Extrema without convexity and stability without Lyapunov

Brian D. O. Anderson (Research School of Electrical, Energy and Material Engineering, Australian National University, Acton, ACT, Australia; Hangzhou Dianzi University, Hangzhou, China; and Data61-CSIRO, Acton, ACT, Australia)

Mengbin Ye (Optus–Curtin Centre of Excellence in Artificial Intelligence, Faculty of Science and Engineering, Curtin University, Perth, WA, Australia)

Abstract

The great majority of optimization problems where there is a global minimum are convex, and a great variety of demonstrations of equilibrium point stability of nonlinear systems involve Lyapunov functions. This work illustrates alternative techniques which may allow dispensing with a convexity assumption, or dispensing with use of a Lyapunov function. The techniques are grounded in topology, in particular Morse Theory, and results of Lefschetz–Hopf and Poincaré–Hopf. Illustrations are provided from the literature.

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B. D. O. Anderson (corresponding author) was supported by the Australian Research Council under grant DP-160104500 and DP190100887, and by Data61-CSIRO.

M. Ye was supported by the European Research Council (ERC-CoG-771687), and by Optus Business.

Received 6 December 2019

Published 2 December 2020