A Galois correspondence between sets of semidefinite solutions of continuous-time algebraic Riccati equations
type: Item
instance of: scholarly article
title: A Galois correspondence between sets of semidefinite solutions of continuous-time algebraic Riccati equations
zbMATH Open document ID: 0809.34016
DOI: 10.1016/0024-3795(94)90386-7
author: Harald K. Wimmer
published in: Linear Algebra and its Applications
publication date: 1994-08-15T00:00:00Z
review text: The paper is devoted to comparison of solutions of Riccati equations. Let \[ {\mathcal R}_ i(X)= A^*_ i X+ X A_ i+ X B_ i B^*_ i X- C_ i^*C_ i= 0\tag{1} \] be an algebraic Riccati equation, where \(A_ i\), \(B_ i\), \(C_ i\) are complex matrices of sizes \(n\times n\), \(n\times p\), \(q\times n\) respectively. Let \({\mathcal T}_ i= \{X\mid X\leq 0,\;{\mathcal R}_ i(X)= 0\}\) denote the set of negative semidefinite solutions of (1) and \[ {\mathcal H}_ i= \left[\begin{matrix} -C^*_ i C_ i & A^*_ i\\ A_ i & B_ i B^*_ i\end{matrix}\right]. \] Under the assumption that \({\mathcal H}_ 1\leq {\mathcal H}_ 2\), the paper compares solutions in \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\). In particular, the author shows that if \(X_{i^ +}\) is the greatest solution of (1) then \(X_{1^ +}\leq X_{2^ +}\). A similar inequality for the least element is also true. The paper also establishes a Galois correspondence between other elements in \({\mathcal T}_ 1\) and \({\mathcal T}_ 2\), rather than the greatest and the least elements.
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34C11
zbMATH DE Number: 614140
zbMATH Keywords: comparison of solutions
zbMATH Keywords: Riccati equations
zbMATH Keywords: Galois correspondence
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90386-7
OpenAlex ID: W2162607105
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description: scientific article; zbMATH DE number 614140