Barriers in impulsive antiperiodic problems

Rachůnková, Irena and Rachůnek, Lukáš: Barriers in impulsive antiperiodic problems. Electronic journal of qualitative theory of differential equations 83. pp. 1-9. (2019)

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Some real world models are described by means of impulse control of nonlinear BVPs, where time instants of impulse actions depend on intersection points of solutions with given barriers. For i = 1, . . . , m, and [a, b] ⊂ R, continuous functions γi : R → [a, b] determine barriers Γi = {(t, z) : t = γi(z), z ∈ R}. A solution (x, y) of a planar BVP on [a, b] is searched such that the graph of its first component x(t) has exactly one intersection point with each barrier, i.e. for each i ∈ {1, . . . , m} there exists a unique root t = tix ∈ [a, b] of the equation t = γi(x(t)). The second component y(t) of the solution has impulses (jumps) at the points t1x, . . . , tmx. Since a size of jumps and especially the points t1x, . . . , tmx depend on x, impulses are called state-dependent. Here we focus our attention on an antiperiodic solution (x, y) of the van der Pol equation with a positive parameter µ and a Lebesgue integrable antiperiodic function f x 0 (t) = y(t), y 0 (t) = µ x(t) − x 3 (t) 3 �0 − x(t) + f(t) for a.e. t ∈ R, t 6∈ {t1x, . . . . , tmx}, where y has impulses at the points from the set {t1x, . . . , tmx}, y(t+) − y(t−) = Ji(x), t = tix, i = 1, . . . , m, and Ji are continuous functionals defining a size of jumps. Previous results in the literature for this antiperiodic problem assume that impulse points are values of given continuous functionals. Such formulation is certain handicap for applications to real world problems where impulse instants depend on barriers. The paper presents conditions which enable to find such functionals from given barriers. Consequently the existence results for impulsive antiperiodic problem to the van der Pol equation formulated in terms of barriers are reached.

Item Type: Article
Journal or Publication Title: Electronic journal of qualitative theory of differential equations
Date: 2019
Number: 83
Page Range: pp. 1-9
ISSN: 1417-3875
Uncontrolled Keywords: Differenciaegyenlet
Additional Information: Bibliogr.: p. 8-9. ; összefoglalás angol nyelven
Date Deposited: 2020. Jan. 27. 12:51
Last Modified: 2020. Jan. 27. 12:51

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