A Converse Theorem for Riemannian Contraction under Uniform Incremental Exponential Stability

AbstractWe prove that uniform exponential stability of the variational dynamics on a forward-invariant convex domain implies the existence of a Riemannian contraction metric. The metric is constructed explicitly via an integral Gramian representation. Explicit bounds on the contraction rate and robustness under bounded disturbances are derived. The result can be viewed as an incremental analogue of classical converse Lyapunov theorems [5].