Canonical state representations and Hilbert functions of multidimensional systems

Acta Applicandae Mathematicae

Volumn 94, Issue 2, 2006, Pages 83-135

  Oberst, Ulrich ^a  

^a UNIVERSITY OF INNSBRUCK   (Austria)

Author keywords

Behavior; Hilbert function; Multidimensional system; Partial difference equation; Partial differential equation; Polynomial module; State

Indexed keywords

ALGORITHMS; LINEAR SYSTEMS; MATRIX ALGEBRA; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; TRANSFER FUNCTIONS;

CANONICAL STATE BEHAVIOR; HILBERT FUNCTIONS; MULTIDIMENSIONAL SYSTEMS; POLYNOMIAL MODULES;

THEOREM PROVING;

EID: 33750545583     PISSN: 01678019     EISSN: 15729036     Source Type: Journal    
DOI: 10.1007/s10440-006-9068-8     Document Type: Article

References (43)

1. Apel, J.: The theory of involutive divisions and an application to Hilbert function computations. J. Symbolic Comput. 25, 683-704 (1998)
2. Apel, J.: On a conjecture of R.P. Stanley II: quotients modulo monomoial ideals. J. Algebraic Combin. 17, 57-74 (2003)
3. Blondel, V.D., Megretski, A. (eds.): Unsolved Problems in Mathematical Systems and Control Theory. Princeton University Press, New Jersey (2004)
4. Egorov, Y.V., Shubin, M.A. (eds.): Partial Differential liquations I. Springer, Berlin Heidelberg New York (1992)
5. Egorov, Y.V., Shubin, M.A. (eds.): Partial Differential Equations III. Springer, Berlin Heidelberg New York (1991)
6. Fröhler, S., Oberst, U.: Continuous time-varying linear systems. Systems Control Lett. 35, 97-110 (1998)
7. Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.): Computer Algebra in Scientific Computing. Springer, Berlin Heidelberg New York (1999)
8. Gelfand, I.M., Shilov, G.E.: Generalized Functions III. Theory of Differential Equations. Academic Press, New York (1967)
9. Gerdt, V.P.: Completion of linear differential systems to involution. In: Ganzha, V.O., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing, pp. 115-137. Springer, Berlin Heidelberg New York (1999)
10. Gluesing-Luerssen, H.: Linear Delay-Differential Systems with Commensurate Delays: An Algebraic Approach. Lecture Notes in Mathematics 1770, Springer. Berlin Heidelberg New York (2002)
11. Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin Heidelberg New York (1983)
12. Hörmander, L.: The Analysis of Linear Partial Differential Operators II. Springer, Berlin Heidelberg New York (1983)
13. Janet, M.: Sur les systèmes d'équations aux dérivées partielles. J. Math. Pures Appl. 8, 65-151 (1920)
14. Kailath, T.: Linear Systems. Prentice-Hall, Englewood Cliffs, New Jersey (1980)
15. Kalman, R.E.: Mathematical description of linear dynamical systems. SIAM J. Control Optim. 1, 152-192 (1963)
16. Kreuzer, M., Robbiano, L.: Computational Commutative Algebra 1. Springer, Berlin Heidelberg New York (2000)
17. Lu, P., Liu, M., Oberst, U.: Linear recurring arrays, linear systems and multidimensional cyclic codes over Quasi-Frobenius rings. Acta Appl. Math. 80, 175-198 (2004)
18. Macaulay, F.S.: The Algebraic Theory of Modular Systems. Cambridge University Press, UK (1916)
19. Malgrange, B.: Systèmes Différentiels Involutifs. Prépublication 636, Institut Fourier Grenoble, France (2004)
20. Oberst, U.: Multidimensional constant linear systems. Acta Appl. Math. 20, 1-175 (1990)
21. Oberst, U.: Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients. AAECC 6, 211-243 (1995)
22. Oberst, U.: Canonical State Representations and Hilbert Functions of Multidimensional Systems (with historical comments and a discussion of Professor Pommaret's remarks). Talk at the conference D3, Linz, Austria (May 2006)(homepage of the Grubner-Semester)
23. Oberst, U., Pauer, F.: The constructive solution of linear systems of partial difference and differential equations with constant coefficients. Multidimens. Systems Signal Process. 12, 253-308 (2001)
24. Palamodov, V.P.: Linear Differential Operators. Springer, Berlin Heidelberg New York (1970)
25. Polderman, J.W., Willems, J.C.: Introduction to Mathematical Systems Theory. Springer, Berlin Heidelberg New York (1998)
26. Pommaret, J.-F.: Systems of Partial Differential Equations and Lie Pseudogroups. Gordon and Breach, New York (1978)
27. Pommaret, J.-F.: New perspectives in control theory for partial differential equations. IMA J. Math. Control Inform. 9, 305-330 (1992)
28. Pommaret, J.-F.: Partial Differential Control Theory, Volume I: Mathematical Tools, Volume II: Control Systems. Kluwer, Dordrecht (2001)
29. Pommaret, J.-F.: Localization and transfer matrix computation for linear multidimensional control systems. Proceedings MTNS, Leuven, Belgium (2004)
30. Pommaret, J.-F.: Lecture Notes of the mini-course on algebraic analysis of control systems defined by partial differential equations, pp. 1-45. Gröbner Semester Linz, Sections D2 and D3, RISC, Hagenberg, and RICAM, Linz, Austria (May 2006)
31. Rapisarda, P., Willems, J.C.: State maps for linear systems. SIAM J. Control Optim. 35, 1053-1091 (1997)
32. Riquier, C.: Les Systèmes d'Équations aux Dérivées Partielles. Gauthiers-Villars, Paris, France (1910)
33. Rocha, P., Willems, J.C.: Markov properties for systems described by PDEs and first-order representations. Systems Control Lett. 55, 538-542 (2006)
34. Stanley, R.P.: Combinatorics and Commutative Algebra, 2nd edn., Birkhäuser, Boston (1996)
35. Sturmfels, B.: Solving Systems of Polynomial Equations. American Mathematical Society, Providence, Rhode Island (2002)
36. Treves, F.: Basic Linear Partial Differential Equations. Academic Press, New York (1975)
37. Vardulakis, A.I.G.: Linear Multivariable Control. Wiley, Chichester, West Sussex, England (1991)
38. Vasconcelos, W.V.: Computational Methods in Commutative Algebra and Algebraic Geometry. Springer, Berlin Heidelberg New York (1998)
39. Willems, J.C.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Automat. Control 36, 259-294 (1991)
40. Willems, J.C.: State and first order representations. In: Blondel, V. D. and Megretski, A. (eds.) Unsolved Problems in Mathematical Systems and Control Theory, pp. 54-57. Princeton University Press, New Jersey (2004)
41. Wolovich, W.A.: Linear Multivariable Systems. Springer, Berlin Heidelberg New York (1974)
42. Wood, J., Rocha, P., Rogers, E., Owens, D.H.: Structure indices for multidimensional systems. IMA J. Math. Control Inform. 17, 227-256 (2000)
43. Zerz, E.: Topics in Multidimensional Linear Systems Theory. Lecture Notes in Control and Information Sciences 256, Springer, Berlin Heidelberg New York (2000)