The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases. The Riemannian metric can be expressed by a matrix-valued function on the phase space.

The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF). However, so far no rigorous verification that the computed metric is indeed a contraction metric has been provided.

In this paper, we combine the RBF method to compute a contraction metric with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification. We apply the method to two examples.

可以通过收缩度量来确定由一般自治常微分方程给出的动力系统的指数稳定平衡及其吸引域。收缩度量是黎曼度量，相对而言，相邻解决方案之间的距离随时间增加而减小。黎曼度量可以由相空间上的矩阵值函数表示。

可以通过使用径向基函数（RBF）通过无网格搭配近似求解矩阵值的偏微分方程来实现收缩度量的确定。但是，到目前为止，尚未提供关于所计算的度量确实是收缩度量的严格验证。

在本文中，我们结合了RBF方法和CPA方法来计算收缩率，以对其进行严格验证。特别是，在固定三角剖分的顶点处通过连续分段仿射（CPA）度量对计算的收缩度量进行插值，并且通过有限地检查不等式，我们可以验证该插值是收缩度量。而且，我们表明，使用足够密集的搭配点和足够精细的三角剖分，我们始终能够成功进行构造和验证。我们将该方法应用于两个示例。