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首页> 外文期刊>Journal of Differential Equations >The contraction rate in Thompson's part metric of order-preserving flows on a cone - Application to generalized Riccati equations
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The contraction rate in Thompson's part metric of order-preserving flows on a cone - Application to generalized Riccati equations

机译：锥上保序流的汤普森分度量中的收缩率-在广义Riccati方程中的应用

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摘要

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 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.
机译：我们给出了汤普森分量度量中Lipschitz常数的公式，该常数是在（可能是无穷维的）闭合凸尖锥内部上任何保序流的。这表明在特殊情况下，Nussbaum给出了汤普森分量度量中收缩率的一般特征，从而得出了一个明确的公式。作为一个应用，我们表明随机线性二次控制中产生的广义Riccati方程的流是在正定矩阵的圆锥上的局部收缩，并通过矩阵不等式表征其Lipschitz常数。我们还表明，相同的流不再是此锥上其他不变Finsler度量（包括标准不变黎曼度量）的收缩。这是由Bougerol，Liverani，Wojtkowski，Lawson，Lee和Lim建立的与标准Riccati方程有关的一系列收缩特性引起的：我们证明了其中一些特性有效，而另一些则不适用Riccati方程。

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