# Differential Equations with Hysteresis Operators. Existence of Solutions, Stability, and Oscillations

Differential Equations with Hysteresis Operators. Existence of Solutions, Stability, and... ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 13, pp. 1764–1816.  Pleiades Publishing, Ltd., 2017. CONTROL THEORY Diﬀerential Equations with Hysteresis Operators. Existence of Solutions, Stability, and Oscillations 1,2∗ 3 3 1 G. A. Leonov ,M.M. Shumafov ,V.A. Teshev , 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 e-mail: g.leonov@spbu.ru DOI: 10.1134/S0012266117130055 INTRODUCTION 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 dx = Ax + bξ(t), dt (1) ∗ t σ = c x, ξ(t)= ϕ[σ(τ)| ,ξ ](t), τ =0 where x ∈ R ; A, b,and c are constant matrices of dimensions n×n, n×m,and n×m, respectively; σ is an m-dimensional vector; ϕ[σ(τ)| ,ξ ](t) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Differential Equations Springer Journals

# Differential Equations with Hysteresis Operators. Existence of Solutions, Stability, and Oscillations

, Volume 53 (13) – Mar 14, 2018
53 pages

/lp/springer_journal/differential-equations-with-hysteresis-operators-existence-of-ai5m9sSqs8
Publisher
Springer Journals
Subject
Mathematics; Ordinary Differential Equations; Partial Differential Equations; Difference and Functional Equations
ISSN
0012-2661
eISSN
1608-3083
D.O.I.
10.1134/S0012266117130055
Publisher site
See Article on Publisher Site

### Abstract

ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 13, pp. 1764–1816.  Pleiades Publishing, Ltd., 2017. CONTROL THEORY Diﬀerential Equations with Hysteresis Operators. Existence of Solutions, Stability, and Oscillations 1,2∗ 3 3 1 G. A. Leonov ,M.M. Shumafov ,V.A. Teshev , 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 e-mail: g.leonov@spbu.ru DOI: 10.1134/S0012266117130055 INTRODUCTION 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 dx = Ax + bξ(t), dt (1) ∗ t σ = c x, ξ(t)= ϕ[σ(τ)| ,ξ ](t), τ =0 where x ∈ R ; A, b,and c are constant matrices of dimensions n×n, n×m,and n×m, respectively; σ is an m-dimensional vector; ϕ[σ(τ)| ,ξ ](t)

### Journal

Differential EquationsSpringer Journals

Published: Mar 14, 2018

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