In nonlinear control, Aizerman's conjecture or Aizerman problem states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of the sector. This conjecture, proposed by Mark Aronovich Aizerman in 1949, was proven false but led to the (valid) sufficient criteria on absolute stability.

Mathematical statement of Aizerman's conjecture (Aizerman problem)

Consider a system with one scalar nonlinearity ''where P is a constant n×n-matrix, q, r are constant n-dimensional vectors, ∗ is an operation of transposition, f(e) is scalar function, and f(0)=0. Suppose that the nonlinearity f is sector bounded, meaning that for some real'' k_1 and k_2 with k_1 <k_2, the function f 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. However, under stronger assumptions on the system, such as positivity, Aizerman's conjecture is known to hold true.

Variants