Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture
Kuznetsov, N. V., Kuznetsova, O. A., Mokaev, T. N., Mokaev, R. N., Yuldashev, M. V., & Yuldashev, R. V. (2019). Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture. In L. Jadachowski (Ed.), 11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019 (52, pp. 7-12). IFAC; Elsevier. IFAC-PapersOnLine. https://doi.org/10.1016/j.ifacol.2019.11.747
© 2019 IFAC
The Aizerman and Kalman conjectures played an important role in the theory of global stability for control systems and set two directions for its further development – the search and formulation of sufficient stability conditions, as well as the construction of counterexamples for these conjectures. From the computational perspective the latter problem is nontrivial, since the oscillations in counterexamples are hidden, i.e. their basin of attraction does not intersect with a small neighborhood of an equilibrium. Numerical calculation of initial data of such oscillations for their visualization is a challenging problem. Up to now all known counterexamples to the Kalman conjecture were constructed in such a way that one locally stable limit cycle (hidden oscillation) co-exists with a locally stable equilibrium. In this paper we demonstrate a multistable configuration of three co-existing hidden oscillations (limit cycles) and a locally stable equilibrium in the phase space of the fourth-order system, which provides a new class of counterexamples to the Kalman conjecture. ...
ConferenceIFAC Symposium on Nonlinear Control Systems
Is part of publication11th IFAC Symposium on Nonlinear Control Systems NOLCOS 2019 Vienna, Austria, 4-6 September 2019
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Additional information about fundingWe acknowledge support form the Russian Scientific Foundation (project 19-41-02002, sections 2-4) and the Leading Scientific Schools of Russia (project NSh-2858.2018.1, section 1).
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