Bulletin of Dnipropetrovsk University. Series: Communications in Mathematical Modeling and Differential Equations Theory

3-38

Authors: Т. Horsin, P. I. Kogut

We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain Ω. We take the matrixvalued coecients A(x) of such system as a control in L1(Ω;RN RN). One of the important features of the admissible controls is the fact that the coecient matrices A(x) are non-symmetric, unbounded on Ω, and eigenvalues of the symmetric part Asym = (A + At)=2 may vanish in Ω.

88-98

Authors: Yu. L. Menshikov, M. V. Polyakov

In this paper we propose the definition algorithm of the gravitational mass of un ­known origin on the results of astronomical observations of the movements of other gravity bodies. Unlike a similar problem, which has been solved at first by Leverrier and J. D. Adams, in the paper a more universal algorithm was proposed. To obtain of useful estimates of the exact solutions the hypothesis was proposed, which eliminates from use the size of operator error. Conditions for the existence of estimations of exact solutions are considered. Several non-standard statements of inverse problems is investigated.

111-120

Authors: N. N. Osipchuk, V. I. Perechrect

A comparative study of two fundamental solutions of the space-axisymmetric Euler equations, one of which is regular throughout space and describes the regular system of vortex rings, and the other is singular at the origin. It is shown that despite the feature zero singular solution generates a flow with finite integral characteristics of vortex rings and systems similar to a regular case and close to it on the global structure.

137-149

Authors: T. A. Boganova

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 apply the so-called vector-valued approximation of the original optimal vector problem. Conditions under which the solutions to approximation problems will be admissible for the original problem are derived.

159-171

Authors: S. O. Gorbonos, P. I. Kogut

The necessary optimality conditions to optimal control problem for parabolic system with unbounded coefficients were developed using the concept of the generalized right hand directional derivative and quasi-adjoint state.

3-30

Authors: N. V. Polyakov, Yu. L. Menshikov, G. I. Skorohod

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.

47-61

Authors: I. G. Balanenko, P. I. Kogut

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.

69-83

Authors: S. O. Gorbonos, P. I. Kogut

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.

98-106

Authors: V. I. Perehrest, N. N. Osipchuk, L. V. Klychinska

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.

118-129

Authors: T. S. Zelenskaya, A. V. Siasiev

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.

143-154

Authors: P. I. Tkatchenko

It is discussed the alternative approach for the construction of decreasing rearrange­ment for the correct definition of the Lorenz spaces.

162-173

Authors: N. A. Maruchno, V. A. Ostapenko

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.

3-18

Authors: I. G. Balanenko, P. I. Kogut

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.

40-52

Authors: V. M. Bogomaz, P. I. Kogut

We study vector optimization problems in partially ordered Banach Spaces. We sup­pose that the objective mapping possesses a weakened property of lower semicontinuity and make no assumptions on the interior of the ordering cone. We discuss the ”clas­sical” scalarization of vector optimization problems in the form of weighted sum and also we propose other type of scalarization for vector optimization problem, the socalled adaptive scalarization, which inherits some ideas of Pascoletti-Serafini approach. As a result, we show that the scalar nonlinear optimization problems can byturn approxi­mated by the quadratic minimization problems. The advantage of such regularization is especially interesting from a numerical point of view because it gives a possibility to apply rather simple computational methods for the approximation of the whole set of efficient solutions.

63-82

Authors: P. I. Kogut, O. P. Kupenko

We study an optimal boundary control problem (OCP) associated to a linear elliptic equation —div (Vj/ + A(x)Vy) = f describing diffusion in a turbulent flow. The characteristic feature of this equation is the fact that, in applications, the stream matrix A(x) = [a,ij(x)]i,j=i,...,N is skew-symmetric, ац(х) = —a,ji(x), measurable, and belongs to L -space (rather than L°°). An optimal solution to such problem can inherit a singular character of the original stream matrix A. We show that optimal solutions can be attainable by solutions of special optimal boundary control problems.

95-115

Authors: V. A. Ostapenko

The boundary-value problem about construction of the displacement waves and the strain waves arising in ropes of elevating devices, such as lifts, mine lifts and so on is considered. The rope at lifting of loads is reeled up on a drum. In a case when the friction coefficient of a rope about a drum is not too big, occurs frictional sliding a rope on a drum. Therefore the behavior of a rope on a drum is described by the telegraph equation. The behavior of a hanging part of a rope is described by the wave equation. It means, that in different parts of a rope the displacements are solutions of the different equations. That is from this point of view the rope is shared on two zones. Thus owing to reeling of a rope on a drum the border which shares these two zones is a variable. In such model the waves not only reflect from ending points of a rope. There is also their reflection and refraction on moving border of the sharing of zones. Is developed methods for obtaining of exact solutions for the boundary-value problems with mobile borders for both the wave and telegraph equations. They are based on maintenance of a continuity of the displacements in points of reflection of waves. The exact solution of such problem is obtained for the case of sagging a rope prior to the beginning of rise.

128-138

Authors: V. A. Tychynin, O. N. Tertyshnik

The finite nonlocal integro-differential transformation linearizing the nonlinear tele­graph equation uu — dx(—u~1 + u~2ux) = 0 is received. The formula of nonlinear superposition and formula for generating of solutions are constructed. New solutions of this equation are found. The Lie symmetry for the equations connected with initial by means of the potential system are investigated and exact solutions for them are received. Potential symmetry of the linear equation are used for construction of new potential symmetry of the nonlinear telegraph equation and for generating of its solutions.

3-20

Authors: P. I. Kogut

The stability of linear Volterra integral equations are discussed from the Lyapunov Direct Method point of view. The characteristic feature of these equations is the fact that their solutions are not continuous functions. We consider the case when the solution class is the Bochner space Lp loc(0;1;X) of locally p-integrable functions.

30-54

Authors: V. A. Ostapenko

The first boundary-value problem for the telegraph equation on an interval which one end is mobile is considered. The method of the solution of such problem is developed and its exact solution is obtained. This method is based on integrated representation of solutions of the telegraph equation and generalization of a method of re°ections with reference to areas with variable border. Variants of movement of the mobile end with subsonic, sound and supersonic speeds, and also with arbitrary speed are considered.

74-77

Authors: V. A. Ostapenko

The third boundary-value problem for the telegraph equation in semi-bounded do­main is considered. The solution of this problem in quadratures is obtained. Construction of the exact solution to this problem is based on application of the method of extensions and on development of the method of integral representation for rather wide class of solutions to the telegraph equation.

86-98

Authors: O. P. Kogut

In this paper we study an optimal control problem for a nonlinear elliptic variational inequality with generalized solenoidal coefficients which we adopt as controls in L°°(fi). We prove the existence of optimal solution of the stated problem.

114-127

Authors: T. A. Boganova

We consider the traffic flow models in vector-valued optimization statement, where the flow is controlled on the edges of network. We study the topological properties of the set of all admissible pairs to the problem. The existence of efficient solutions of vector optimization problem for traffic flow on network are proved.

143-159

Authors: V. A. Tychynin, O. N. Tertyshnik

On the basis of known potential symmetry the finite nonlocal integro-differential transformation leaving invariant the nonlinear telegraph equation is constructed. The algorithms generating its solutions are obtained. New solutions are present there among generated. Equations connected with the given telegraph equa­tion by means of potential system are received. Lie symmetries for them are investigated and exact solutions are constructed. It is shown, that potential symmetries are a special case of nonlocal symmetries - the invariance under finite Lie-Baklund transformations de­pending on integral variable. The characteristic equations corresponding to the potential symmetry of the telegraph equation are deduced. They define the nonlocal symmetries of equations connected by means of potential system. They also are used for searching of exact solutions of the specified equations.

3-26

Authors: V. Ye. Belozyorov, А. V. Belozyorov

The new existence conditions of homoclinic and heteroclinic orbits for the system of ordinary quadratic differential equations are founded. Further, the realization of these conditions together with the Shilnikov Homoclinic Theorem guarantees the existence of a chaotic attractor at 3-D autonomous quadratic system. Examples of the homoclinic orbits are given.

36-44

Authors: A. V. Didinskii, D. V. Evdokimov, O. O. Kochubey, M. V. Polyakov

Onsagers equation system describes complex systems in liquids and gases with suf­ficient coupled conduction and diffusion processes. It attracts increasing interests of in­vestigators last decades due to development a lot of modern technologies. Conventional numerical methods cannot be applied to Onsager’s equation system directly because of difficulties with correct calculation of non-diagonal terms which several orders less than diagonal terms. To overcome this difficulty correspondent asymptotic algorithm is pro­posed in the present paper.

64-72

Authors: N. N. Osipchuk, V. I. Perekhrest

The problem of the existence of resonances between circular, meridional and az-imuthal motions of particles flow on toroidal surfaces in the planetary vortex is re­searched. The numeral-analytical selection method for resonance trajectories of motion is offered accoding to the initial conditions and task parameters.

90-102

Authors: V. N. Bogomaz

For a design vibrosystem builting into the sealing machine the optimization problem is considered. The optimality conditions are derived.

114-123

Authors: V. V. Gotsulenko

The thesis deals with the boundary velocity suboptimal control of incompressible flow in cylindrical perforated domains (in the so-called generalized Couette cell). It is supposed that the mathematical model of above control object is the non-linear Bernard.

137-148

Authors: Yu. L. Menshikov

In the given work the inverse problem of restoration of the form of the bottom of the channel on experimental measurements of a level of a free surface of a liquid is considered. The problem is investigated in two-dimension statement for a case of stationary current of an incompressible liquid along the channel of final depth. This problem is reduced to the solution of the integrated Fredgolm’s equation of the first sort with the kernel which is defined approximately. For obtaining of the steady solution the regularization method with a choice of special mathematical model is used.