Riccati equations for strongly stabilizable bounded linear systems
type: Item
instance of: scholarly article
title: Riccati equations for strongly stabilizable bounded linear systems
zbMATH Open document ID: 0979.93092
DOI: 10.1016/S0005-1098(98)00035-1
author: Job C. Oostveen
author: Ruth Curtain
published in: Automatica
publication date: 2002-02-19T00:00:00Z
review text: The authors study the Riccati equation \[ A^* Xx+ XAx- (B^* X+ NC)^*\;R^{-1}(B^* X+ NC)x+ C^* QCx= 0.\tag{1} \] It is assumed that it generates a \(C_0\) semigroup \(T(t)\). \(B\) is a linear operator between Hilbert spaces \(U\) and \(Z\), where \(U\) is separable, \(C\) between \(Z\) and \(Y\), where \(Y\) is also a separable Hilbert space. In the last ten years several papers appeared considering spectral factorization of this type of Riccati equation. All of them allow the operators \(B\) and \(C\) to be unbounded. Here, the authors consider only the special case when they are bounded, but they relax other conditions assumed by other authors and also obtain much stronger results. They address existence and uniqueness of strongly stabilizable self-adjoint solutions. As the author state, the more recent papers on strong stabilizability and detectability obtained only sufficient conditions for the existence of solutions. The authors conditions are now necessary and sufficient. The present study is influenced by the papers of Weiss. The class of problems which is studied here arises in mathematical modelling of large structures with collocated actuators and sensors, and in general studies of structural dynamics. The authors consider a vibrating square plate with two parallel edges simply supported, a third one is clamped and the remaining free edge is subject to a control force and a control torque. They quote a paper by Y. You, where it was shown that \(A\) is a closed, densely defined skew-adjoint operator with compact support and compact resolvent. The reviewer comments that this is a true statement for most structural dynamics operators without internal dissipation. The system is reduced by well known procedure to a first-order vector formulation \[ dz/dt= Az(t)+ Bu(t),\quad y(t)= B^*z(t). \] It is also shown that \(A\)-\(BB^*\) generates a strongly stable contraction semigroup \(T_B(t)\). The authors proceed to discuss some properties of linear bounded operator…
reviewed by: Vadim Komkov
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C25
Mathematics Subject Classification ID: 74M05
Mathematics Subject Classification ID: 15A23
Mathematics Subject Classification ID: 93B28
zbMATH DE Number: 1325616
zbMATH Keywords: Riccati equation
zbMATH Keywords: operator between Hilbert spaces
zbMATH Keywords: spectral factorization
zbMATH Keywords: existence and uniqueness
zbMATH Keywords: strongly stabilizable self-adjoint solutions
zbMATH Keywords: strong stabilizability
zbMATH Keywords: large structures
zbMATH Keywords: collocated actuators and sensors
zbMATH Keywords: structural dynamics
zbMATH Keywords: bounded operators
zbMATH Keywords: \(\tau\) concatenation
zbMATH Keywords: input-output maps
zbMATH Keywords: Popov function
zbMATH Keywords: Wiener-Hopf factorization
zbMATH Keywords: \(J\)-spectral factorization
zbMATH Keywords: stabilizing solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1325616