Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems (Q656381)

The paper is devoted to the investigation of the following system with one scalar nonlinearity \[ \frac{dx}{dt} = Px + q\psi(r^*x), \;x \in \mathbb{R}^n, \tag{1} \] where \(P\) is a constant \(n \times n\) matrix, \(q\) and \(r\) are constant \(n\)-vectors, \(^*\) denotes transposition, \(\psi(\sigma)\) is a piecewise continuous scalar function, \(\psi(0) = 0.\) The complete controllability of the pair \((P,q)\) and the complete observability of the pair \((P,r)\) are assumed. The authors propose an analytical criterion for the existence and analytical-numerical algorithm for localization of periodic solutions for (1), a basin of attraction of which does not contain neighborhoods of equilibria. Using this algorithm, counterexamples to Aizerman's and Kalman's problems are constructed [\textit{M. A. Ajzerman}, Usp. Mat. Nauk 4, No. 4 (32), 187--188 (1949; Zbl 0040.19601), \textit{R. E. Kalman}, ``Physical and mathematical mechanisms of instability in nonlinear automatic control systems'', Trans. ASME 79, 553--566 (1957)].