Regularity of distributional differential algebraic equations

Original Article

Abstract

Time-varying differential algebraic equations (DAEs) of the form \({E\dot{x}=Ax+f}\) are considered. The solutions x and the inhomogeneities f are assumed to be distributions (generalized functions). As a new approach, distributional entries in the time-varying coefficient matrices E and A are allowed as well. Since a multiplication for general distributions is not possible, the smaller space of piecewise-smooth distributions is introduced. This space consists of distributions which could be written as the sum of a piecewise-smooth function and locally finite Dirac impulses and derivatives of Dirac impulses. A restriction can be defined for the space of piecewise-smooth distributions, this restriction is used to study DAEs with inconsistent initial values; basically, it is assumed that some past trajectory for x is given and the DAE is activated at some initial time. If this initial trajectory problem has a unique solution for all initial trajectories and all inhomogeneities, then the DAE is called regular. This generalizes the regularity for classical DAEs (i.e. a DAE with constant coefficients). Sufficient and necessary conditions for the regularity of distributional DAEs are given.

Keywords

Differential algebraic equations Distributional solutions Regularity 

References

  1. 1.
    Campbell SL (1982) Singular Systems of differential equations II. Pitman, New YorkMATHGoogle Scholar
  2. 2.
    Cobb JD (1984) Controllability, observability and duality in singular systems. IEEE Trans Autom Control AC-29: 1076–1082CrossRefMathSciNetGoogle Scholar
  3. 3.
    Colombeau JF (1992) Multiplication of distributions. Lecture Notes in Mathematics, vol 1532. Springer, BerlinGoogle Scholar
  4. 4.
    Doležal V (1964) The existence of a continuous basis of a certain linear subspace of E r, which depends on a parameter. Časopis pro pestovani matematiky 89: 466–468MATHGoogle Scholar
  5. 5.
    Fisher B, Lin-Zhi C (1992) The product of distributions on R n. Commentat Math Univ Carol 33(4): 605–614MATHGoogle Scholar
  6. 6.
    Fuchssteiner B (1968) Eine assoziative Algebra über einen Unterraum der Distributionen. Math Ann 178: 302–314MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Fuchssteiner B (1984) Algebraic foundation of some distribution algebras. Stud Math 76: 439–453MathSciNetGoogle Scholar
  8. 8.
    Geerts AHWT, Schumacher JM (1996) Impulsive-smooth behavior in multimode systems. Part I: State-space and polynomial representations. Automatica 32(5): 747–758MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Geerts AHWT, Schumacher JM (1996) Impulsive-smooth behavior in multimode systems. Part II: Minimality and equivalence. Automatica 32(6): 819–832MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Geerts T (1993) Solvability conditions, consistency and weak consistency for linear differential-algebraic equations and time-invariant linear systems: the general case. Linear Algebra Appl 181: 111–130MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Geerts T (1996) Higher-order continuous-time implicit systems: Consistency and weak consistency, impulse controllability, geometric concepts, and invertibility properties. Linear Algebra Appl 244: 203–253MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hautus MLJ (1976) The formal Laplace transform for smooth linear systems. In: Marchesini G, Mitter SK (eds) Mathematical systems theory. Lecture notes in economics and mathematical systems, vol. 131. Springer, New York, pp 29–47Google Scholar
  13. 13.
    Hautus MLJ, Silverman LM (1983) System structure and singular control. Linear Algebra Appl 50: 369–402MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hinrichsen D, Prätzel-Wolters D (1980) Solution modules and system equivalence. Int J Control 32: 777–802MATHCrossRefGoogle Scholar
  15. 15.
    Hoskins RF, Sousa Pinto JJM (1994) Distributions, ultradistributions and other generalized functions. Mathematics and its applications. Ellis Horwood, Chichester, West Sussex, UKGoogle Scholar
  16. 16.
    Hou M, Pugh AC, Hayton GE (2000) A test for behavioral equivalence. IEEE Trans Autom Control 45(11): 2177–2182MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    König H (1955) Multiplikation von Distributionen, I. Math Ann 128: 420–452MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    König H, Raeder R (1995) Vorlesung über die Theorie der Distributionen. No. 6-1 in Annales Universitatis Saraviensis - Series Mathematicae. Universität des Saarlandes, SaarbrückenGoogle Scholar
  19. 19.
    Kunkel P, Mehrmann V (2006) Differential-Algebraic Equations. Analysis and Numerical Solution. EMS Publishing House, Zürich, SwitzerlandGoogle Scholar
  20. 20.
    Lakshmikantham V, Baĭnov DD, Simeonov PS (1989) Theory of Impulsive Differential Equations. No. 6 in Series in Modern Applied Mathematics. World Scientific, SingaporeGoogle Scholar
  21. 21.
    Liberzon D (2003) Switching in systems and control. Systems and control: foundations and applications. Birkhäuser, BostonGoogle Scholar
  22. 22.
    Mikusiński J (1966) On the square of the Dirac delta-distribution. Bull Acad Pol Sci, Sér Sci Math Astr Phys XIV(9):511–513Google Scholar
  23. 23.
    Oberguggenberger M (1992) Multiplication of distributions and applications to partial differential equations. Pitman research notes in mathematics, vol 259. Longman Scientific & Technical, HarlowGoogle Scholar
  24. 24.
    Rabier PJ, Rheinboldt WC (1996) Time-dependent linear DAEs with discontinuous inputs. Linear Algebra Appl 247: 1–29MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Rabier PJ, Rheinboldt WC (2002) Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet PG, Lions JL (eds) Handbook of numerical analysis, vol VIII. Elsevier Science, Amsterdam, pp 183–537Google Scholar
  26. 26.
    Schwartz L (1957, 1959) Théorie des Distributions. Hermann, ParisGoogle Scholar
  27. 27.
    Sontag ED (1998) Mathematical control theory, deterministic finite dimensional systems, 2nd edn. Springer, New YorkMATHGoogle Scholar
  28. 28.
    Vardulakis AIG, Antoniou EN, Karampetakis N (1999) On the solution and impulsive behaviour of polynomial matrix descriptions of free linear multivariable systems. Int J Control 72(3): 215–228MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Vardulakis AIG, Fragulis G (1989) Infinite elementary divisors of polynomial matrices and impulsive solutions of linear homogeneous matrix differential equations. Circuits Syst Signal Process 8(3): 357–373MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Verghese GC, Levy BC, Kailath T (1981) A generalized state-space for singular systems. IEEE Trans Autom Control AC-26(4): 811–831CrossRefMathSciNetGoogle Scholar
  31. 31.
    Walter HF (1970) Über die Multiplikation von Distributionen in einem Folgenmodell. Math Ann 189: 211–221MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Walter W (1994) Einführung in die Theorie der Distributionen, 3rd edn. BI Wissenschaftsverlag, MannheimGoogle Scholar
  33. 33.
    Weierstraß K (1868) Zur Theorie der bilinearen und quadratischen Formen. Monatsh. Akademie. Wiss., pp 310–338Google Scholar
  34. 34.
    Willems JC (2007) The behavioral approach to open and interconnected systems. IEEE Control Syst Magazine 27(6): 46–99CrossRefMathSciNetGoogle Scholar
  35. 35.
    Yip EL, Sincovec RF (1981) Solvability, controllability and observability of continuous descriptor systems. IEEE Trans Autom Control AC-26: 702–707CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.Institute for MathematicsIlmenau University of TechnologyIlmenauGermany

Personalised recommendations