Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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Towards Disease Eradication: Long-Term Control with Constant Vaccination Rates in the Normalized SIR Model
Get PDFIn this paper, we investigate a normalized SIR model incorporating exponential natural birth and death rates, as well as disease-induced mortality and a constant vaccination control parameter denoted as u. This entails vaccinating a fixed percentage of susceptibles in each campaign, a pragmatic approach considering that available economic and human resources often correlate with population size at any given time. Then, we identify a bifurcation value, ubv, determined by other parameters and show that, for u in the interval [0, ubv), the system converges to a steady state with a positive proportion of infected individuals, while for u in (ubv, 1], this proportion approaches zero asymptotically. Notably, the threshold ubv serves as the minimum percentage of the population that should be vaccinated in each campaign to effectively pursue disease eradication. Additionally, we explore the cost implications of a two-phase control strategy. Initially, we employ optimal control techniques to expedite the system’s transition to a state where the infected population proportion stabilizes. Subsequently, we implement a constant-rate vaccination policy to drive the proportion of infected individuals to zero. Our analysis utilizes generic parameters, as we do not focus on a specific disease or population
Approximation of an Optimal BV -Control Problem in the Coefficient for the p(x)-Laplace Equation
Get PDFWe study a Dirichlet optimal control problem for a quasilinear monotone elliptic equation with the so-called weighted p(x)-Laplace operator. The coefficient of the p(x)-Laplacian, the weight u, we take as a control in BV (Ω) ∩ L∞(Ω). In this article, we use box-type constraints for the admissible controls. In order to handle the inherent degeneracy of the p(x)-Laplacian, we use a special two-parametric regularization scheme. We derive existence and uniqueness of variational V -solutions to the underlying boundary value problem and the corresponding optimal control problem. Further we discuss the asymptotic behaviour of the solutions to regularized problems on each (ε, k)-level as the parameters tend to zero and infinity, respectively. The characteristic feature of the considered OCP is the fact that the exponent p(x) is assumed to be Lebesgue-measurable, and we do not impose any additional assumptions on p(x) like to be a Lipschitz function or satisfy the so-called log-H‥older continuity conditio
Qualitative Analysis of a Control-Volume Fibre Coating Model
Get PDFThis paper presents qualitative analysis of the solutions to a control-volume model for liquid films flowing down a vertical fibre. Time evolution of the free surface is governed by a coupled system of degenerate nonlinear partial differential equations, which describe the fluid film’s radius and axial velocity. We demonstrate the existence of weak solutions to an approximation of this system coupled with Dirichlet-Neumann boundary conditions. Additionally, we compare numerical solutions of the model with an approximation for the curvature related term with the original full curvature term model
On a Control Problem and a Pursuit Game of Transferring States Described by An Infinite Three-Systems of Differential Equations
Get PDFIn this paper, we devote to study a pursuit game described by an infinite three-systems of differential equations in Hilbert space. The game involves transferring of the states as the pursuit is said to be completed if the state ζ(・) of the system is shifted to another non zero state ζ1 of the system at some finite time. The control functions of the players are constrained by geometric constraints. We first find the control function that transfers the control system’s state to the state ζ1 at some time. We then extend to solve the pursuit problem where an admissible pursuer’s strategy is constructed and a guaranteed pursuit time is determine
Optimal Stabilization in Systems of Linear Differential Equations
Get PDFThis article considers the optimal stabilization problems for complex dynamical systems, which can be described in terms of linear differential equations. At the beginning of the article, general provisions on optimal stabilization and the application of the apparatus of optimal Lyapunov functions for the purpose of solving the formulated problem are given. To ensure consistency and easier understanding of the obtained results, the systems with scalar control are considered first. The main results were obtained for systems with n-dimensional control and the presence of a diagonal matrix in the quality criteria. Finally, the conditions are extended to the case when a matrix of the general form is used in the quality criterio
Encryption of Color Images Based on Chaotic Attractors Generated by ODE Systems Containing Module Nonlinearities
Get PDFThe main objective of this work is to construct an algorithm for modeling chaotic attractors using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and modular power nonlinearities. These attractors are generated by a cascade of bifurcations of homoclinic orbits existing in the specified ODE system. This approach makes it possible to construct complex chaotic attractors containing many wings or many scrolls. The latter circumstance allows us to significantly complicate the encoding of documents with images transmitted over networks, which is the main criterion for the security of the information contained in these documents. This paper presents a method for encrypting color images using a cryptosystem based on on the properties of the well-known Lorenz attractor and new chaotic attractors generated by ODE systems with modulus-type nonlinearities. When encrypting color images, these attractors can generate parameters for changing pixel colors, which makes it difficult to restore the original image without knowing the exact characteristics of the attractors and starting points. To enhance the resilience of the encryption and protect against predictable patterns, a Block Cipher Mode of Operation is employed. The performance of the algorithm will be demonstrated both with and without the application of these modes, allowing for an assessment of their impact on the overall security of the syste
Univariate Time Series Analysis with Hyper Neural ODE
Get PDFNeural ordinary differential equations (NODE) are ordinary differential equations whose right-hand side is determined by a neural network. Hyper NODE (hNODE) is a special type of neural network architecture, which is aimed at creating such NODE system that regulates its own parameters based on known input data. The article uses a new approach to the study of one-dimensional time series, the basis of which is the hNODE system. This system takes into account the relationship between the input data and its latent representation in the network and uses an explicit parametrization when controlling the latent flow. The proposed model is tested on artificial time series of data. The influence of some activation functions (besides sigmoid and hyperbolic tangent) on the quality of the forecast is also considered
On the Asymptotic Equivalence of Ordinary and Functional Stochastic Differential Equations
Get PDFThis paper studies the asymptotic behavior of solutions of linear stochastic functional-differential equations. This behavior is investigated using the method of asymptotic equivalence, according to which an ordinary system of linear differential equations is constructed based on the initial stochastic system, and the asymptotic behavior of the solutions of this system is analogous to the behavior of the solutions of the initial system
The Analytical View of Solution of the First Boundary Value Problem for the Nonlinear Equation of Heat Conduction with Deviation of the Argument
Get PDFIn this article, for the first time, the first boundary value problem for the equation of thermal conductivity with a variable diffusion coefficient and with a nonlinear term, which depends on the sought function with the deviation of the argument, is solved. For such equations, the initial condition is set on a certain interval. Physical and technical reasons for delays can be transport delays, delays in information transmission, delays in decision-making, etc. The most natural are delays when modeling objects in ecology, medicine, population dynamics, etc. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (solid bodies, liquids, etc.) is modeled by heat conduction equations. The Green’s function of the first boundary value problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated, and the formula for the solution is established
Optimal Control Problem for Non-linear Degenerate Parabolic Variation Inequality: Solvability and Attainability Issues
Get PDFWe investigate the optimal control problem with respect to coefficients of the degenerate parabolic variational inequality. Since problems of this type can have the Lavrentieff effect, we consider the optimal control problem in a class of so-called Hadmissible solutions. We substantiate the attainability of H-optimal pairs via optimal solutions of some nondegenerate perturbed optimal control problems