Nonnegative solutions of algebraic Riccati equations
type: Item
instance of: scholarly article
title: Nonnegative solutions of algebraic Riccati equations
zbMATH Open document ID: 0881.15017
DOI: 10.1016/S0024-3795(96)00444-2
published in: Linear Algebra and its Applications
publication date: 1997-08-05T00:00:00Z
review text: For the Riccati equation (RE) \(XBJB^*X -XA -A^*X -C^*C =0\) one studies the Hermitian solution which is stabilizing for the pair \((A,-BJB^*)\). The approach relies on notions and results from the theory of indefinite inner product spaces and on the Hamiltonian associated to the \(RE\). For the case \(C=0\) a very explicit criterion for nonnegativity of the stabilizing solution in terms of certain Jordan chains of \(A\) is formulated. For the case \(J= I\) and \((A,B)\) controllable the full set of nonnegative solutions is considered. A parametrization of this set is discussed and the inertia of Hermitian solutions is studied. Applications to topics arising from \(LQ\) optimal control like isolatedness, order structure, and stability are given. Finally the discrete-time counterpart of the results are discussed.
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93D15
zbMATH DE Number: 1044263
zbMATH Keywords: algebraic Riccati equation
zbMATH Keywords: nonnegative Hermitian solution
zbMATH Keywords: stabilizing solution
zbMATH Keywords: indefinite inner product space
zbMATH Keywords: optimal control
zbMATH Keywords: Hamiltonian
zbMATH Keywords: Jordan chains
zbMATH Keywords: order structure
zbMATH Keywords: stability
author: Heinz Langer
author: André C. M. Ran
author: Dirk Temme
reviewed by: Mihail Voicu
MaRDI profile type: Publication
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Wikidata QID: Q127352940
description: scientific article; zbMATH DE number 1044263