Юсипів, Тарас ВасильовичТарас ВасильовичЮсипівКапустян, Олексій Володимирович2024-11-042024-11-042024Юсипiв Т. В. Робастна стiйкiсть глобальних атракторiв еволюцiйних систем без єдності : дис. д-ра філософії : 111 Математика / наук. кер. О.В. Капустян. Київ, 2024. 128 с.УДК 517.9https://ir.library.knu.ua/handle/15071834/5188Дисертацiйна робота присвячена дослiдженню стiйкостi глобальних атракторiв нелiнiйних еволюцiйних систем без єдиностi по вiдношенню до зовнiшнiх збурень. Дисертацiя складається з анотацiй українською та англiйською мовами, вступу, чотирьох роздiлiв основної частини, висновкiв, списку використаних джерел та додатку. У дисертацiйнiй роботi отримано такi новi науковi результати: – одержано достатнi умови локальної стiйкостi щодо неавтономних збурень для глобального атрактору абстрактної нескiнченновимiрної еволюцiйної системи без єдиностi; – встановлено робастну оцiнку типу асимптотичного пiдсилення для неавтономних еволюцiйних систем без єдиностi вiдносно атракторiв автономних незбурених систем; – для параболiчної системи типу реакцiя-дифузiя з гладкою нелiнiйнiстю та збуренням, що мiстить фазову змiнну та не гарантує єдинiсть розв’язку задачi Кошi, доведено результат про локальну стiйкiсть глобального атрактору незбуреної системи по вiдношенню до збурень; – для загальної системи типу реакцiя-дифузiя з негладкою нелiнiйнiстю та неавтономних збуренням доведено властивiсть асимптотичного пiдсилення для глобального атрактору незбуреної системи по вiдношенню до збурень; – доведено робастну стiйкiсть атрактору дисипативної еволюцiйної си стеми, що складається з параболiчної нелiнiйної системи та системи звичайних диференцiальних рiвнянь, що збурюються вхiдними обмеженими сигналами; – для загального нелiнiйного гiперболiчного рiвняння з негладкою функцiєю взаємодiї та неавтономних збуренням доведено властивiсть асимптотичного пiдсилення для глобального атрактору незбуреної системи по вiдношенню до збурень; – для гiперболiчного рiвняння з гладкою нелiнiйнiстю та збуренням, що мiстить фазову змiнну та не гарантує єдинiсть розв’язку задачi Кошi, доведено результат про локальну стiйкiсть та асимптотичне пiдсилення для глобального атрактору незбуреної системи по вiдношенню до збурень. Результати роботи доповнюють абстрактну теорiю стiйкостi еволюцiйних нескiнченновимiрних систем без єдиностi розв’язку та можуть бути використанi в подальшому для дослiдження якiсної поведiнки розв’язкiв дисипативних диференцiальних рiвнянь в частинних похiдних. Одержанi в роботi результати також можуть мати прикладне значення, зокрема, при дослiдженнi стiйкостi граничних режимiв в системах з зовнiшнiми сигналами.The dissertation is devoted to the study of the stability of global attractors of nonlinear evolutionary systems without unity in relation to external disturbances. The dissertation consists of annotations in Ukrainian and English, an introduction, four sections of the main part, conclusions, a list of used sources and an appendix. The introduction substantiates the relevance of the topic, indicates the connection of the work with scientific programs, plans, topics, establishes the goal and task, object, subject and methods of research, provides the scientific novelty and practical significance of the obtained results, characterizes the personal contribution of the recipient, provides a list of conferences and scientific seminars at which the dissertation passed approval. The first section of the work is devoted to the review of the literature on this topic and the methodology of dissertation research. The main provisions of the theory of input to state stability for systems of differential equations are given. Also highlighted are the works of authors who applied this theory to the study of the qualitative behavior of solutions of both finite-dimensional and infinite-dimensional dissipative systems. In the second chapter, the main provisions of the theory of global attractors of nonlinear evolutionary equations in the case of non-uniqueness of the solution of the Cauchy problem are outlined. Both the autonomous case (theory of m- semiflows) and the non-autonomous case, where families of m-semiprocesses are the main object of research, are analyzed in detail. The concept of local input to the state stability and properties of asymptotic gain with respect to the global attractor for an infinite-dimensional evolutionary system are introduced. The property of asymptotic stability of the global attractors of m-semiflows is proved in the form of a robust estimate. Based on this result, sufficient conditions of local input to the state stability for an abstract non-autonomous perturbed system with respect to the attractor of the unperturbed system are obtained. For uniform attractors of m-semiprocesses, a result is obtained about the upper- semicontinuous dependence on the parameter. On the basis of this result, the property of asymptotic gain is established for non-autonomous evolutionary systems without uniqueness with respect to the attractors of autonomous undi- sturbed systems. In the third chapter, the asymptotic behavior of the solutions of nonlinear parabolic systems with perturbations in the right-hand side and the stability of global attractors with respect to these perturbations are investigated. Results on global resolvability, regularity and derived a priori estimates of solutions for general nonlinear non-autonomous parabolic equations are proved. For a parabolic system of the reaction-diffusion type with a smooth nonlinearity and a disturbance containing a phase variable and not guaranteeing the uniqueness of the solution of the Cauchy problem, a result on the local stability of the global attractor of the undisturbed system with respect to disturbances is proved. For a general system of the reaction-diffusion type with non-smooth nonlinearity and non-autonomous disturbances, the property of asymptotic gain for the global attractor of the undisturbed system with respect to disturbances is proved. The dissipative evolutionary system consisting of of a parabolic nonlinear system and a system of ordinary differential equations perturbed by input bounded signals is considered. It is proved that the global attractor of the unperturbed system is stable in the sense of ISS regarding the magnitude of disturbances. The fourth chapter investigates the qualitative behavior of the solutions of the nonlinear hyperbolic equation with perturbations in the right-hand side and the stability of the global attractor with respect to these perturbations. The results on global resolvability, regularity and derived a priori estimates of solutions for a general nonlinear non-autonomous hyperbolic equation are proved. For the general wave equation with non-smooth nonlinearity and non-autonomous perturbations, the asymptotic gain property for the global attractor of the unperturbed equation with respect to perturbations is proved. For a hyperbolic equation with a smooth nonlinearity and a perturbation, which contains a phase variable and does not guarantee the uniqueness of the solution of the Cauchy problem, a result on the local stability and asymptotic gain of the global attractor of the unperturbed equation with respect to perturbations is proved. The following new scientific results were obtained in the dissertation: – sufficient conditions of local stability with respect to non-autonomous perturbations are obtained for the global attractor of an abstract infinite- dimensional evolutionary system without uniqueness; – a robust estimate of the type of asymptotic gain is established for non- autonomous evolutionary systems without uniqueness with respect to the attractors of autonomous undisturbed systems; – for a parabolic system of the reaction-diffusion type with a smooth nonlinearity and a disturbance that contains a phase variable and does not guarantee the uniqueness of the solution of the Cauchy problem, a result on the local stability of the global attractor of the undisturbed system with respect to disturbances is proved; – for a general system of the reaction-diffusion type with non-smooth nonlinearity and non-autonomous perturbations, the asymptotic gain property for the global attractor of the undisturbed system with respect to perturbations is proved; – the robust stability of the attractor of the dissipative evolutionary system consisting of parabolic nonlinear system and system of ordinary differential equations perturbed by input bounded signals is obtained; – for a general nonlinear hyperbolic equation with a non-smooth interaction function and non-autonomous perturbations, the asymptotic gain property for the global attractor of the undisturbed system with respect to perturbations is proved. – for a hyperbolic equation with a smooth nonlinearity and a disturbance, which contains a phase variable and does not guarantee the uniqueness of the solution of the Cauchy problem, a result on the local stability and asymptotic gain of the global attractor of an undisturbed system with respect to disturbances is proved. The results of the work supplement the abstract stability theory of evolutionary infinite-dimensional systems without uniqueness of the solution and can be used in the future to study the qualitative behavior of solutions of dissipative partial differential equations. The results obtained in the work can also be of applied value, in particular, when studying the stability of limit regimes in systems with external signals.ukдиференцiальне рiвнянняасимптотична поведiнкастiйкiстьнелiнiйне еволюцiйне рiвняннязбуреннядинамiчна системанапiвпотiкомега-гранична множинаатракторпараболiчне рiвняннягiперболiчне рiвняннясистема реакцiї-дифузiїdifferential equationasymptotic behaviorstabilitynonlinear evolution equationperturbationdynamical systemsemiflowomega-limit setattractorparabolic equationhyperbolic equationreaction-diffusion systemДисертація