On incremental and semi-global exponential stability of gradient flows satisfying generalized {\L}ojasiewicz inequalities

The {\L}ojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly give rates of convergence in the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized {\L}ojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized {\L}ojasiewicz inequality implies semi-global exponential stability; on arbitrary compact subsets, this yields exponential stability. We then give two curvature-based sufficient conditions, together with constraints on the {\L}ojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable.