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

32-41

Authors: Volodymyr V. Pichkur, Valentyn V. Sobchuk

The paper suggests an approach to modeling of industrial enterprises providing production according to the set standard with admissible tolerances and requirements. The mathematical model has the form of a discrete control system. We use the properties of generalized inverse matrices to design the control. We present an algorithm of the control of a production process providing release of production. This approach allows to simulate the technological processes (including metallurgical, chemical, energy, etc.) and gives the operating conditions under the constant influence of internal and external destabilizing factors.

54-78

Authors: Peter I. Kogut, Olga P. Kupenko, Mykola V. Uvarov

In this paper we propose a new statement of the spatial increasing resolution problem of MODIS-like multi-spectral images via their fusion with Lansat-like imagery at higher resolution. We give a precise definition of the solution to the indicated problem, postulate assumptions that we impose at the initial data, establish existence and uniqueness result, and derive the corresponding necessary optimality conditions. For illustration, we supply the proposed approach by results of numerical simulations with real-life satellite images.

21-46

Authors: Vasiliy Ye. Belozyorov, Danylo V. Dantsev

The article presents a study of solutions of ODEs system with a special nonlinear part, which is a continuous analogue of an arbitrary recurrent neural network (neural ODEs). As a nonlinear part of the mentioned system of differential equations, we used sums of piecewise continuous functions, where each term is a power function. (These are activation functions.) The use of power activation functions (PAF) in neural networks is a generalization of well-known the rectified linear units (ReLU). In the present time ReLU are commonly used to increase the depth of trained of a neural network. Therefore, the introduction of PAF into neural networks significantly expands the possibilities of ReLU. Note that the purpose of introducing power activation functions is that they allow one to obtain verifiable Lyapunov stability conditions for solutions of the system differential equations simulating the corresponding dynamic processes. In turn, Lyapunov stability is one of the guarantees of the adequacy of the neural network model for the process under study. In addition, from the global stability (or at least the boundedness) of continuous analog solutions it follows that learning process of the corresponding neural network will not diverge for any training sample.

43-97

Authors: Vasiliy Ye. Belozyorov, Danylo V. Dantsev, Svetlana A. Volkova

A problem of description of algebraic invariants for a linear control system that determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.

23-43

Authors: Vladimir V. Borsch

Initial boundary value problems in space-time rectangle for the following linear inhomogeneous degenerate wave equation of the second order smooth coefficient function a(x) vanishes in single points of segment. The well-posedness of the initial boundary value problems is achieved using some approaches to regularization of the equation and the theory of characteristics. The problems for wave equation are then reduced to problems for hyperbolic balance laws 2 and 3 of partial differential equations of the first order. Weak solutions to the problems are obtained using proper numerical methods. Results obtained for some approaches to regularization are presented.

60-95

Authors: Volodymyr V. Hnatushenko, Peter I. Kogut, Mykola V. Uvarov

In this paper we propose a new technique for the solution of the image segmentation problem which is based on the concept of a piecewise smooth approximation of some target functional. We discuss in details the consistency of the new statement of segmentation problem and its solvability. We focus our main intension on the rigor mathematical substantiation of the proposed approach, deriving the corresponding optimality conditions, and show that the new optimization problem is rather flexible and powerful model to the study of variational image segmentation problems. We illustrate the accuracy and efficiency of the proposed algorithm by numerical experiences.

23-57

Authors: Vasiliy Ye. Belozyorov, Yevhen M. Kosariev, Mykola M. Pulin, Viktor G. Sychenko, Vadym G. Zaytsev

A new autonomous 4D nonlinear model with two nonlinearities describing the dynamics of change of voltage and current in the contact railway electric network is offered. This model is a connection of two 2D oscillatory circuits for current and voltage in the contact electric network. In the found system for the defined values of parameters an existence of limit cycles is proved. By introduction of new variables this system can be reduced to 5D system only with one quadratic nonlinearity. The constructed model may be used for the control by voltage stability in a direct current power supply system.