From the introduction: This review is aimed at exploring ordinary differential equations with hysteresis as viewed by the contemporary nonlinear control theory. The right-hand sides of differential equations are usually treated in the nonlinear control theory as the sums of linear and nonlinear parts in the form \[ \begin{gathered} {dx\over dt}+ Ax+ b\xi(t),\\ \sigma= c^*x,\;\xi(t)= \varphi[\sigma(\tau)|^t_{\tau= 0},\,\xi_0](t),\end{gathered}\tag{1} \] where \(x\in\mathbb{R}^n\); \(A\), \(b\), and \(c\) are constant matrices of dimensions \(n\times n\), \(n\times m\), and \(n\times m\), respectively; \(\sigma\) is an \(m\)-dimensional vector; \(\varphi[\sigma(\tau)|^t_{\tau=0},\;\xi_0](t)\) is a nonlinear vector-operator, and \(\xi_0= \xi(0)\). In particular, for \(m=1\), \(b\) and \(c\) are \(n\)-dimensional vectors.
Let us consider two well-known operators of “play” \(\varphi_1[\sigma(\tau)|^t_{\tau=0}, \xi_{10}](t)\) and “stop” \(\varphi_2[\sigma(\tau)|^t_{\tau= 0},\) \(\chi_{20}](t)\) (the same operator is called the Prandtl operator in the theory of elastic-plastic deformations). It can be easily shown and it is well-known that \[ \varphi_1[\sigma(\tau)||^t_{\tau=0}, \xi_{10}](t)= \sigma(t)- \varphi_2[\sigma(\tau)|^t_{\tau=0}, \xi_{20}](t)\quad\text{for }\xi_{10}= \sigma(0)- \xi_{20}. \] Therefore, results obtained for system (1) with the “play” operator can be immediately reformulated for system (1) with the “stop” operator and vice versa.
The theory of existence and uniqueness of solutions has been thoroughly developed for differential equations with continuous hysteresis operators. In this paper, we provide the main results of this theory. In case of discontinuous dynamic systems a generalized consideration is usually used, viz. differential inclusions. No such theory is currently available for discontinuous systems with hysteresis operators. The existence of solutions for such systems is easy to prove unless there exist the so-called sets of sliding modes.
The outstanding scientists Andronov and Bautin and Feldbaum published the first mathematically rigorous results on the stability and oscillations of two-dimensional systems with hysteresis operators of the “play type and with the “non-ideal relay with dead zone” type. Herein we will provide these results and amplify them with results by Zheleztsov on the oscillations on a two-dimensional system with a “non-ideal relay” hysteresis operator and with out theorem on the global stability of similar systems with the Prandtl operator.
Yakubovich contributed remarkable to the theory of the absolute stability of multidimensional systems with hysteresis operators. In this paper, we provide one of Yakubovich’s frequency criteria with a proof, where one can get acquainted with the methods and techniques of applying Lyapunov functions and frequency theorems to the analysis of the global stability of hysteresis systems. Using the Andronov-Bautin results, we construct a counterexample to the Barabanov-Yakubovich frequency criterion.
To analyse the stabilizability of system with “play” and Prandtl operator, we provide new frequency criteria of stability an compare them with the well-known Logemann-Ryan results.
Studying oscillations in multidimensional hysteresis systems presents a complicated problem. In this paper, we demonstrate the application of one of the efficient nonlocal study methods, viz. the method of nonlocal reduction, which unifies the secon Lyapunov method and the theory of nonlocal bifuractions in two-dimensional systems. Based on this method, a frequency criterion of oscillation is derived that extends the results of A. A. Andronov, N. N. Bautin, A. A. Feldbaum, and N. A. Zheleztsov to multidimensional systems.