Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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How Can We Manage Repairing a Broken Finite Vibrating String? Formulations of the Problem
Get PDFThree approaches to solve a well-known IBVP posed for vibrating composite string with piece-wise constant properties have been applied. The main issue of the stufy is the number of the matching conditions to be imposed for the solution to the IBVP to be obtained
Singular Differential Equations and their Applications for Modeling Strongly Oscillating Processes
Get PDFThe normal system of ordinary differential equations, whose right-hand sides are the ratios of linear and nonlinear positive functions, is considered. A feature of these ratios is that some of their denominators can take on arbitrarily small nonzero values. (Thus, the modules of the corresponding derivatives can take arbitrarily large value.) In the sequel, the constructed system of differential equations is used to model strongly oscillating processes (for example, processes determined by the rhythms of electroencephalograms measured at certain points in the cerebral cortex). The obtained results can be used to diagnose human brain diseases
Qualitative Analysis of an Optimal Sparse Control Problem for Quasi-Linear Parabolic Equation with Variable Order of Nonlinearity
Get PDFIn this work, we study a sparse optimal control problem involving a quasilinear parabolic equation with variable order of nonlinearity as a state equation and with a pointwise control constraints. We show that in the case if the cost functional contains the terminal term of the tracking type, the proposed optimal control problem is ill-posed, in general. In view of this, we provide a sufficiently mild relaxation of the proposed problem and establish the existence of optimal solutions for the relaxed version. Using the compensated compactness technique and the consept of variational convergence of minimization problems, we study the attainability of optimal pairs to the relaxed problem by optimal solutions of the special approximating problems. We also discuss the optimality conditions for approximating problems and provide their substantiation
Systems of Singular Differential Equations as the Basis for Neural Network Modeling of Chaotic Processes
Get PDFCurrently, systems of neural ordinary differential equations (ODEs) have become widespread for modeling various dynamic processes. However, in forecasting tasks, priority remains with the classical neural network approach to building a model. This is due to the fact that by choosing the neural network architecture, a more accurate approximation of the trajectories of a dynamic system can be achieved. It is known that the accuracy of the mentioned approximation significantly depends on the settings of the neural network parameters and their initial values. In this regard, the main idea of the article is that the initial values of the neural network parameters are taken to be equal to the parameters of the neural ODE system obtained by modeling the same process, which will then be simulated using a neural network. Subsequently, the singular ODE system was used to adjust the parameters of the LSTM (Long Short Term Memory) neural network. The results obtained were used to model the process of epilepsy
Stability w.r.t. Disturbances for the Global Attractor of Multi-Valued Semiflow Generated by Nonlinear Wave Equation
Get PDFThe paper investigates the issue of stability with respect to external disturbances for the global attractor of the wave equation under conditions that do not ensure the uniqueness of the solution to the initial problem. Under general conditions for nonlinear terms, it is proved that the global attractor of the undisturbed problem is locally stable in the sense of ISS and has the AG property with respect to disturbances
Homogeneous Approximation of One- Dimensional Series of Iterated Integrals and Time Optimality
Get PDFIn the paper we consider nonlinear systems depending linearly on control with one-dimensional output. As is well known, under the analyticity requirement, the output can be expressed as a series of iterated integrals of controls with scalar coefficients. Since iterated integrals generate a free associative algebra, algebraic and combinatorial tools can be applied. Developing ideas of the algebraic approach to homogeneous approximation of nonlinear control systems, we propose the definition of a homogeneous approximation for series of iterated integrals with one-dimensional coefficients and study its algebraic properties. In addition, we describe relations between the homogeneous approximation of one-dimensional series of iterated integrals and an approximation in the sense of time optimality. Namely, we give conditions under which the optimal time and optimal controls for the minimal realization of the initial series are approximated by the optimal time and optimal controls for the minimal realization of its homogeneous approximation in a neighborhood of the origin
Can a Finite Degenerate ‘String’ Hear Itself? Numerical Solutions to a Simplified IBVP
Get PDFSome discrete models for a simplified (compared to that published earlier in JODEA, 28 (1) (2020), 1 – 42) initial boundary value problem for a 1D linear degenerate wave equation, posed in a space-time rectangle and solved earlier exactly (JODEA, 30 (1) (2022), 89 – 121), have been considered. It has been demonstrated that the correct evaluation of the degenerate grid flux can be possible
Nonlinear Parabolic Variational Inequalities with Variable Time-Delay in Time Unbounded Domains
Get PDFWe research the well-posedness of the problem without initial condition for nonlinear parabolic variational inequalities with variable time-delay. To justify our results, we impose some assumptions on the solution behavior and growth of the data-in as time variable tends to −∞. Also, we obtain estimates for weak solutions of this problem
Optimal Distributed Globally Bounded Control for Parabolic–Hyperbolic Equations with Nonlocal Boundary Conditions and a Linear Quality Criterion
Get PDFFor the problem of optimal control of a parabolic-hyperbolic process with nonlocal point boundary conditions, an explicit form of the solution is obtained in the form of formal series according to the system of eigenfunctions, which are generated by the spatial differential operator and boundary conditions. At the same time, the unequivocal solvability of the intermediate problems is established for each iteration. In addition, sufficient conditions for the convergence of the series are established, which determine the obtained formal solution of the optimal control problem, which justifies its correctness system of differential equations; Lyapunov exponents; fractal dimensio
Model of Non-Isothermal Consolidation in the Presence of Geobarriers and the Total Approximation Properties of its Finite Element Solutions
Get PDFThe boundary value problem for the system of quasi-linear parabolic equations in the presence of integral conjugation conditions is considered. The boundary value problem is a mathematical model of the process of non-isothermal filtration consolidation of the soil mass which contains a thin geobarrier. Geobarriers exposed to non-isothermal conditions are a component of waste storage facilities. The change of hydromechanical and thermal properties the geobarriers, as well as the phenomenon of thermal osmosis, require modification of both the equations in the mathematical model and the conjugation conditions. The finite element method is used to find approximate solutions of the corresponding system of quasi-linear parabolic equations. The existence and uniqueness of the approximate generalized solution is proved. The accuracy of finite element solutions in the sense of total approximation are also estimated. The differences in the values of pressure and temperature distributions for the classical case and the case considered in the article were analyzed on the test model example