Numerische Mathematik

, Volume 135, Issue 2, pp 431–458 | Cite as

The ADI method for bounded real and positive real Lur’e equations

  • Arash Massoudi
  • Mark R. Opmeer
  • Timo ReisEmail author


We propose an algorithm for the numerical solution of the Lur’e equations in the bounded real and positive real lemma for stable systems. The algorithm provides approximate solutions in low-rank factored form. We prove that the sequence of approximate solutions is monotonically increasing with respect to definiteness. If the shift parameters are chosen appropriately, the sequence is proven to be convergent to the minimal solution of the Lur’e equations. The algorithm is based on the ideas of the recently developed ADI iteration for algebraic Riccati equations (Massoudi et al., SIAM J Matrix Anal Appl, 2016). In particular, the matrices obtained in our iteration express the optimal cost in a certain projected optimal control problem.

Mathematics Subject Classification

15A24 49N10 47J20 65F30 49M30 93B52 65K10 



This work was supported by the Klaus-Tschira-Stiftung.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität HamburgHamburgGermany
  2. 2.Department of Mathematical SciencesUniversity of BathBathUK

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