Multidimensional Systems Theory : Progress, Directions and Open Problems in Multidimensional Systems

1. Trends in Multidimensional Systems Theory.- 1.1 Introduction.- 1.2 Multidimensional Systems Stability.- 1.3 Multivariate Realization Theory.- 1.4 n-D Problem of Moments and Its Applications in Multidimensional Systems Theory.- 1.5 Role of Irreducible Polynomials in Multidimensional Systems Theory.- 1.6 Hilbert Transform and Spectral Factorization.- 1.7 Conclusions.- References.- 2. Multivariate Rational Approximants of the Pade-Type in Systems Theory.- 2.1 Introduction and Motivation.- 2.2 Multivariate Pade-Type Approximants (Scalar Case).- 2.3 Pade-Type Matrix Approximants.- 2.4 Conclusions.- References.- 3. Causal and Weakly Causal 2-D Filters with Applications in Stabilization.- 3.1 Scalar 2-D Input/Output Systems.- 3.2 Stability.- 3.3 Structural Stability.- 3.4 Multi-Input/Multi-Output Systems.- 3.5 Stabilization of Scalar Feedback Systems.- 3.6 Characterization of Stabilizers for Scalar Systems.- 3.7 Stabilization of Strictly Causal Transfer Matrices.- 3.8 Characterization of Stabilizers for MIMO Systems.- 3.9 Stabilization of Weakly Causal Systems.- 3.10 Stabilization of MIMO Weakly Causal Systems.- 3.11 Conclusions.- References.- 4. Stabilization of Linear Spatially-Distributed Continuous- Time and Discrete- Time Systems.- 4.1 Introduction.- 4.2 The State Representation and Input/Output Description.- 4.3 Discretizations in Time.- 4.4 Representation in Terms of a Family of Finite-Dimensional Systems.- 4.5 Stability.- 4.6 Reachability and Stabilizability.- 4.7 The Riccati Equation and Stabilizability.- 4.8 Stabilization by Dynamic Output Feedback.- 4.9 Application to Tracking.- Acknowledgement.- References.- 5. Linear Shift-Variant Multidimensional Systems.- 5.1 Introduction.- 5.2 2-D Quarter Plane State-Space Model.- 5.3 k-D State-Space Model.- 5.4 State-Space Model for the Inverse System.- 5.5 Examples of Applications.- 5.6 Conclusions.- References.- 6. Grobner Bases: An Algorithmic Method in Polynomial Ideal Theory.- 6.1 Introduction.- 6.2 Grobner Bases.- 6.3 Algorithmic Construction of Grobner Bases.- 6.4 An Improved Version of the Algorithm.- 6.5 Application: Canonical Simplification, Decision of Ideal Congruence and Membership, Computation in Residue Class Rings.- 6.6 Application: Solvability and Exact Solution of Systems of Algebraic Equations.- 6.7 Application: Solution of Linear Homogeneous Equations with Polynomial Coefficients.- 6.8 Grobner Bases for Polynomial Ideals over the Integers.- 6.9 Other Applications.- 6.10 Specializations, Generalizations, Implementations, Complexity.- Acknowledgement.- References.- 7. The Equation Ax = b Over the Ring C [z, w].- 7.1 Introduction.- 7.2 Sufficient Condition for Solution.- Appendix A: Zero-Dimensional Polynomial Ideals.- References.- 8. Open Problems.