Article is devoted to the memory of the known mechanic, the mathematician, the teacher, professor Victor Aleksandrovich Ostapenko who was one of differential equations chair founders at Dnipropetrovsk National University.
Existence conditions of homoclinic orbits for some systems of ordinary quadratic differential equations with singular linear part are founded. A realization of these conditions guarantees the existence of chaotic attractors at 3-D autonomous quadratic systems. In addition, a chaotic behavior of the solutions of these systems is determined by one-dimensional discrete map at some values of parameters. Examples are given.
An optimal control problem for degenerate parabolic equation with mixed boundary conditions are considered. Having applied the Hardy - Poincare inequality, it is shown that this problem has a unique optimal solution in the correspondence weighted Sobolev space. The necessary optimality conditions are derived and substantiated.
The article is devoted to the optimal control problem of parabolical system with unbounded coefficient. The definition of variational solution of the optimal control problem is given and the conditions of its existence are determined.
We study the bilinear forms on the space of measurable square-integrable functions
which are generated by skew-symmetric matrices with unbounded coecients.
We show that in the case when a skew-symmetric matrix contains L2-elements, the corresponding quadratic forms can be alternative. Since these questions are closely related with the existence of a unique solution for linear elliptic equations with unbounded coecients, we show that the energy identities for weak solutions can be studied in the framework of the corresponding alternative quadratic forms. To this end, we discuss the problems of integration by parts for measurable functions and give a generalization of some formulae for the non-Lipschitz case.
Determined resistance of circular motion in the central circles vortex rings for the first approach and Lyapunov theory of systems, such stability is a necessary condition for the possibility of formation of these circles of solid planets with dust and gases planetary nebula. The existence of resonance relations between circular and meridional motions at auction can be the key to explain the reasons for the formation of vortex rings of planetary satellites.
Several important properties of the particle motion flow in the planetary vortex are set. Its relate to distributions of their angular moments and angular velocities. It has been found that these properties, when gravitational force effect on the vortex star, involve flattening of rings and their significant migration from the primary positions. With it rings are intersect, that contribute to pushing and particle aggregation and the formation of solid planets.
Initial boundary value problem definition for a steel rope of lifting installation is considered in the article.The solution of initial-boundary problem for elastic filament as the area with mobile border is found.Program realization of results of influence of the reflected waves on stress in rope sections is presented.
We study one class of nonlinear fluid dynamic models with controls in the initial condition and the source term. The model is described by a nonlinear inhomogeneous hyperbolic conservation law with state and control constraints. We consider the case when the greatest lower bound of the cost functional can be unattainable on the set S of admissible pairs or the set S is possible empty. We apply the so-called vector-valued approximation of the original optimal vector problem. We consider a special vector optimization problem and show that this problem has non-empty set of efficient solutions.
The inverse problems of measurement are investigated in this paper. The main
hypothesis was suggested for estimation from below of exact solutions of such
problems. Two practical inverse problems have been considered as examples: inverse problem of Krylov, identication of moment of technological resistance on rolling mill.
The problem of construction asymptotic decomposition in an electrical field in a vicinity of border is considered. The solution of a problem is under construction as product of functions, so that the boundary conditions on border of area were satisfied. In an obvious kind are constructed asymptotic decomposition for areas limited to a circle.
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