# Volume 61, № 8, 2009

### Locally nilpotent derivations and Nagata-type utomorphisms of a polynomial algebra

Bodnarchuk Yu. V., Prokof’ev P. H.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1011-1024

We study locally nilpotent derivations belonging to a Lie algebra $sa_n$ of a special affine Cremona group in connection with the root decompositions of sa n relative to the maximum standard torus. It is proved that all root locally nilpotent derivations are elementary. As a continuation of this research, we describe two- and three-root derivations. By using the results obtained by Shestakov and Umirbaev, it is shown that the exponents of almost all obtained three-root derivations are wild automorphisms of a polynomial algebra in three variables.

### On the ε-sufficient control in one merton problem with “physical” white noise

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1025-1039

We consider the Merton problem of finding the strategies of investment and consumption in the case where the evolution of risk assets is described by the exponential model and the role of the main process is played by the integral of a certain stationary “physical” white noise generated by the centered Poisson process. It is shown that the optimal controls computed for the limiting case are ε-sufficient controls for the original system.

### A modular transformation for a generalized theta function with multiple parameters

Bhargava S., Mahadeva Naika M. S., Maheshkumar M. C.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1040-1052

We obtain a modular transformation for the theta function $$∑_{-∞}^{∞}∑_{-∞}^{∞}q^{a(m^2 + m^n) + cn^2 + λm + μn + ν_{ς}Am + Bn_{Z}Cm + Dn},$$ which enables us to unify and extend several modular transformations known in literature.

### Optimal control with impulsive component for systems described by implicit parabolic operator differential equations

Samoilenko A. M., Vlasenko L. A.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1053-1065

We study the problem of optimal control with impulsive component for systems described by abstract Sobolev-type differential equations with unbounded operator coefficients in Hilbert spaces. The operator coefficient of the time derivative may be noninvertible. The main assumption is a restriction imposed on the resolvent of the characteristic operator pencil in a certain right half plane. Applications to Sobolevtype partial differential equations are discussed.

### Cauchy problem for one class of degenerate parabolic equations of Kolmogorov type with positive genus

Ivasyshen S. D., Litovchenko V. A.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1066-1087

We investigate properties of a fundamental solution and establish the correct solvability of the Cauchy problem for one class of degenerate Kolmogorov-type equations with \( \left\{ {\overrightarrow p, \overrightarrow h } \right\} \)-parabolic part with respect to the main group of variables and with positive vector genus in the case where solutions are infinitely differentiable functions and their initial values may be generalized functions of Gevrey ultradistribution type.

### Problems for equations with special parabolic operator of fractional differentiation

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1088-1097

We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.

### Tikhonov regularization method for a system of equilibrium problems in Banach spaces

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1098-1105

The purpose of this paper is to investigate the Tikhonov regularization method for solving a system of ill-posed equilibrium problems in Banach spaces with *a posteriori* regularization-parameter choice. An application to convex minimization problems with coupled constraints is also given.

### Nonexistence theorem except the out-of-phase and in-phase solutions in the coupled van der Pol equation system

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1106-1129

We consider a coupled van der Pol equation system. Our coupled system consists of two van der Pol equations that are connected with each other by linear terms. We assume that two distinctive solutions (out-of-phase and in-phase solutions) exist in the dynamical system of coupled equations and give answers to some problems.

### Life and work of N. N. Bogolyubov (on his 100th birthday)

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1130-1141

### Systems of control over set-valued trajectories with terminal quality criterion

Arsirii A. V., Plotnikov A. V.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1142-1147

We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria.

### Oscillations of a diaphragm under the action of pulse forces

Kirilich V. M., Myshkis A. D., Prokhorenko M. V.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1148-1153

We investigate the problem of the existence of periodic solutions of the problem of oscillations of a diaphragm with friction and pulse feedback in the case where the times of pulse action are determined by a solution of the system.