Converse theorems on contraction metrics for an equilibrium

The stability and basin of attraction of an equilibrium can be determined by a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent trajectories decreases. The advantage of a contraction metric over, e.g., a Lyapunov function is that the contraction condition is robust under perturbations of the system. While the sufficiency of a contraction metric for the existence, stability and basin of attraction of an equilibrium has been extensively studied, in this paper we will prove converse theorems, showing the existence of several different contraction metrics. This will be useful to develop algorithms for the construction of contraction metrics