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The Index of an Infinite Dimensional Implicit System

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  • S.L. Campbell
  • W. Marszalek

Abstract

The idea of the index of a differential algebraic equation (DAE) (or implicit differential equation) has played a fundamental role in both the analysis of DAEs and the development of numerical algorithms for DAEs. DAEs frequently arise as partial discretizations of partial differential equations (PDEs). In order to relate properties of the PDE to those of the resulting DAE it is necessary to have a concept of the index of a possibly constrained PDE. Using the finite dimensional theory as motivation, this paper will examine what one appropriate analogue is for infinite dimensional systems. A general definition approach will be given motivated by the desire to consider numerical methods. Specific examples illustrating several kinds of behavior will be considered in some detail. It is seen that our definition differs from purely algebraic definitions. Numerical solutions, and simulation difficulties, can be misinterpreted if this index information is missing.

Suggested Citation

  • S.L. Campbell & W. Marszalek, 1999. "The Index of an Infinite Dimensional Implicit System," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 5(1), pages 18-42, March.
    • Handle: RePEc:taf:nmcmxx:v:5:y:1999:i:1:p:18-42
    DOI: 10.1076/mcmd.5.1.18.3625
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    Cited by:

    1. K. Sternberg & K. Chudej & H. J. Pesch, 2007. "Suboptimal control of a 2D molten carbonate fuel cell PDAE model," Mathematical and Computer Modelling of Dynamical Systems, Taylor & Francis Journals, vol. 13(5), pages 471-485, October.
    2. Günther, Michael, 2000. "Semidiscretization may act like a deregularization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 53(4), pages 293-301.
    3. Strömgren, Magnus & Hanke, Michael, 2009. "On the numerical approximation of a degenerated hyperbolic system," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(5), pages 1585-1602.
    4. Hanke, Michael & Henrik, K. & Olsson, A. & Strömgren, Magnus, 2007. "Stability analysis of a degenerate hyperbolic system modelling a heat exchanger," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 74(1), pages 8-19.

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