Skip to main content
SpringerLink
Log in

Analysis of Linear Variable Coefficient Delay Differential-Algebraic Equations

  • Published:
Journal of Dynamics and Differential Equations Aims and scope Submit manuscript

Abstract

The analysis of general linear variable coefficient delay differential-algebraic systems (DDAEs) is presented. The solvability for DDAEs is investigated and a reformulation procedure to regularize a given DDAE is developed. Based on this regularization procedure existence and uniqueness of solutions and consistency of initial functions is analyzed as well as other structural properties of DDAEs like smoothness requirements. We also present some examples to demonstrate that for the numerical solution of a DDAE, a reformulation of the system before applying numerical methods is essential.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Ascher, U.M., Petzold, L.R.: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. SIAM Publications, Philadelphia (1998)

  2. Ascher, U.M., Petzold, L.R.: The numerical solution of delay-differential algebraic equations of retarded and neutral type. SIAM J. Numer. Anal. 32, 1635–1657 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Baker, C.T.H., Paul, C.A.H., Tian, H.: Differential algebraic equations with after-effect. J. Comput. Appl. Math. 140(1–2), 63–80 (Mar. 2002)

  4. Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003)

    Book  MATH  Google Scholar 

  5. Brenan, K.E., Campbell, S.L., Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations, 2nd edn. SIAM Publications, Philadelphia (1996)

    MATH  Google Scholar 

  6. Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 2, 129–136 (1980)

    Article  Google Scholar 

  7. Campbell, S.L.: Nonregular 2D descriptor delay systems. IMA J. Math. Control Appl. 12, 57–67 (1995)

    Article  MATH  Google Scholar 

  8. Campbell, S.L., Linh, V.H.: Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions. Appl. Math Comput. 208(2), 397–415 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  9. Caraballo, T., Real, J.: Navier–Stokes equations with delays. Proc. R. Soc. A 457, 2441–2453 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dieci, L., Eirola, T.: On smooth decompositions of matrices. SIAM J. Matrix Anal. Appl. 20, 800–819 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Garrido-Atienza, M.J., Marín-Rubio, P.: Navier-Stokes equations with delays on unbounded domains. Nonlinear Anal. 64(5), 1100–1118 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gear, C. W.: Simulation: conflicts between real-time and software. In Mathematical Software III, Proceedings of the Symposium at the University of Wisconsin, March 28–30, 1977, (ed. J.R. Rice), pp. 121–138. Academic Press, New York (1977)

  13. Griepentrog, E., März, R.: Differential-Algebraic Equations and their Numerical Treatment. Teubner Verlag, Leipzig (1986)

    MATH  Google Scholar 

  14. Grossmann, C., Roos, H., Stynes, M.: Numerical Treatment of Partial Differential Equations. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  15. Guglielmi, N., Hairer, E.: Computing breaking points in implicit delay differential equations. Adv. Comput. Math. 29, 229–247 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ha, P., Mehrmann, V.: Analysis and reformulation of linear delay differential-algebraic equations. Electron. J. Linear Alg. 23, 703–730 (2012)

    MATH  MathSciNet  Google Scholar 

  17. Hairer, E., Wanner, G.: Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, 2nd edn. Springer, Berlin (1996)

    Book  MATH  Google Scholar 

  18. Heeb, H., Ruehli, A.: Retarded models for PC board interconnects-or how the speed of light affects your spice circuit simulation. In IEEE International Conference on Computer-Aided Design, 1991. ICCAD-91, pp. 70–73 (1991)

  19. Hollot, C., Misra, V., Towsley, D., Gong, W.: Analysis and design of controllers for AQM routers supporting TCP flows. IEEE Trans. Automat. Control 47(6), 945–959 (2002)

    Article  MathSciNet  Google Scholar 

  20. Kelly, F.: Mathematical modelling of the internet. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited-2001 and Beyond, pp. 685–702. Springer, Berlin (2001)

    Chapter  Google Scholar 

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

    Book  Google Scholar 

  22. Liu, W.: Asymptotic behavior of solutions of time-delayed Burgers’ equation. Discrete Contin. Dyn. Syst. Ser. B. 2(1), 47–56 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Mehrmann, V., Shi, C.: Transformation of high order linear differential-algebraic systems to first order. Numer. Alg. 42, 281–307 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Misra, V., Gong, W.B., Towsley, D.: Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to red. SIGCOMM Comput. Commun. Rev. 30(4), 151–160 (Aug. 2000)

  25. Planas, G., Hernández, E.: Asymptotic behaviour of two-dimensional time-delayed Navier–Stokes equations. Discrete Contin. Dyn. Syst. Ser. S. 21(4), 1245–1258 (2008)

    Article  MATH  Google Scholar 

  26. Riaza, R.: Differential-Algebraic Systems: Analytical Aspects and Circuit Applications. World Scientific Publishing Co., Pvt. Ltd, Hackensack (2008)

    Book  Google Scholar 

  27. Shampine, L.F., Gahinet, P.: Delay-differential-algebraic equations in control theory. Appl. Numer. Math. 56(3–4), 574–588 (Mar. 2006)

  28. Steinbrecher, A.: Regularization of quasi-linear differential-algebraic equations in multibody dynamics. In Proceedings of Multibody Dynamics 2011-ECCOMAS Thematic Conference Brussels (Bruxelles, Belgium, July 4–7, 2011) (2011)

  29. Wim, M., Silviu-Iulian, N.: Stability and Stabilization of Time-Delay Systems: An Eigenvalue-Based Approach. SIAM, Philadelphia (2007)

    Google Scholar 

  30. Zhong, Q.: Robust Control of Time-Delay Systems. Springer, Berlin (2006)

    MATH  Google Scholar 

Download references

Acknowledgments

We thank Vinh Tho Ma for carrying the numerical simulations in Examples 5.1 and 5.2. Phi Ha has been supported by Deutsche Forschungsgemeinschaft through Sonderforschungsbereich 910 Control of self-organizing nonlinear systems: Theoretical methods and application concepts. V. Mehrmann and A. Steinbrecher has been supported by European Research Council through Advanced Grant MODSIMCONMP.

Author information

Authors and Affiliations

  1. Institut für Mathematik, MA 4-5, TU Berlin, Straße des 17. Juni 136, 10623 , Berlin, Germany

    Phi Ha, Volker Mehrmann & Andreas Steinbrecher

Authors
  1. Phi Ha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Volker Mehrmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Andreas Steinbrecher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Volker Mehrmann.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ha, P., Mehrmann, V. & Steinbrecher, A. Analysis of Linear Variable Coefficient Delay Differential-Algebraic Equations. J Dyn Diff Equat 26, 889–914 (2014). https://doi.org/10.1007/s10884-014-9386-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10884-014-9386-x

Keywords

Mathematics Subject Classification

Search

Navigation