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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 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

\( {\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—hidden oscillation .[1][2][3][4]

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.
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References

Leonov G.A.; Kuznetsov N.V. (2011). "Algorithms for Searching for Hidden Oscillations in the Aizerman and Kalman Problems" (PDF). Doklady Mathematics. 84 (1): 475–481. doi:10.1134/S1064562411040120.
Bragin V.O.; Vagaitsev V.I.; Kuznetsov N.V.; Leonov G.A. (2011). "Algorithms for Finding Hidden Oscillations in Nonlinear Systems. The Aizerman and Kalman Conjectures and Chua's Circuits" (PDF). Journal of Computer and Systems Sciences International. 50 (5): 511–543. doi:10.1134/S106423071104006X.
Leonov G.A.; Kuznetsov N.V. (2011). "Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems" (PDF). IFAC Proceedings Volumes (IFAC-PapersOnline). 18 (1): 2494–2505. doi:10.3182/20110828-6-IT-1002.03315.

Leonov G.A.; Kuznetsov N.V. (2013). "Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits". International Journal of Bifurcation and Chaos. 23 (1): 1330002–219. Bibcode:2013IJBC...2330002L. doi:10.1142/S0218127413300024.

Further reading
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.

 

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