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IMA Journal of Mathematical Control and Information 2008 25(4):515-546; doi:10.1093/imamci/dnn011
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Weakly coprime factorization and continuous-time systems

Kalle M. Mikkola{dagger}

Helsinki University of Technology, Institute of Mathematics, PO Box 1100, FIN-02015 HUT, Finland

{dagger} Email: kalle.mikkola{at}iki.fi

We give many necessary and sufficient conditions for the existence of a weakly coprime or Bézout coprime factorization of a transfer function, possibly operator valued. Some of these conditions are given in terms of the output- or state-feedback stabilizability of realizations. Our realizations are well-posed linear systems—continuous-time linear time-invariant infinite-dimensional systems. We also study further properties of such factorizations, their relations to discrete-time weakly coprime factorizations, counterexamples and (weak) left invertibility. Moreover, analogous discrete-time results are obtained. Control-theoretic consequences are indicated.

Keywords: weak coprimeness; stabilizable realization; stabilizable and detectable realization; LQ optimal state feedback; infinite-dimensional systems theory; well-posed linear systems; holomorphic fractions.


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