Algebraic Theory of Linear Systems: A Survey

Chapter
First Online:
Part of the Differential-Algebraic Equations Forum book series (DAEF)

Abstract

An introduction into the algebraic theory of several types of linear systems is given. In particular, linear ordinary and partial differential and difference equations are covered. Special emphasis is given to the formulation of formally well-posed initial value problem for treating solvability questions for general, i.e. also under- and over-determined, systems. A general framework for analysing abstract linear systems with algebraic and homological methods is outlined. The presentation uses throughout Gröbner bases and thus immediately leads to algorithms.

Keywords

Algebraic methods Autonomy and controllability Behavioural approach Gröbner bases Initial value problems Linear systems Over- and under-determined systems Symbolic computation 

Mathematics Subject Classification (2010)

13N10 13P10 13P25 34A09 34A12 35G40 35N10 68W30 93B25 93B40 93C05 93C15 93C20 93C55 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Adams, W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)Google Scholar
  2. 2.
    Brenan, K., Campbell, S., Petzold, L.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics, vol. 14. SIAM, Philadelphia (1996)Google Scholar
  3. 3.
    Bronstein, M., Petkovšek, M.: An introduction to pseudo-linear algebra. Theor. Comput. Sci. 157, 3–33 (1996)CrossRefMATHGoogle Scholar
  4. 4.
    Buchberger, B.: Ein algorithmus zum auffinden der basiselemente des restklassenringes nach einem nulldimensionalen polynomideal. Ph.D. thesis, Universität Innsbruck (1965) [English translation: J. Symb. Comput. 41, 475–511 (2006)]Google Scholar
  5. 5.
    Campbell, S., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Chyzak, F., Quadrat, A., Robertz, D.: Effective algorithms for parametrizing linear control systems over Ore algebras. Appl. Algebra Eng. Commun. Comput. 16, 319–376 (2005)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Cohn, P.: Free Rings and Their Relations. Academic, New York (1971)MATHGoogle Scholar
  8. 8.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, New York (1992)CrossRefMATHGoogle Scholar
  9. 9.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. Springer, New York (1998)Google Scholar
  10. 10.
    Drach, J.: Sur les systèmes complètement orthogonaux dans l’espace à n dimensions et sur la réduction des systèmes différentielles les plus généraux. C. R. Acad. Sci. 125, 598–601 (1897)MATHGoogle Scholar
  11. 11.
    Gerdt, V., Blinkov, Y.: Involutive bases of polynomial ideals. Math. Comp. Simul. 45, 519–542 (1998)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Gómez-Torrecillas, J.: Basic module theory over non-commutative rings with computational aspects of operator algebras. In: Barkatou, M., Cluzeau, T., Regensburger, G., Rosenkranz, M. (eds.) Algebraic and Algorithmic Aspects of Differential and Integral Operators. Lecture Notes in Computer Science, vol. 8372, pp. 23–82. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  13. 13.
    Goodearl, K., Warfield, R.: An Introduction to Noncommutative Noetherian Rings. London Mathematical Society Student Texts, vol. 61, 2nd edn. Cambridge University Press, Cambridge (2004)Google Scholar
  14. 14.
    Hairer, E., Lubich, C., Roche, M.: The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods. Lecture Notes in Mathematics, vol. 1409. Springer, Berlin (1989)Google Scholar
  15. 15.
    Hausdorf, M., Seiler, W.: Perturbation versus differentiation indices. In: Ghanza, V., Mayr, E., Vorozhtsov, E. (eds.) Computer Algebra in Scientific Computing—CASC 2001, pp. 323–337. Springer, Berlin (2001)CrossRefGoogle Scholar
  16. 16.
    Hausdorf, M., Seiler, W.: An efficient algebraic algorithm for the geometric completion to involution. Appl. Algebra Eng. Commun. Comput. 13, 163–207 (2002)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Jacobson, N.: The Theory of Rings. Americal Mathematical Society, Providence (1943)CrossRefMATHGoogle Scholar
  18. 18.
    Janet, M.: Leçons sur les Systèmes d’Équations aux Dérivées Partielles. Cahiers Scientifiques, Fascicule IV. Gauthier-Villars, Paris (1929)MATHGoogle Scholar
  19. 19.
    Kalman, R.: Algebraic structure of linear dynamical systems. Proc. Natl. Acad. Sci. USA 54, 1503–1508 (1965)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Kandry-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput. 9, 1–26 (1990)CrossRefGoogle Scholar
  21. 21.
    Kashiwara, M., Kawai, T., Kimura, T.: Foundations of Algebraic Analysis. Princeton University Press, Princeton (1986)MATHGoogle Scholar
  22. 22.
    Kato, G., Struppa, D.: Fundamentals of Algebraic Microlocal Analysis. Pure and Applied Mathematics, vol. 217. Dekker, New York (1999)Google Scholar
  23. 23.
    Kredel, H.: Solvable Polynomial Rings. Verlag Shaker, Aachen (1993)MATHGoogle Scholar
  24. 24.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Textbooks in Mathematics. EMS Publishing House, Zürich (2006)CrossRefGoogle Scholar
  25. 25.
    Lam, T.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer, New York (1999)Google Scholar
  26. 26.
    Lam, T.: On the equality of row rank and column rank. Expo. Math. 18, 161–163 (2000)MATHMathSciNetGoogle Scholar
  27. 27.
    Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin/Heidelberg (2013)CrossRefGoogle Scholar
  28. 28.
    Lemaire, F.: An orderly linear PDE with analytic initial conditions with a non-analytic solution. J. Symb. Comput. 35, 487–498 (2003)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Levandovskyy, V.: Non-commutative computer algebra for polynomial algebras: Gröbner bases, applications and implementation. Ph.D. thesis, Fachbereich Mathematik, Universität Kaiserslautern (2005)Google Scholar
  30. 30.
    Levandovskyy, V., Schindelar, K.: Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases. J. Symb. Comput. 46, 595–608 (2011)CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Li, 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)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Malgrange, B.: Systemes différentiels à coefficients constants. Semin. Bourbaki 15, 1–11 (1964)Google Scholar
  33. 33.
    Mora, T., Robbiano, L.: The Gröbner fan of an ideal. J. Symb. Comput. 6, 183–208 (1988)CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Morimoto, M.: An Introduction to Sato’s Hyperfunctions. Translation of Mathematical Monographs, vol. 129. American Mathematical Society, Providence (1993)Google Scholar
  35. 35.
    Noether, E., Schmeidler, W.: Moduln in nichtkommutativen Bereichen, insbesondere aus Differential- und Differenzausdrücken. Math. Z. 8, 1–35 (1920)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Oberst, U.: Multidimensional constant linear systems. Acta Appl. Math. 20, 1–175 (1990)CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Oberst, U., Pauer, F.: The constructive solution of linear systems of partial difference and differential equations with constant coefficients. Multidim. Syst. Sign. Process. 12, 253–308 (2001)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933)CrossRefMATHMathSciNetGoogle Scholar
  39. 39.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)Google Scholar
  40. 40.
    Pillai, H., Shankar, S.: A behavioral approach to control of distributed systems. SIAM J. Control Optim. 37, 388–408 (1999)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Pommaret, J., Quadrat, A.: Generalized Bezout identity. Appl. Algebra Eng. Commun. Comput. 9, 91–116 (1998)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Pommaret, J., Quadrat, A.: Algebraic analysis of linear multidimensional control systems. IMA J. Math. Control Inf. 16, 275–297 (1999)CrossRefMATHMathSciNetGoogle Scholar
  43. 43.
    Pommaret, J., Quadrat, A.: Localization and parametrization of linear multidimensional control systems. Syst. Control Lett. 37, 247–260 (1999)CrossRefMATHMathSciNetGoogle Scholar
  44. 44.
    Rabier, P., Rheinboldt, W.: Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet, P., Lions, J. (eds.) Handbook of Numerical Analysis, vol. VIII, pp. 183–540. North-Holland, Amsterdam (2002)Google Scholar
  45. 45.
    Riquier, C.: Les Systèmes d’Équations aux Derivées Partielles. Gauthier-Villars, Paris (1910)Google Scholar
  46. 46.
    Robertz, D.: Recent progress in an algebraic analysis approach to linear systems. Multidimensional System Signal Processing (2015, to appear). doi: 10.007/s11045-014-0280-9Google Scholar
  47. 47.
    Rocha, P., Zerz, E.: Strong controllability and extendibility of discrete multidimensional behaviors. Syst. Control Lett. 54, 375–380 (2005)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Seiler, W.: On the arbitrariness of the general solution of an involutive partial differential equation. J. Math. Phys. 35, 486–498 (1994)CrossRefMATHMathSciNetGoogle Scholar
  49. 49.
    Seiler, W.: Indices and solvability for general systems of differential equations. In: Ghanza, V., Mayr, E., Vorozhtsov, E. (eds.) Computer Algebra in Scientific Computing—CASC ‘99, pp. 365–385. Springer, Berlin (1999)Google Scholar
  50. 50.
    Seiler, W.: A combinatorial approach to involution and δ-regularity I: involutive bases in polynomial algebras of solvable type. Appl. Algebra Eng. Commun. Comput. 20, 207–259 (2009)CrossRefMATHMathSciNetGoogle Scholar
  51. 51.
    Seiler, W.: A combinatorial approach to involution and δ-regularity II: structure analysis of polynomial modules with Pommaret bases. Appl. Algebra Eng. Commun. Comput. 20, 261–338 (2009)CrossRefMATHMathSciNetGoogle Scholar
  52. 52.
    Seiler, W.: Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2009)Google Scholar
  53. 53.
    Willems, J.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Autom. Control 36, 259–294 (1991)CrossRefMATHMathSciNetGoogle Scholar
  54. 54.
    Wood, J., Rogers, E., Owens, D.: A formal theory of matrix primeness. Math. Control Signals Syst. 11, 40–78 (1998)CrossRefMATHMathSciNetGoogle Scholar
  55. 55.
    Zerz, E.: Extension modules in behavioral linear systems theory. Multidim. Syst. Sign. Process. 12, 309–327 (2001)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Zerz, E.: Multidimensional behaviours: an algebraic approach to control theory for PDE. Int. J. Control 77, 812–820 (2004)CrossRefMATHMathSciNetGoogle Scholar
  57. 57.
    Zerz, E.: An algebraic analysis approach to linear time-varying systems. IMA J. Math. Control Inf. 23, 113–126 (2006)CrossRefMATHMathSciNetGoogle Scholar
  58. 58.
    Zerz, E.: Discrete multidimensional systems over \(\mathbb{Z}_{n}\). Syst. Control Lett. 56, 702–708 (2007)CrossRefMATHMathSciNetGoogle Scholar
  59. 59.
    Zerz, E., Rocha, P.: Controllability and extendibility of continuous multidimensional behaviors. Multidim. Syst. Sign. Process. 17, 97–106 (2006)CrossRefMATHMathSciNetGoogle Scholar
  60. 60.
    Zerz, E., Seiler, W., Hausdorf, M.: On the inverse syzygy problem. Commun. Algebra 38, 2037–2047 (2010)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Institut für Mathematik, Universität KasselKasselGermany
  2. 2.Lehrstuhl D für Mathematik, RWTH AachenAachenGermany

Personalised recommendations