1,721,061 research outputs found
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
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
On the solvability of systems of inclusions involving noncompact operators
No full textWe consider the solvability of a system [GRAPHICS]
of set-valued maps in two different caws. In the first one, the map (x, y)-o F(x, y)BAR is supposed to be closed graph with convex values and condensing in the second variable and (x, y)-o G(x, y)BAR is supposed to be a permissible map (i.e. composition of an upper semicontinuous map with acyclic values and a continuous, single-valued map), satisfying a condensivity condition in the first variable. In the second case FBAR is as before with compact, not necessarily convex, values and GBAR is an admissible map (i.e. it is composition of upper semicontinuous acyclic maps). In the latter case, in order to apply a fixed point theorem for admissible maps, we have to assume that the solution set x-o S(x) of the first equation is acyclic. Two examples of applications of the abstract results are given. The first is a control problem for a neutral functional differential equation on a finite time interval; the second one deals with a semilinear differential inclusion in a Banach space and sufficient conditions are given to show that it has periodic solutions of a prescribed period
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)
Positive solutions of elliptic non-positone problems
No full textWe give conditions for the existence or nonexistence of positive solutions of second-order subcritical elliptic nonpositone problems. We do not assume that the problems are radial, nor that they satisfy a variational structure. Our chief tools are Degree Theory, a priori estimates, and Maximum Principle arguments. © 1992, Khayyam Publishing. All rights reserved
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
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