Controllability of Linear Differential-Algebraic Systems—A Survey

  • Thomas BergerEmail author
  • Timo Reis
Part of the Differential-Algebraic Equations Forum book series (DAEF)


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.


Differential-algebraic equations Controllability Stabilizability Kalman decomposition Canonical form Feedback Hautus criterion Invariant subspaces 

Mathematics Subject Classification (2010)

34A09 15A22 93B05 15A21 93B25 93B27 93B52 



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


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für MathematikTechnische Universität IlmenauIlmenauGermany
  2. 2.Fachbereich MathematikUniversität HamburgHamburgGermany

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