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)
Differential Equations – Springer Journals
Published: Mar 14, 2018
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