Absolute stability approach to stochastic stability of infinite-dimensional nonlinear systems
type: Item
instance of: scholarly article
title: Absolute stability approach to stochastic stability of infinite-dimensional nonlinear systems
zbMATH Open document ID: 0840.93094
DOI: 10.1016/0005-1098(95)00063-3
author: V. A. Brusin
author: Valery Ugrinovskii
published in: Automatica
publication date: 1996-07-04T00:00:00Z
review text: The authors consider a system described by the semilinear stochastic evolution equation \[ dx= [Ax+ b\varphi(x)]dt+ [Cx+ D\varphi(x)] dw(t),\quad x(s)= h,\tag{1} \] where \(A\) is an operator with domain \(D(A)\subseteq H\), \(b\in {\mathcal L}(U, H)\), \(C\in {\mathcal L}(H, {\mathcal L}(W, H))\), \(D\in {\mathcal L}(U, {\mathcal L}(W, H))\) and \(\varphi: H\to U\) is a Lipschitz continuous nonlinear operator with \(\varphi(0)= 0\), \(W\), \(H\) and \(U\) are real separable Hilbert spaces and \({\mathcal L}(W, H)\) is the space of bounded linear operators \(W\to H\). To obtain sufficient conditions for absolute stability of the system (1), the authors derived a stochastic Lyapunov function and introduced the special stochastic infinite-dimensional counter-part of the Kalman-Yakubovich lemma. Examples show extensions of the Popov and circle criteria to the feedback stochastic heat and delay equations, respectively.
Mathematics Subject Classification ID: 93E15
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 93D10
Mathematics Subject Classification ID: 93D30
zbMATH DE Number: 819362
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: operator
zbMATH Keywords: absolute stability
zbMATH Keywords: Lyapunov function
zbMATH Keywords: Kalman-Yakubovich lemma
reviewed by: L. Socha
MaRDI profile type: Publication
full work available at URL: 0005-1098(95)00063-3
OpenAlex ID: W2059594444
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description: scientific article; zbMATH DE number 819362