Aizerman's conjecture

Mathematical statement of Aizerman's conjecture (Aizerman problem)

Consider a system with one scalar nonlinearity : \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 k_1 and k_2 with k_1 , the function f satisfies : k_1
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.{{cite journal author1=Kuznetsov N.V. | year = 2020| title = Theory of hidden oscillations and stability of control systems| journal = Journal of Computer and Systems Sciences International| volume = 59| number = 5|pages = 647–668| url = https://link.springer.com/content/pdf/10.1134/S1064230720050093.pdf| doi = 10.1134/S1064230720050093| s2cid = 225304463}} year = 2013 | title = Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractor in Chua circuits | journal = International Journal of Bifurcation and Chaos | volume = 23 | issue = 1 | pages = 1330002–219| doi = 10.1142/S0218127413300024|bibcode=2013IJBC...2330002L | doi-access = }} However, under stronger assumptions on the system, such as positivity, Aizerman's conjecture is known to hold true.

Variants

• 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, and it can be used to derive the circle criterion for the stability of nonlinear time-varying systems.

References

References

1. (2001). "Mathematical Problems of Control Theory". World Scientific.
2. Liberzon, M. R.. (2006). "Essays on the absolute stability theory". Automation and Remote Control.
3. (2023). "Aizerman Conjectures for a Class of Multivariate Positive Systems". IEEE Transactions on Automatic Control.
4. (1992-01-01). "Destabilization by output feedback". Differential and Integral Equations.
5. (2011). "The Circle Criterion and Input-to-State Stability". IEEE Control Systems Magazine.