ω-limit sets for impulsive-perturbed parabolic equation in the space of continuous functions
DOI:
https://doi.org/10.17721/1812-5409.2022/4.5Keywords:
parabolic equation, impulsive perturbation, attracting setAbstract
We consider an important class of disturbed infinite-dimensional systems provided by evolutionary equations with impulsive effects. An impulsive dynamical system acting in the phase space H is a hybrid system described by an evolutionary equation together with impulsive disturbances which, in turn, are characterized by a closed set M ⊂ H (impulsive set) and a mapping I: M -> H (impulsive mapping). A solution of such a system, by definition, is a left-continuous function y satisfying evolutionary equation for all times t for which y(t) \notin M and jumping to a new state I(y(t)) at all times t for which y(t) reaches M. The main problem in the study of qualitative behavior of impulsive dynamical systems by classical methods is that solutions usually do not depend continuously on initial data. A number of examples illustrate already in the finite-dimensional case that new complex scenarios of possible limit behaviors of impulsive trajectories can occur, which are fundamentally impossible in the non-impulsive case. Results concerning attracting sets for infinite-dimensional impulsive systems have appeared only a few years ago. But all those results concern the limit behavior of impulsive semiflows in Hilbert space and the presence of scalar product played crucial role in those construction. In this paper we want to investigate ω-limit attracting sets for impulsive-disturbed parabolic system in the sup-norm of Banach space of continuous functions.
Pages of the article in the issue: 42 - 48
Language of the article: Ukrainian
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