Communications in Information and Systems

Volume 20 (2020)

Number 3

Mathematical Engineering: A special issue at the occasion of the 85th birthday of Prof. Thomas Kailath

Guest Editors: Ali H. Sayed, Helmut Bölcskei, Patrick Dewilde, Vwani Roychowdhury, and Stephen Shing-Toung Yau

Extrema without convexity and stability without Lyapunov

Pages: 253 – 281



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)


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.

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