Construction of a contraction metric by meshless collocation

Abstract

<p style='text-indent:20px;'>A contraction metric for an autonomous ordinary differential equation is a Riemannian metric such that the distance between adjacent solutions contracts over time. A contraction metric can be used to determine the basin of attraction of an equilibrium and it is robust to small perturbations of the system, including those varying the position of the equilibrium. <p style='text-indent:20px;'>The contraction metric is described by a matrix-valued function <inline-formula><tex-math id="M1">\begin{document}$ M(x) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M2">\begin{document}$ M(x) $\end{document}</tex-math></inline-formula> is positive definite and <inline-formula><tex-math id="M3">\begin{document}$ F(M)(x) $\end{document}</tex-math></inline-formula> is negative definite, where <inline-formula><tex-math id="M4">\begin{document}$ F $\end{document}</tex-math></inline-formula> denotes a certain first-order differential operator. In this paper, we show existence, uniqueness and continuous dependence on the right-hand side of the matrix-valued partial differential equation <inline-formula><tex-math id="M5">\begin{document}$ F(M)(x) = -C(x) $\end{document}</tex-math></inline-formula>. We then use a construction method based on meshless collocation, developed in the companion paper [<xref ref-type="bibr" rid="b12">12</xref>], to approximate the solution of the matrix-valued PDE. In this paper, we justify error estimates showing that the approximate solution itself is a contraction metric. The method is applied to several examples.