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

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• 作者：Arash Massoudi ; Mark R. Opmeer ; Timo Reis
• 关键词：Mathematics Subject Classification15A24 ; 49N10 ; 47J20 ; 65F30 ; 49M30 ; 93B52 ; 65K10
• 刊名：Numerische Mathematik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：135
• 期：2
• 页码：431-458
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Numerical Analysis; Mathematics, general; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation; Appl.Mathematics/Comput
• 出版者：Springer Berlin Heidelberg
• ISSN：0945-3245
• 卷排序：135

文摘

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.