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Surveys in Differential-Algebraic Equations I pp 1–61Cite as

Controllability of Linear Differential-Algebraic Systems—A Survey

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Part of the book series: Differential-Algebraic Equations Forum ((DAEF))

Abstract

Different concepts related to controllability of differential-algebraic equations are described. The class of systems considered consists of linear differential-algebraic equations with constant coefficients. Regularity, which is, loosely speaking, a concept related to existence and uniqueness of solutions for any inhomogeneity, is not required in this article. The concepts of impulse controllability, controllability at infinity, behavioral controllability, and strong and complete controllability are described and defined in the time domain. Equivalent criteria that generalize the Hautus test are presented and proved.

Special emphasis is placed on normal forms under state space transformation and, further, under state space, input and feedback transformations. Special forms generalizing the Kalman decomposition and Brunovský form are presented. Consequences for state feedback design and geometric interpretation of the space of reachable states in terms of invariant subspaces are proved.

Thomas Berger was supported by DFG grant IL 25/9 and partially supported by the DAAD.

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Acknowledgements

We are indebted to Harry L. Trentelman (University of Groningen) for providing helpful comments on the behavioral approach.

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  1. Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, 98693, Ilmenau, Germany

    Thomas Berger

  2. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, 20146, Hamburg, Germany

    Timo Reis

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  1. Institut für Mathematik, Technische Universität Ilmenau, Weimarer Straße 25, Ilmenau, 98693, Germany

    Achim Ilchmann

  2. Fachbereich Mathematik, Universität Hamburg, Bundesstraße 55, Hamburg, 20146, Germany

    Timo Reis

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Berger, T., Reis, T. (2013). Controllability of Linear Differential-Algebraic Systems—A Survey. In: Ilchmann, A., Reis, T. (eds) Surveys in Differential-Algebraic Equations I. Differential-Algebraic Equations Forum. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34928-7_1

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