Aizerman's conjecture

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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, proposed by Mark Aronovich Aizerman in 1949, [1] was proven false but led to the (valid) sufficient criteria on absolute stability. [2]

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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 and with , the function 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. [7]

Variants

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References

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  3. 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. S2CID   120692391.
  4. 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. S2CID   21657305.
  5. Kuznetsov N.V. (2020). "Theory of hidden oscillations and stability of control systems" (PDF). Journal of Computer and Systems Sciences International. 59 (5): 647–668. doi:10.1134/S1064230720050093. S2CID   225304463.
  6. 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.
  7. Drummond, Ross; Guiver, Chris; Turner, Matthew C. (2023). "Aizerman Conjectures for a Class of Multivariate Positive Systems". IEEE Transactions on Automatic Control. 68 (8): 5073–5080. doi:10.1109/TAC.2022.3217740. ISSN   0018-9286. S2CID   260255282.
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