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

Stéphane Gaubert 1, 2 Zheng Qu 3
2 MAXPLUS - Max-plus algebras and mathematics of decision
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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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https://hal.inria.fr/hal-00783972
Contributor : Stephane Gaubert <>
Submitted on : Saturday, February 2, 2013 - 11:47:23 AM
Last modification on : Thursday, March 5, 2020 - 6:28:12 PM

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Stéphane Gaubert, Zheng Qu. The contraction rate in Thompson part metric of order-preserving flows on a cone - application to generalized Riccati equations. Journal of Differential Equations, Elsevier, 2014, 256 (8), pp.2902-2948. ⟨10.1016/j.jde.2014.01.024⟩. ⟨hal-00783972⟩

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