ISSN 0012-2661, Differential Equ ations, 2017, Vol. 53, No. 13, pp. 1764–1816.
Pleiades Publishing, Ltd., 2017.
Diﬀerential Equations with Hysteresis Operators.
Existence of Solutions, Stability, and Oscillations
G. A. Leonov
, and K. D. Aleksandrov
Saint Petersburg State University, St. Petersburg, 199034 Russia
Peoples’ Friendship University of Russia, Moscow, 117198 Russia
Adygeya State University, Maikop, 385000 Russia
This review is aimed at exploring ordinary diﬀerential equations with hysteresis as viewed by
the contemporary nonlinear control theory. Currently, there are books and reviews devoted to
hysteresis operators; however, the latter are mainly described from the standpoint of physical-
mechanical consideration. The approach suggested in this paper has some advantages to be brieﬂy
described in what follows. 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
= Ax + bξ(t),
σ = c
x, ξ(t)=ϕ[σ(τ )|
where x ∈
; A, b,andc are constant matrices of dimensions n×n, n×m,andn×m, respectively;
σ is an m-dimensional vector; ϕ[σ(τ)|
](t) is a nonlinear vector-operator, and ξ
In particular, for m =1,b and c are n-dimensional vectors.
Let us consider two well-known operators of “play” ϕ
](t) and “stop”
](t) (the same operator is called the Prandtl operator in the theory of elastic-plastic
deformations). The graphs of these operators are displayed in Figs. 1 a and 1 b. It can be easily
shown and it is well-known [1, 2] that
](t)=σ(t) − ϕ
= σ(0) − ξ
Therefore, results obtained for system (1) with the “play” operator can be immediately refor-
mulated for system (1) with the “stop” operator and vice versa. In other words, if a certain result
holds true for system (1) with the “play” operator for some domain in the space of parameters of
system (1), then one can specify a domain in the space of parameters within which the result is
valid with the “play” replaced by the “stop.”
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 [3, 4]. 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 math-
ematically rigorous results on the stability and oscillations of two-dimensional systems with hys-
teresis 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 in