Aizerman's conjecture

Consider a system with one scalar nonlinearity

${\displaystyle {\frac {dx}{dt}}=Px+qf(e),\quad e=r^{*}x\quad x\in \mathbb {R} ^{n},}$

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 ${\displaystyle k_{1}}$ and ${\displaystyle k_{2}}$ with ${\displaystyle k_{1}<k_{2}}$, the function ${\displaystyle f}$ satisfies

${\displaystyle k_{1}<{\frac {f(e)}{e}}<k_{2},\quad \forall \;e\neq 0.}$

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.[2][3][4][5] However, under stronger assumptions on the system, such as positivity, Aizerman's conjecture is known to hold true.[6]

• Strengthening of Aizerman's conjecture is Kalman's conjecture (or Kalman problem) where in place of condition on the nonlinearity it is required that the derivative of nonlinearity belongs to linear stability sector.
• A multivariate version of Aizerman's conjecture holds true over the complex field,[7] and it can be used to derive the circle criterion for the stability of nonlinear time-varying systems.[8]

• Atherton, D.P.; Siouris, G.M. (1977). "Nonlinear Control Engineering". IEEE Transactions on Systems, Man, and Cybernetics. 7 (7): 567–568. doi:10.1109/TSMC.1977.4309773.

• "Counterexamples to Aizerman's and Kalman's conjectures and describing function method" (PDF).