On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917363492847616 |
|---|---|
| author | Oliveira, Andreas de Oliveira, Arthur C. B. Sznaier, Mario Sontag, Eduardo |
| author_facet | Oliveira, Andreas de Oliveira, Arthur C. B. Sznaier, Mario Sontag, Eduardo |
| contents | The Łojasiewicz 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 Łojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Łojasiewicz 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 Łojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25822 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities Oliveira, Andreas de Oliveira, Arthur C. B. Sznaier, Mario Sontag, Eduardo Optimization and Control Systems and Control The Łojasiewicz 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 Łojasiewicz inequalities. We first show that, when the objective has a unique strongly convex minimizer, the generalized Łojasiewicz 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 Łojasiewicz rate, under which the nonconvex gradient flow is globally incrementally exponentially stable. |
| title | On incremental and semi-global exponential stability of gradient flows satisfying generalized Łojasiewicz inequalities |
| topic | Optimization and Control Systems and Control |
| url | https://arxiv.org/abs/2603.25822 |