TY - JOUR

T1 - Coexistence of hidden attractors and multistability in counterexamples to the Kalman conjecture

AU - Kuznetsov, N. V.

AU - Kuznetsova, O. A.

AU - Mokaev, T. N.

AU - Mokaev, R. N.

AU - Yuldashev, M. V.

AU - Yuldashev, R. V.

PY - 2019/9

Y1 - 2019/9

N2 - 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.

AB - 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.

KW - Global stability

KW - Hidden attractors

KW - Kalman conjecture

KW - Multistability

KW - Periodic oscillations

UR - http://www.scopus.com/inward/record.url?scp=85077441612&partnerID=8YFLogxK

U2 - 10.1016/j.ifacol.2019.11.747

DO - 10.1016/j.ifacol.2019.11.747

M3 - Conference article

AN - SCOPUS:85077441612

VL - 52

SP - 7

EP - 12

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

SN - 2405-8971

IS - 16

T2 - 11th IFAC Symposium on Nonlinear Control Systems, NOLCOS 2019

Y2 - 4 September 2019 through 6 September 2019

ER -