Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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Mathematical Model and Control Design of a Functionally Stable Technological Process
Get PDFThe paper suggests an approach to modeling of industrial enterprises providingproduction 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.The paper suggests an approach to modeling of industrial enterprises providingproduction 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
Mathematical Simulations of Deformation for the Rotation Shells with Variable Wall Thickness
Get PDFWell-posed boundary value problems are constructed for calculating rotation shells of with a stiffness variable along the meridian in two directions, and also with variable bilateral with respect to the reference surface with the shell wall thickness. Algorithms for the numerical integration of systems of differential equations with variable coefficients are discussed.Well-posed boundary value problems are constructed for calculating rotation shells of with a stiffness variable along the meridian in two directions, and also with variable bilateral with respect to the reference surface with the shell wall thickness. Algorithms for the numerical integration of systems of differential equations with variable coefficients are discussed
On Weak and Strong Solutions of Paired Stochastic Functional Differential Equations in Infinite-Dimensional Spaces
Get PDFIn this paper, we study the questions of the existence of global weak solutions and local strong solutions of paired stochastic functional differential equations in a Hilbert space, one of which is an equation with an unbounded operator, and the other is an ordinary differential equation. We proved the existence and uniqueness theorems in the case of coefficients with polynomial growth.In this paper, we study the questions of the existence of global weak solutions and local strong solutions of paired stochastic functional differential equations in a Hilbert space, one of which is an equation with an unbounded operator, and the other is an ordinary differential equation. We proved the existence and uniqueness theorems in the case of coefficients with polynomial growth
To the memory of Professor Mykola Poliakov
Get PDFTo the memory of Professor Mykola PoliakovTo the memory of Professor Mykola Poliako
The Exact Bounded Solution to an Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy. I. Separation of Variables
Get PDFStability of Neural Ordinary Differential Equations with Power Nonlinearities
Get PDFThe article presents a study of solutions of ODEs system with a specialnonlinear 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, weused sums of piecewise continuous functions, where each term is a power function. (Theseare activation functions.) The use of power activation functions (PAF) in neural networksis a generalization of well-known the rectified linear units (ReLU). In the present timeReLU 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 ofReLU. Note that the purpose of introducing power activation functions is that theyallow one to obtain verifiable Lyapunov stability conditions for solutions of the systemdifferential equations simulating the corresponding dynamic processes. In turn, Lyapunovstability is one of the guarantees of the adequacy of the neural network model for theprocess 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 neuralnetwork will not diverge for any training sample.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 ofReLU. 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
Variational Approach for the Reconstruction of Damaged Optical Satellite Images Through Their Co-Registration with Synthetic Aperture Radar
Get PDFIn this paper the problem of reconstruction of damaged multi-band opticalimages is studied in the case where we have no information about brightness of suchimages in the damage region. Mostly motivated by the crop field monitoring problem,we propose a new variational approach for exact reconstruction of damaged multi-bandimages using results of their co-registration with Synthetic Aperture Radar (SAR) imagesof the same regions. We discuss the consistency of the proposed problem, give the schemefor its regularization, derive the corresponding optimality system, and describe in detailthe algorithm for the practical implementation of the reconstruction procedure.In this paper the problem of reconstruction of damaged multi-band opticalimages is studied in the case where we have no information about brightness of suchimages in the damage region. Mostly motivated by the crop field monitoring problem,we propose a new variational approach for exact reconstruction of damaged multi-bandimages using results of their co-registration with Synthetic Aperture Radar (SAR) imagesof the same regions. We discuss the consistency of the proposed problem, give the schemefor its regularization, derive the corresponding optimality system, and describe in detailthe algorithm for the practical implementation of the reconstruction procedure
ON AN INITIAL BOUNDARY-VALUE PROBLEM FOR 1D HYPERBOLIC EQUATION WITH INTERIOR DEGENERACY: SERIES SOLUTIONS WITH THE CONTINUOUSLY DIFFERENTIABLE FLUXES
Get PDFON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS
Get PDFA problem of description of algebraic invariants for a linear control system thatdetermine 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
FOURIER PROBLEM FOR WEAKLY NONLINEAR EVOLUTION INCLUSIONS WITH FUNCTIONALS
Get PDFThe Fourier problem or, in other words, the problem without initial conditionsfor evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained.The Fourier problem or, in other words, the problem without initial conditionsfor evolution equations and inclusions arise in modeling different nonstationary processes in nature, that started a long time ago and initial conditions do not affect on them in the actual time moment. The Fourier problem for evolution variational inequalities (inclusions) with functionals is considered in this paper. The conditions for existence and uniqueness of weak solutions of the problem are set. Also the estimates of weak solutions are obtained