On Professor Smale's legacy for asymptotic stability theory

on YouTube ↗27m

" summary

Gives a concrete classification: the space of smooth asymptotically stable vector fields on R^n and the space of proper smooth Lyapunov functions are, for n ≠ 4,5, path-connected and simply connected and are weakly contractible when n < 3. Proofs combine Lyapunov theory with differential topology—using Wilson, the disk theorem, Smale’s h‑cobordism, Perelman, Cerf and Hatcher—to show the sublevel‑set map is a fiber bundle with weakly contractible fibers, yielding applications like a parametric Hartman‑Grobman for nonhyperbolic equilibria and a partial answer to Conley.

AI-written summary · about

" chapters (18)

1. 0:00Honoring Professor Smale’s 95th birthday
2. 0:20Asymptotic stability
3. 1:42Why engineers care
4. 2:49Controllers and open questions
5. 3:53Stable vector fields topology
6. 5:17Dimensions 4 and 5
7. 6:20Conley’s question and uses
8. 7:45Hartman-Grobman theorem
9. 8:49Almost non-singular fields
10. 9:12Proof outline
11. 10:14Lyapunov functions
12. 12:00Smale’s level sets
13. 13:46Nonlinear Grassmannian
14. 15:30Fiber bundle proof
15. 18:02Smale’s conjecture
16. 21:22Contractibility n<4
17. 23:53Conclusion
18. 24:18Q&A unsolved cases