D Systems
12. Vector Bundles on Orbit Spaces
13. Vector Bundles Defined by Systems of Partial Differential Equations and Linear Groups of Symmetries
14. SO(3.C)-Bundles
15. The Compactified Vector Bundles Determined by the Helmholtz Equation
16. Linear Filters and Groups
17. Generalization of the Pseudo-Inverse Construction to Riemannian Manifolds: Slices and Canonica1 Form for Group Actions
18. Deformation of Exterior Differential Systems, Sinqular Perturbation Theory and Simplification of Complicated Systems
References
FREQUENCY RESPONSE FOR 1-D TIME-VARYING SYSTEMS
1. Introduction
2. A General setting for Linear System Theory
3. Systems and Symbols Associated with Commutative Banach Alqebras
4. The Gelfand Representation as Gauqe Transformations
5. Linear Systems and the Generalized Gelfand Representation
6. Discrete-Time Linear Systems from the point of View of the Generalized Gelfand Representation
7. Zadeh's Definition of the Frequency Response of a Linear, Time-Varying, Scalar Input-Output System
8. The Frequency Response as a vector Bundle/Sheaf
9. The Frequency Response for Linear. Time-Varyinq 1-D Systems in State Space Form
10. The Grassmann Vector Bundle Construction
11. Final Remarks
References
THE MIKUSINSKI PSEUDODIFFERENTIAL OPERATORS AND DIFFERENTIAL ALGEBRAS
1. Introduction
2. The Heaviside Operators as a Differential Field
3. Examples of Fields of Heaviside Operators which are Algebraic over the Field of Rational Operators
4. Curves in Grassmann Manifolds Defined by Certain Linear Systems
References
THE HEAVISIDE-MIKUSINSKI POINT OF VIEW IN LINEAR SYSTEM THEORY
1. Introduction
2. The Titchmarsh Alqebra
3. Picard-Vessiot Integral Operators and the Galois Group
4. Laplace Transform
5. Hille-Yosida Theory and the Growth Conditions
6. Existence of the Laplace Transform for Functions Defined via Linear, Time-Dependent Systems
7. Laplace Transform and the Weyl Alqebra of Linear Differential Operators
8. Monodromv Properties of Linear Ordinary Differential Equations
9. Monodromy Conditions that a Solution of a Weyl Equation be Algebraic
10. The Galois Croup of a Picard-Vessiot Extension of the Rationals and the Monodromy Group
11. Realization Theory for Infinitely Differentiable Functions and the Special Punctions ll8
12. Rea1izatlon Theory in Terms of the Laplace Transform and Riemann Surface Theory
13. The Riemann-Rach Theorem Applied to Finite Dimensional Systems
14. Embeddinq of the "Dynamic" Operator of the Standard Time-Domain State-Space Realization in a Lie Alqebra via the Picard-Vessiot-Infeld-Hull Factorization
15. Examples of Elements of the Titchmarsh Algebra which are Algebraic over the Differential Field of Exponential Polynomials
References
SCATTERING SYSTEMS AND GRASSMANN VARIETIES
l. The Frequency Variety of Finite-Dimensional, Linear, Time-Invariant Input-Output Systems
2. The Frequency Variety of Linear, Time-Invariant Filters
3. Alqebraic and Analytic Varieties Associated with One-Dimensional Wave Equations and Boundary values
4. An Illustrative Linear Wave System
5. Scattering Theory
6. Scattering Theory for the l-D Wave Equations
7. Final Remarks
References
FORMAL SCATTERING THEORY AND GRASSMANNIANS
l. Introduction
2. Some Aspects of Formal Scattering Theory
3. Schrodinqer Potential Scattering
4. The Vector Bundle Settinq
5. Formal Scatterinq Theory on the Frequency Domain
6. Linear Ordinary Differential Operators Defined by Variable Coefficient Output Systems
7. The "Scattering Matrix" for Linear Ordinary Differential Equation Systems l6i
8. Final Remarks
References
FURTHER REMARKS ON SCATTERING SYSTEMS
1. Introduction
2. Linear Lie Systems
3. Constant Coefficient and Piecewise Constant Lie Systems
4. Prolongations of Linear Partial Differential Operators
SYSTEM THEORY AND GEOMETRTC ANALYSIS
1. Introduction
2. The Heaviside-Titchmarsh-Mikusinski Theory and Differential Algebra
3. The Laplace Transform as a "Generalized Gelfand Representation" of the Titchmarsh Algebra and Mikusinski Field
4. Lie Differential Algebras 18i
LINEAR TIME-INVARIANT SYSTEMS AND THE HERMITE FORMULA
1. Introduction
2. The Limit of a Sequency of Finite Dimensional Linear Svstems
3. Hermite's Contour Integral Formula for Solutions of Linear Ordinary Differential Equations, and Scalar Input-Output Linear Systems
4. Sequences of Rational Functions and the Linear Systems They Generate
5. Scalar Input-Output Systems Whose State Space is the Functions on the Circle and State Evolution Operations is an Invariant Linear Differential Operator
LINEAR SYSTEMS AND VECTOR BUNDLE CONNECTIONS
1. Introduction
2. Vector Bundles and Connections Associated with Input-Output Systems
3. Reformulation in Coordinate-Free Terms
4. Input-Output Systems Associated with Linear Connections in Vector Bundles
5. Zero-Curvature Deformations of one-Dimensional Systems
6. Formal Definition of the External Description of a System Associated with a Connection
7. External Description of Scalar Input-Output, One-Dimensional, Time-lnvariant Systems
8. The Internal Description of the Sturm-Liouville Systems
9. The Observation Map as a Connection Homomorphism
10. Controllability and Observability of One-Dimensional Input-Output Maps and Jet Spaces
CONNECTIONS IN VECTOR BUNDLES. LINEAR SYSTEMS AND SCATTERING
1. Introduction
2. Linear Connections in Vector Bundles and Linear Ordinary Differential Equations
3. The Construction of a Vector Bundle and Connection Associated to a Linear Differential Operator
4. Linear Input-Output Systems
5. Deformation and Flows for Connection
6. Isomonodromy as a Curvature-Zero Condition
7. Partial Solution of the Isomonodromy Differential Equation
8. Final Remarks
References
INFINITE DIMENSIONAL LINEAR SYSTEMS IN TERMS OF FUNCTIONS ON MANIFOLDS
1. Introduction
2. Multiplier Actions of the Real Line on Manifolds
3. Scalar Input-Output Linear Systems Constructed from Multiplier Actions of R
4. Coordinate-Free Computation of the Exponential of a First Order Linear Differential Operator
5. Exponential Growth and Existence of the Laplace Transform
INFELD-HULL FACTORIZATION AND LINEAR EVOLUTION EOUATIONS
1. Introduction
2. An Algebraic Form of the Infeld-Hull Idea Applied to Linear Evolution Equations
3. Factorizinq Second Order Differential Operators in Commutative Differential Algebras
4. Differential Equations whose Lie Algebras are Finite Dimensional
5. Generalization of the Fock-Bargmann-Seqal Construction
References
A DIFFERENTTIAL-GEOMETRIC VIEW OF THE POLE-PLACEMENT THEOREM
1. Introduction
2. The Basic Mapping φ and its Differential
3. Condition that φ be a Submersion
4. Remarks on Extensions to Infinite Dimensional Systems
References
INFINITE DIMENSIONAL LINEAR SYSTEMS AND INFINITE ORDER DIFFERENTIAL SYSTEMS
1. Introduction
2. Infinite Order Constant-Coefficient Linear Differential Equations Satisfied by the Bessel Function t->J₀(t)
References
LINEAR FILTERS AND LIE SEMI-GROUPS
1. Introduction
2. The Relation between the Convolution and Lie Generator on the Lie Semiqroup of Positive Real Numbers
3. Multidimensional Filter on R^n
4. Filters with Finite Dimensional State Space and Commuting Generators
LINEAR SYSTEM THEORY AND THE RIEMANN-ROCH THEOREM
1. Introduction
2. The Laplace Transform and Meromorphic Differential Forms
3. Realization Theory in the Time-Domain
4. The Frequencv Domain Description of the Realization
5. The Realization in Terms of the Valuation of the Meromorphic Differential Forms
6. The Realization and the Riemann-Roch Theorem
References
APPROXIMATION OF LINEAR SYSTEMS IN THE FREQUENCY DOMAIN
1. Introduction
2. Closeness in the Frequencv Domain
3. Comparison of the Legendre and Bessel Filtered Linear Systems
4. Linear Systems Defined by Inteqration, Generalized for the Legendre Function