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arXiv · 1206.0448

The contraction rate in Thompson metric of order-preserving flows on a cone - application to generalized Riccati equations

Stéphane Gaubert·Zheng Qu
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Abstract

We 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 provides an explicit form of a characterization of Nussbaum concerning non order-preserving flows. As an application of this formula, 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 natural 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, Wojtowski, 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.

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Stéphane Gaubert, Zheng Qu. 2012-06-03. The contraction rate in Thompson metric of order-preserving flows on a cone - application to generalized Riccati equations. https://doi.org/10.1016/j.jde.2014.01.024

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