Perturbation bounds and characterisation of the solution of the associated algebraic Riccati equation
type: Item
instance of: scholarly article
title: Perturbation bounds and characterisation of the solution of the associated algebraic Riccati equation
zbMATH Open document ID: 0933.15025
DOI: 10.1016/S0024-3795(98)10094-0
author: M. O. Stanislavova
author: Mihail M. Konstantinov
author: Petko Hr. Petkov
published in: Linear Algebra and its Applications
publication date: 1999-11-28T00:00:00Z
review text: The following complex algebraic matrix quadratic equation is studied: \[ Q+A^*X+X^*A-X^*MX=0,\tag{1} \] where \(Q,M,A\) are given complex \(n\times n\) matrices, and \(X\) is an unknown \(n\times n\) matrix. The standing hypotheses are: \(Q\) and \(M\) are positive semidefinite, the pair \((Q,A)\) is detectable, and the pair \((A,M)\) is stabilizable. The above equation is closely related to the ubiquitous algebraic matrix Riccati equation which has the same form (1), but only Hermitian solutions \(X\) are sought. An effective parametrization of the solutions of (1) is obtained: The parameter set is the group of \(n\times n\) unitary matrices with a complex algebraic set removed; in particular, the set of solutions of (1) has no isolated points. Allowing for a small perturbations of the coefficients \(Q,M\) and \(A\), local linear perturbation bounds are obtained for solutions of the perturbed equation (1). These bounds are based on computation of partial Fréchet derivatives, and are valid asymptotically when the perturbations of \(Q,M\), and \(A\) tend to zero. Another type of perturbation bounds is derived as well, which is applicable in well-defined neighborhoods of the coefficients.
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 1371030
zbMATH Keywords: perturbation bounds
zbMATH Keywords: complex algebraic matrix quadratic equation
zbMATH Keywords: algebraic matrix Riccati equation
zbMATH Keywords: Hermitian solutions
reviewed by: Leiba Rodman
describes a project that uses: EISPACK
describes a project that uses: LINPACK
describes a project that uses: LAPACK
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1371030