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  • Differential-algebraic  (1)
  • Mathematics Subject Classification (1991):65L20  (1)
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  • 1
    Electronic Resource
    Electronic Resource
    The index of general nonlinear DAEs (1995)
    Springer
    Numerische Mathematik 72 (1995), S. 173-196 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65L20
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary. In the last few years there has been considerable research on differential algebraic equations (DAEs) $F(t, y, y') = 0$ where $F_{y'}$ is identically singular. Much of the mathematical effort has focused on computing a solution that is assumed to exist. More recently there has been some discussion of solvability of DAEs. There has historically been some imprecision in the use of the two key concepts of solvability and index for DAEs. The index is also important in control and systems theory but with different terminology. The consideration of increasingly complex nonlinear DAEs makes a clear and correct development necessary. This paper will try to clarify several points concerning the index. After establishing some new and more precise terminology that we need, some inaccuracies in the literature will be corrected. The two types of indices most frequently used, the differentiation index and the perturbation index, are defined with respect to solutions of unperturbed problems. Examples are given to show that these indices can be very different for the same problem. We define new "maximum indices," which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other. These indices are also related to an index defined in terms of Jacobians.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050165
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  • 2
    Electronic Resource
    Electronic Resource
    Duality, observability, and controllability for linear time-varying descriptor systems (1991)
    Springer
    Circuits, systems and signal processing 10 (1991), S. 455-470 
    Details
    ISSN: 1531-5878
    Keywords: 34A08 ; 93B07 ; 93B15 ; 93C15 ; 93C50 ; Key words ; Descriptor ; Singular ; Differential-algebraic ; Observability ; Controllability ; Duality ; Structural forms
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology
    Notes: Abstract A characterization of observability for linear time-varying descriptor systemsE(t)x′(t)+F(t)x(t)=B(t)u(t), y(t)=C(t)x(t) was recently developed. NeitherE norC were required to have constant rank. This paper defines a dual system, and a type of controllability so that observability of the original system is equivalent to controllability of the dual system. Criteria for observability and controllability are given in terms of arrays of derivatives of the original coefficients. In addition, the duality results of this paper lead to an improvement on a previous fundamental structure result for solvable systems of the formE(t)x′(t)+F(t)x(t)=f(tt).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01194883
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    Paper (German National Licenses)
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