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

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.