Skip to main content
SpringerLink
Log in

Algebraic Theory of Linear Systems: A Survey

  • Chapter
  • First Online:
Surveys in Differential-Algebraic Equations II

Part of the book series: Differential-Algebraic Equations Forum ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 16.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For arbitrary systems, not even an a priori bound on the maximal order of an integrability condition is known. In the case of linear equations, algebraic complexity theory provides a double exponential bound which is, however, completely useless for computations, as for most systems appearing in applications it grossly overestimates the actual order.

  2. 2.

    Here we are actually dealing with the special case of a homogeneous linear system where consistency simply follows from the fact that u = 0 is a solution.

  3. 3.

    An initial value problem in the strict sense is obtained, if one starts with a complementary Rees decomposition (see Definition 7.5).

  4. 4.

    It is quite instructive to try to transform (4.7) into such a form: one will rapidly notice that this is not possible!

  5. 5.

    We define the order of an operator matrix as the maximal order of an entry.

  6. 6.

    In applications, it is actually quite rare that systems of partial differential equations contain algebraic equations. In this case, no differentiations are required and F is of order γ 1 so that we obtain the same estimate ν ≤ γ 1 + 1 as in the case of ordinary differential equations.

  7. 7.

    http://cocoa.dima.unige.it.

  8. 8.

    http://www.math.uiuc.edu/Macaulay2.

  9. 9.

    http://magma.usyd.edu.au.

  10. 10.

    http://www.singular.uni-kl.de.

  11. 11.

    For us a term is a pure power product x μ whereas a monomial is of the form cx μ with a coefficient \(c \in \daleth \); beware that some text books on Gröbner bases use the words term and monomial with exactly the opposite meaning.

  12. 12.

    Beware that the left and the right ideal generated by a set F are generally different.

  13. 13.

    The term “reduction” refers to the fact that the monomial ct in f is replaced by a linear combination of terms which are all smaller than t with respect to the used term order. It does not imply that \(\tilde{f}\) is simpler in the sense that it has less terms than f. In fact, quite often the opposite is the case!

References

  1. Adams, W., Loustaunau, P.: An Introduction to Gröbner Bases. Graduate Studies in Mathematics, vol. 3. American Mathematical Society, Providence (1994)

    Google Scholar 

  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. Bronstein, M., Petkovšek, M.: An introduction to pseudo-linear algebra. Theor. Comput. Sci. 157, 3–33 (1996)

    Article  MATH  Google Scholar 

  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. Campbell, S., Gear, C.: The index of general nonlinear DAEs. Numer. Math. 72, 173–196 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cohn, P.: Free Rings and Their Relations. Academic, New York (1971)

    MATH  Google Scholar 

  8. Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Undergraduate Texts in Mathematics. Springer, New York (1992)

    Book  MATH  Google Scholar 

  9. Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, vol. 185. Springer, New York (1998)

    Google Scholar 

  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)

    MATH  Google Scholar 

  11. Gerdt, V., Blinkov, Y.: Involutive bases of polynomial ideals. Math. Comp. Simul. 45, 519–542 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  16. Hausdorf, M., Seiler, W.: An efficient algebraic algorithm for the geometric completion to involution. Appl. Algebra Eng. Commun. Comput. 13, 163–207 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Jacobson, N.: The Theory of Rings. Americal Mathematical Society, Providence (1943)

    Book  MATH  Google Scholar 

  18. Janet, M.: Leçons sur les Systèmes d’Équations aux Dérivées Partielles. Cahiers Scientifiques, Fascicule IV. Gauthier-Villars, Paris (1929)

    MATH  Google Scholar 

  19. Kalman, R.: Algebraic structure of linear dynamical systems. Proc. Natl. Acad. Sci. USA 54, 1503–1508 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kandry-Rody, A., Weispfenning, V.: Non-commutative Gröbner bases in algebras of solvable type. J. Symb. Comput. 9, 1–26 (1990)

    Article  Google Scholar 

  21. Kashiwara, M., Kawai, T., Kimura, T.: Foundations of Algebraic Analysis. Princeton University Press, Princeton (1986)

    MATH  Google Scholar 

  22. Kato, G., Struppa, D.: Fundamentals of Algebraic Microlocal Analysis. Pure and Applied Mathematics, vol. 217. Dekker, New York (1999)

    Google Scholar 

  23. Kredel, H.: Solvable Polynomial Rings. Verlag Shaker, Aachen (1993)

    MATH  Google Scholar 

  24. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Textbooks in Mathematics. EMS Publishing House, Zürich (2006)

    Book  Google Scholar 

  25. Lam, T.: Lectures on Modules and Rings. Graduate Texts in Mathematics, vol. 189. Springer, New York (1999)

    Google Scholar 

  26. Lam, T.: On the equality of row rank and column rank. Expo. Math. 18, 161–163 (2000)

    MATH  MathSciNet  Google Scholar 

  27. Lamour, R., März, R., Tischendorf, C.: Differential-Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum. Springer, Berlin/Heidelberg (2013)

    Book  Google Scholar 

  28. Lemaire, F.: An orderly linear PDE with analytic initial conditions with a non-analytic solution. J. Symb. Comput. 35, 487–498 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. Levandovskyy, V., Schindelar, K.: Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases. J. Symb. Comput. 46, 595–608 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  32. Malgrange, B.: Systemes différentiels à coefficients constants. Semin. Bourbaki 15, 1–11 (1964)

    Google Scholar 

  33. Mora, T., Robbiano, L.: The Gröbner fan of an ideal. J. Symb. Comput. 6, 183–208 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  34. Morimoto, M.: An Introduction to Sato’s Hyperfunctions. Translation of Mathematical Monographs, vol. 129. American Mathematical Society, Providence (1993)

    Google Scholar 

  35. Noether, E., Schmeidler, W.: Moduln in nichtkommutativen Bereichen, insbesondere aus Differential- und Differenzausdrücken. Math. Z. 8, 1–35 (1920)

    Article  MATH  MathSciNet  Google Scholar 

  36. Oberst, U.: Multidimensional constant linear systems. Acta Appl. Math. 20, 1–175 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  38. Ore, O.: Theory of non-commutative polynomials. Ann. Math. 34, 480–508 (1933)

    Article  MATH  MathSciNet  Google Scholar 

  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. Pillai, H., Shankar, S.: A behavioral approach to control of distributed systems. SIAM J. Control Optim. 37, 388–408 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  41. Pommaret, J., Quadrat, A.: Generalized Bezout identity. Appl. Algebra Eng. Commun. Comput. 9, 91–116 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  42. Pommaret, J., Quadrat, A.: Algebraic analysis of linear multidimensional control systems. IMA J. Math. Control Inf. 16, 275–297 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  43. Pommaret, J., Quadrat, A.: Localization and parametrization of linear multidimensional control systems. Syst. Control Lett. 37, 247–260 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  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. Riquier, C.: Les Systèmes d’Équations aux Derivées Partielles. Gauthier-Villars, Paris (1910)

    Google Scholar 

  46. Robertz, D.: Recent progress in an algebraic analysis approach to linear systems. Multidimensional System Signal Processing (2015, to appear). doi: 10.007/s11045-014-0280-9

    Google Scholar 

  47. Rocha, P., Zerz, E.: Strong controllability and extendibility of discrete multidimensional behaviors. Syst. Control Lett. 54, 375–380 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  48. Seiler, W.: On the arbitrariness of the general solution of an involutive partial differential equation. J. Math. Phys. 35, 486–498 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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. Willems, J.: Paradigms and puzzles in the theory of dynamical systems. IEEE Trans. Autom. Control 36, 259–294 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  54. Wood, J., Rogers, E., Owens, D.: A formal theory of matrix primeness. Math. Control Signals Syst. 11, 40–78 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  55. Zerz, E.: Extension modules in behavioral linear systems theory. Multidim. Syst. Sign. Process. 12, 309–327 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  56. Zerz, E.: Multidimensional behaviours: an algebraic approach to control theory for PDE. Int. J. Control 77, 812–820 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  57. Zerz, E.: An algebraic analysis approach to linear time-varying systems. IMA J. Math. Control Inf. 23, 113–126 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  58. Zerz, E.: Discrete multidimensional systems over \(\mathbb{Z}_{n}\). Syst. Control Lett. 56, 702–708 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  59. Zerz, E., Rocha, P.: Controllability and extendibility of continuous multidimensional behaviors. Multidim. Syst. Sign. Process. 17, 97–106 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  60. Zerz, E., Seiler, W., Hausdorf, M.: On the inverse syzygy problem. Commun. Algebra 38, 2037–2047 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Mathematik, Universität Kassel, 34109, Kassel, Germany

    Werner M. Seiler

  2. Lehrstuhl D für Mathematik, RWTH Aachen, 52062, Aachen, Germany

    Eva Zerz

Authors
  1. Werner M. Seiler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eva Zerz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Werner M. Seiler .

Editor information

Editors and Affiliations

  1. Technische Universität Ilmenau Institut für Mathematik, Ilmenau, Germany

    Achim Ilchmann

  2. Universität Hamburg Fachbereich Mathematik, Hamburg, Germany

    Timo Reis

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Seiler, W.M., Zerz, E. (2015). Algebraic Theory of Linear Systems: A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations II. Differential-Algebraic Equations Forum. Springer, Cham. https://doi.org/10.1007/978-3-319-11050-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation