1. A. Oliveira, A. C. B. de Oliveira, M. Sznaier, and E. D. Sontag. On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities. In Proc. 65th IEEE Conference on Decision and Control (CDC), 2026. Note: Submitted.Keyword(s): gradient dynamics, gradient descent, gradient systems, numerical methods, dynamics of algorithms, gradient dominance, gradient flows. Abstract:

   The Lojasiewicz inequality characterizes objective-value convergence along gradient flows and, in special cases, yields exponential decay of the cost. However, such results do not directly imply convergence of the state. In this paper, we use contraction theory to derive state-space guarantees for gradient systems satisfying generalized Lojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Lojasiewicz 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 Lojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable, a property strictly stronger than global exponential stability. A few examples are presented at the end of the paper to validate the proposed theory.

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