Aizerman's conjecture

This conjecture, proposed by Mark Aronovich Aizerman in 1949,[1] was proven false but led to the (valid) sufficient criteria on absolute stability.

Suppose that the nonlinearity f is sector bounded, meaning that for some real

satisfies Then Aizerman's conjecture is that the system is stable in large (i.e. unique stationary point is global attractor) if all linear systems with f(e)=ke, k ∈(k1,k2) are asymptotically stable.

There are counterexamples to Aizerman's conjecture such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists with a stable periodic solution, i.e. a hidden oscillation.

[3][4][5][6] However, under stronger assumptions on the system, such as positivity, Aizerman's conjecture is known to hold true.