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Article: The contraction rate in Thompson's part metric of order-preserving flows on a cone - Application to generalized Riccati equations

TitleThe contraction rate in Thompson's part metric of order-preserving flows on a cone - Application to generalized Riccati equations
Authors
KeywordsFinsler metric
Contraction rate
Perron-Frobenius theory
Riccati equation
Stochastic control
Thompson's part metric
Issue Date2014
Citation
Journal of Differential Equations, 2014, v. 256, n. 8, p. 2902-2948 How to Cite?
AbstractWe give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation. © 2014 Elsevier Inc.
Persistent Identifierhttp://hdl.handle.net/10722/219740
ISSN
2015 Impact Factor: 1.821
2015 SCImago Journal Rankings: 2.809

 

DC FieldValueLanguage
dc.contributor.authorGaubert, Stéphane-
dc.contributor.authorQu, Zheng-
dc.date.accessioned2015-09-23T02:57:51Z-
dc.date.available2015-09-23T02:57:51Z-
dc.date.issued2014-
dc.identifier.citationJournal of Differential Equations, 2014, v. 256, n. 8, p. 2902-2948-
dc.identifier.issn0022-0396-
dc.identifier.urihttp://hdl.handle.net/10722/219740-
dc.description.abstractWe give a formula for the Lipschitz constant in Thompson's part metric of any order-preserving flow on the interior of a (possibly infinite dimensional) closed convex pointed cone. This shows that in the special case of order-preserving flows, a general characterization of the contraction rate in Thompson's part metric, given by Nussbaum, leads to an explicit formula. As an application, we show that the flow of the generalized Riccati equation arising in stochastic linear quadratic control is a local contraction on the cone of positive definite matrices and characterize its Lipschitz constant by a matrix inequality. We also show that the same flow is no longer a contraction in other invariant Finsler metrics on this cone, including the standard invariant Riemannian metric. This is motivated by a series of contraction properties concerning the standard Riccati equation, established by Bougerol, Liverani, Wojtkowski, Lawson, Lee and Lim: we show that some of these properties do, and that some other do not, carry over to the generalized Riccati equation. © 2014 Elsevier Inc.-
dc.languageeng-
dc.relation.ispartofJournal of Differential Equations-
dc.subjectFinsler metric-
dc.subjectContraction rate-
dc.subjectPerron-Frobenius theory-
dc.subjectRiccati equation-
dc.subjectStochastic control-
dc.subjectThompson's part metric-
dc.titleThe contraction rate in Thompson's part metric of order-preserving flows on a cone - Application to generalized Riccati equations-
dc.typeArticle-
dc.description.natureLink_to_subscribed_fulltext-
dc.identifier.doi10.1016/j.jde.2014.01.024-
dc.identifier.scopuseid_2-s2.0-84893753755-
dc.identifier.volume256-
dc.identifier.issue8-
dc.identifier.spage2902-
dc.identifier.epage2948-
dc.identifier.eissn1090-2732-

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