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Periodic solutions of a control problem via marginal maps
No full textWe investigate the existence of periodic solutions to the control problem x ̇=f(t,x,u)+g(t),x∈Rn,u∈Rm, (1) with g and f periodic in t with period 1. We form the associated quantities s(t,x)=supu∈Ω(x,f(t,x,u)),i(t,x)=infu∈Ω(x,f(t,x,u)) where (·,·) denotes the inner product inRn and Ω is a nonempty compact set in Rn. If us(t, x), ui(t, x) denote the (in general multivalued) controls for which s(t, x), i(t, x) are respectively attained, then we can form the family of marginal problems x ̇∈λ(t)co ̄ ̄ ̄ ̄ ̄f(t,x,us(t,x))+(1−λ(t))co ̄ ̄ ̄ ̄ ̄f(t,x,ui(t,x))+g(t),λ(⋅)∈L∞([0,1],[0,1]). (2) We give sufficient conditions for the existence of a periodic solution of certain marginal problems, stated in terms of lim inf|x|→∞ and lim sup|x|→∞ of s(t,»)/¦x¦2 and i(t, x)j¦x¦2. Finally we state the relationship between the periodic solutions of the marginal problems and those of the original problem (1)
Periodic oscillations in systems with hysteresis
No full textWe give precise conditions under which the method of harmonic balance will correctly predict the existence of periodic solutions for a system with relay hysteresis. The equation modeling the system is assumed to be of the form L(m)[y](t) = f[y](t), t greater-than-or-equal-to 0, where L(m) is a constant coefficient linear differential operator of order m greater-than-or-equal-to 2 and f is a possibly discontinuous operator with hysteresis
A theoretical justification of the method of harmonic balance for systems with discontinuities
No full textWe prove a theorem which provides a rigorous justification of an intuitive method used by electrical engineers to predict the presence or absence of periodic oscillations in nonlinear systems. Although the literature contains some excellent discussions of the conditions under which the method can be rigorously justified, there are some oversights and there is lacking a completely detailed treatment, particularly for unforced discontinuous systems. By applying the theory of topological degree for differential inclusions, we are able to present a unified rigorous justification in full detail, and we can illustrate how our abstract hypotheses match up, point by point, with the standard hypotheses used by engineers
An approximation method for the existence of periodic solutions to systems with delay
No full textWe describe a method for proving the existence of periodic solutions to n-dimensional systems of the form z'(t) - Az(t) - Bz(t - tau) = F[z(t)]. The proposed method is based on the harmonic balance method and the theory of reproducing kernels
The application of topological methods for the prediction of periodic oscillations in systems with discontinuities and hysteresis
No full textThe existence of periodic solutions to non autonomous differential inclusions
No full textFor an m-dimensional differential inclusion of the for;...#x220A;A(t)x(t;...[t, x(t)], wit;...n;...-periodic in t, we prove the existence o;...onconstant periodic solution. Our hypotheses requir;...o be odd, and requir;...o have different growth behavior for |x| small and |x| large (often the case in applications). The idea is to guarantee that the topological degree associated with the system has different values on two distinct neighborhoods of the origin. © 1988 American Mathematical Society
Solution sets of differential equations in abstract spaces
No full textAim of the monograph is to analyze the structure of the solution set of a differential equations in the more general case of abstract topological (infinite dimensional) spaces. Specifically, we consider both the spaces with a paucity of useful properties, e.g. locally convex spaces, and the more richly endowed spaces, e.g. Banach or Hilbert spaces. The related results are presented both for the single-valued case and multi-valued (differential inclusions) case
Mathematical models for hysteresis
Full text linkThe various existing classical models for hysteresis, Preisach, Ishlinskii, Duhem-Madelung, are surveyed, as well a more modern treatments by contemporary workers. The emphasis is on a clear mathematical description of the formulation and properties of each model. In addition the authors try to make the reader aware of the many open questions in the study of hysteresis
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