Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
Not a member yet
196 research outputs found
Sort by
Higher-Order Optimality Conditions for Degenerate Unconstrained Optimization Problems
Get PDFIn this paper necessary and sufficient conditions of a minimum for the unconstrained degenerate optimization problem are presented. These conditions generalize the well-known optimality conditions. The new optimality conditions are presented in terms of polylinear forms and Hesse’s pseudoinverse matrix. The results are illustrated by examples.The formulation and appearance of these conditions differ from high-order optimality conditions by other authors. The suggested representation of high-order optimality conditions makes them convenient for the evaluation of the convergence rate for unconstrained optimization methods in the case of a singular minimum point, for example, for the analysis of Newton’s and quasi-Newton’s methods
Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs
Get PDFThe main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. Inaddition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given
Can a Finite Degenerate ‘String’ Hear Itself The Exact Solution to a Simplified IBVP
Get PDFThe separation of variables based solution to 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, has been presented in a fully complete form. Degeneracy of the equation is due to vanishing its coefficient in an interior point of the spatial segment being the side of the rectangle. For the sake of convenience, the solution is interpreted as a vibrating ‘string’. The solution obtained in the case of weak degeneracy is smooth and bounded, whereas that in the case of strong degeneracy is piece-wise smooth, piece-wise continuous and unbounded in a neighborhood of the point of degeneracy, nevertheless being satisfied some regularity conditions, including square-integrability. In both cases the travelling waves pass through the point of degeneracy, and this phenomenon is referred to as an ability of the ‘string’ to hear itself. The total energy of the ‘string’ is shown to conserve in both cases of degeneracy, provided the ends of the ‘string’ are fixed, though the above vibrating ‘string’ analogy fails in the case of strong degeneracy. The total energy conservation implies the uniqueness of the solution to the problem in both cases of degenerac
Solvability Issues for Some Noncoercive and Nonmonotone Parabolic Equations Arising in the Image Denoising Problems
Get PDFThis paper is devoted to the solvability of an initial-boundary value problem for second-order parabolic equations in divergence form with variable order of nonlinearity. The characteristic feature of the considered class of Cauchy-Neumann parabolic problem is the fact that the variable exponent p(t, x) and the anisotropic diffusion tensor D(t, x) are not well predefined a priori, but instead these characteristic depend on a solution of this problem, i.e., p = p(t, x, u) and D = D(t, x, u). Recently, it has been shown that the similar models appear in a natural way as the optimality conditions for some variational problems related to the image restoration technique. However, from practical point of view, in this case some principle difficulties can appear because of the absence of the corresponding rigorous mathematically substantiation. Thus, in this paper, we study the solvability issues of the Cauchy-Neumann parabolic boundary value problem for which the corresponding principle operator is strongly non-linear, non-monotone, has a variable order of nonlinearity, and satisfies a nonstandard coercivity and boundedness conditions that do not fall within the scope of the classical method of monotone operators. To construct a weak solution, we apply the technique of passing to the limit in a special approximation scheme and the Schauder fixed-point theorem
Homogenized Models with Memory Effect for Heterogeneous Periodic Media
Get PDFThe homogenization of initial boundary value problems for heat conduction equations with asymptotically degenerate rapidly oscillating periodic coefficients are considered. Such problems model thermal processes in heterogeneous periodic media. Homogenized problems (whose solutions determine approximate asymptotics for solutions of the original problems) are presented. Estimates for the accuracy of the asymptotics and relevant convergence theorem are discussed. The homogenized problems have the form of initial boundary value problems for integro-differential equations in convolutions. The presence of convolutions in models for media is called the memory effect. Statements about the solvability and regularity for the problems and the homogenized problems are proved. These results are optimal even in the case of zero convolutions, when the homogenized problems coincide with the classical heat conduction problems
Fictitious Controls and Approximation of an Optimal Control Problem for Perona-Malik Equation
Get PDFWe discuss the existence of solutions to an optimal control problem for the Cauchy-Neumann boundary value problem for the evolutionary Perona-Malik equations. The control variable v is taken as a distributed control. The optimal control problem is to minimize the discrepancy between a given distribution ud 2 L2( ) and the current system state. We deal with such case of non-linearity when we cannot expect to have a solution of the original boundary value problem for each admissible control. Instead of this we make use of a variant of its approximation using the model with fictitious control in coefficients of the principle elliptic operator. We introduce a special family of regularized optimization problems for linear parabolic equations and show that each of these problems is consistent, well-posed, and their solutions allow to attain (in the limit) an optimal solution of the original problem as the parameter of regularization tends to zero
Initial-Boundary Value Problems for Anisotropic Parabolic Equations with Variable Exponents of the Nonlinearity in Unbounded Domains with Conditions at Infinity
Get PDFWe deal with the initial-boundary value problems with some restrictions atinfinity for linear and nonlinear anisotropic parabolic second-order equations in unbounded domains with respect to the spatial variables. The weak solutions of our problem in Lebesgue and Sobolev spaces with variable exponents is considered. We prove theorems on the existence and uniqueness of the weak solutions using the method based on Saint- Venant principle, and the monotonicity method. Moreover, we obtain estimate of the weak solutions
Nonlinear Evolutionary Problem of Filtration Consolidation With the Non-Classical Conjugation Condition
Get PDFFinite-element solutions of the initial-boundary value problem for a nonlinear parabolic equation in an inhomogeneous domain with the conjugation condition of a non-ideal contact were found. The initial boundary value problem is a mathematical model of an important technical problem of filtration consolidation of inhomogeneous soils. Inhomogeneity is considered in terms of the presence of thin inclusions, physicochemical characteristics of which differ from those of the main soil. The problem of longterm consolidation is especially pronounced in soils with low filtration coefficient. Low permeability of the porous medium causes deviation from the linear relationship between the pressure gradient and the filtration rate.Weak formulation of the problem is suggested, and the accuracy of the approximate finite element solution, its existence and uniqueness are substantiated for the case of Darcy’s nonlinear law. A test example and the effect of the nonlinear filtration law for thin inclusion on the dynamics of scattering of excess pressures in the entire area of the problem are considered
Shape, Velocity, and Exact Controllability for the Wave Equation
Get PDFA new method to prove exact controllability for the wave equation is demonstrated and discussed on several examples. The method of proof first uses a dynamical argument to prove shape controllability and velocity controllability, thereby solving their associated moment problems. This enables one to solve the moment problem associated to exact controllability
Robust Stability of Global Attractors for Evolutionary Systems without Uniqueness
Get PDFWe establish the local input-to-state stability of multi-valued evolutionary systems with bounded disturbances with respect to the global attractor of the respective undisturbed system. We apply obtained results to disturbed reaction-diffusion equation