Solution Concepts for Linear DAEs: A Survey

  • Stephan TrennEmail author
Part of the Differential-Algebraic Equations Forum book series (DAEF)


This survey aims at giving a comprehensive overview of the solution theory of linear differential-algebraic equations (DAEs). For classical solutions a complete solution characterization is presented including explicit solution formulas similar to the ones known for linear ordinary differential equations (ODEs). The problem of inconsistent initial values is treated and different approaches are discussed. In particular, the common Laplace-transform approach is discussed in the light of more recent distributional solution frameworks.


Differential algebraic equations Descriptor systems Distributional solution theory Laplace transform 

Mathematics Subject Classification (2010)

34A09 34A12 34A05 34A25 


  1. 1.
    Aplevich, J.D.: Implicit Linear Systems. Lecture Notes in Control and Information Sciences, vol. 152. Springer, Berlin (1991) zbMATHCrossRefGoogle Scholar
  2. 2.
    Armentano, V.A.: The pencil (sEA) and controllability-observability for generalized linear systems: a geometric approach. SIAM J. Control Optim. 24, 616–638 (1986) MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berger, T., Trenn, S.: Addition to: “The quasi-Kronecker form for matrix pencils”. SIAM. J. Matrix Anal. Appl. 34(1), 94–101 (2013) CrossRefGoogle Scholar
  4. 4.
    Berger, T., Trenn, S.: The quasi-Kronecker form for matrix pencils. SIAM J. Matrix Anal. Appl. 33(2), 336–368 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Berger, T., Ilchmann, A., Trenn, S.: The quasi-Weierstraß form for regular matrix pencils. Linear Algebra Appl. 436(10), 4052–4069 (2012). doi: 10.1016/j.laa.2009.12.036 MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bernhard, P.: On singular implicit linear dynamical systems. SIAM J. Control Optim. 20(5), 612–633 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. North-Holland, Amsterdam (1989) zbMATHGoogle Scholar
  8. 8.
    Campbell, S.L.: Singular Systems of Differential Equations I. Pitman, New York (1980) Google Scholar
  9. 9.
    Campbell, S.L.: Singular Systems of Differential Equations II. Pitman, New York (1982) zbMATHGoogle Scholar
  10. 10.
    Campbell, S.L., Petzold, L.R.: Canonical forms and solvable singular systems of differential equations. SIAM J. Algebr. Discrete Methods 4, 517–521 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Campbell, S.L., Meyer, C.D. Jr., Rose, N.J.: Applications of the Drazin inverse to linear systems of differential equations with singular constant coefficients. SIAM J. Appl. Math. 31(3), 411–425 (1976). doi: 10.1137/0131035 MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Cobb, J.D.: On the solution of linear differential equations with singular coefficients. J. Differ. Equ. 46, 310–323 (1982) MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Cobb, J.D.: Controllability, observability and duality in singular systems. IEEE Trans. Autom. Control AC-29, 1076–1082 (1984) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Dai, L.: Singular Control Systems. Lecture Notes in Control and Information Sciences, vol. 118. Springer, Berlin (1989) zbMATHCrossRefGoogle Scholar
  15. 15.
    Dieudonné, J.: Sur la réduction canonique des couples des matrices. Bull. Soc. Math. Fr. 74, 130–146 (1946) zbMATHGoogle Scholar
  16. 16.
    Doetsch, G.: Introduction to the Theory and Application of the Laplace Transformation. Springer, Berlin (1974) zbMATHCrossRefGoogle Scholar
  17. 17.
    Drazin, M.P.: Pseudo-inverses in associative rings and semigroups. Am. Math. Mon. 65(7), 506–514 (1958) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Frasca, R., Çamlıbel, M.K., Goknar, I.C., Iannelli, L., Vasca, F.: Linear passive networks with ideal switches: consistent initial conditions and state discontinuities. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 57(12), 3138–3151 (2010) CrossRefGoogle Scholar
  19. 19.
    Fuchssteiner, B.: Eine assoziative Algebra über einen Unterraum der Distributionen. Math. Ann. 178, 302–314 (1968) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Fuchssteiner, B.: Algebraic foundation of some distribution algebras. Stud. Math. 76, 439–453 (1984) MathSciNetGoogle Scholar
  21. 21.
    Gantmacher, F.R.: The Theory of Matrices, vols. I & II. Chelsea, New York (1959) zbMATHGoogle Scholar
  22. 22.
    Geerts, A.H.W.T.: Invariant subspaces and invertibility properties for singular systems: the general case. Linear Algebra Appl. 183, 61–88 (1993). doi: 10.1016/0024-3795(93)90424-M MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Geerts, A.H.W.T.: Solvability conditions, consistency and weak consistency for linear differential-algebraic equations and time-invariant linear systems: the general case. Linear Algebra Appl. 181, 111–130 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Geerts, A.H.W.T.: Regularity and singularity in linear-quadratic control subject to implicit continuous-time systems. IEEE Proc. Circuits Syst. Signal Process. 13, 19–30 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Geerts, A.H.W.T., Schumacher, J.M.H.: Impulsive-smooth behavior in multimode systems. Part I: state-space and polynomial representations. Automatica 32(5), 747–758 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Hautus, M.L.J.: The formal Laplace transform for smooth linear systems. In: Marchesini, G., Mitter, S.K. (eds.) Mathematical Systems Theory. Lecture Notes in Economics and Mathematical Systems, vol. 131, pp. 29–47. Springer, New York (1976) CrossRefGoogle Scholar
  27. 27.
    Hautus, M.L.J., Silverman, L.M.: System structure and singular control. Linear Algebra Appl. 50, 369–402 (1983) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kronecker, L.: Algebraische Reduction der Schaaren bilinearer Formen. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin, pp. 1225–1237 (1890) Google Scholar
  29. 29.
    Kuijper, M.: First-Order Representations of Linear Systems. Birkhäuser, Boston (1994) zbMATHCrossRefGoogle Scholar
  30. 30.
    Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich (2006) zbMATHCrossRefGoogle Scholar
  31. 31.
    Lamour, R., März, R., Tischendorf, C.: Differential Algebraic Equations: A Projector Based Analysis. Differential-Algebraic Equations Forum, vol. 1. Springer, Heidelberg (2013) zbMATHCrossRefGoogle Scholar
  32. 32.
    Lewis, F.L.: A survey of linear singular systems. IEEE Proc. Circuits Syst. Signal Process. 5(1), 3–36 (1986) zbMATHCrossRefGoogle Scholar
  33. 33.
    Liberzon, D., Trenn, S.: Switched nonlinear differential algebraic equations: solution theory, Lyapunov functions, and stability. Automatica 48(5), 954–963 (2012). doi: 10.1016/j.automatica.2012.02.041 MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Lundberg, K.H., Miller, H.R., Trumper, D.L.: Initial conditions, generalized functions, and the Laplace transform. IEEE Control Syst. Mag. 27(1), 22–35 (2007). doi: 10.1109/MCS.2007.284506 CrossRefGoogle Scholar
  35. 35.
    Opal, A., Vlach, J.: Consistent initial conditions of linear switched networks. IEEE Trans. Circuits Syst. 37(3), 364–372 (1990) MathSciNetCrossRefGoogle Scholar
  36. 36.
    Owens, D.H., Debeljkovic, D.L.: Consistency and Liapunov stability of linear descriptor systems: a geometric analysis. IMA J. Math. Control Inf. 2, 139–151 (1985) zbMATHCrossRefGoogle Scholar
  37. 37.
    Rabier, P.J., Rheinboldt, W.C.: Time-dependent linear DAEs with discontinuous inputs. Linear Algebra Appl. 247, 1–29 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Rabier, P.J., Rheinboldt, W.C.: Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VIII, pp. 183–537. Elsevier, Amsterdam (2002) Google Scholar
  39. 39.
    Reißig, G., Boche, H., Barton, P.I.: On inconsistent initial conditions for linear time-invariant differential-algebraic equations. IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 49(11), 1646–1648 (2002) CrossRefGoogle Scholar
  40. 40.
    Rosenbrock, H.H.: State Space and Multivariable Theory. Wiley, New York (1970) zbMATHGoogle Scholar
  41. 41.
    Schwartz, L.: Théorie des Distributions. Hermann, Paris (1957, 1959) Google Scholar
  42. 42.
    Sincovec, R.F., Erisman, A.M., Yip, E.L., Epton, M.A.: Analysis of descriptor systems using numerical algorithms. IEEE Trans. Autom. Control AC-26, 139–147 (1981) MathSciNetCrossRefGoogle Scholar
  43. 43.
    Tanwani, A., Trenn, S.: On observability of switched differential-algebraic equations. In: Proc. 49th IEEE Conf. Decis. Control, Atlanta, USA, pp. 5656–5661 (2010) Google Scholar
  44. 44.
    Tolsa, J., Salichs, M.: Analysis of linear networks with inconsistent initial conditions. IEEE Trans. Circuits Syst. 40(12), 885–894 (1993). doi: 10.1109/81.269029 MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Trenn, S.: Distributional differential algebraic equations. Ph.D. thesis, Institut für Mathematik, Technische Universität Ilmenau, Universitätsverlag Ilmenau, Ilmenau, Germany (2009).
  46. 46.
    Trenn, S.: A normal form for pure differential algebraic systems. Linear Algebra Appl. 430(4), 1070–1084 (2009). doi: 10.1016/j.laa.2008.10.004 MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Trenn, S.: Switched differential algebraic equations. In: Vasca, F., Iannelli, L. (eds.) Dynamics and Control of Switched Electronic Systems—Advanced Perspectives for Modeling, Simulation and Control of Power Converters, pp. 189–216. Springer, London (2012). Chap. 6 CrossRefGoogle Scholar
  48. 48.
    Verghese, G.C., Levy, B.C., Kailath, T.: A generalized state-space for singular systems. IEEE Trans. Autom. Control AC-26(4), 811–831 (1981) MathSciNetCrossRefGoogle Scholar
  49. 49.
    Weierstraß, K.: Zur Theorie der bilinearen und quadratischen Formen. Berl. Monatsb., pp. 310–338 (1868) Google Scholar
  50. 50.
    Wilkinson, J.H.: Linear differential equations and Kronecker’s canonical form. In: de Boor, C., Golub, G.H. (eds.) Recent Advances in Numerical Analysis, pp. 231–265. Academic Press, New York (1978) Google Scholar
  51. 51.
    Wong, K.T.: The eigenvalue problem λTx+Sx. J. Differ. Equ. 16, 270–280 (1974) zbMATHCrossRefGoogle Scholar
  52. 52.
    Yip, E.L., Sincovec, R.F.: Solvability, controllability and observability of continuous descriptor systems. IEEE Trans. Autom. Control AC-26, 702–707 (1981) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.University of KaiserslauternKaiserslauternGermany

Personalised recommendations