Journal of Optimization, Differential Equations and their Applications (JODEA - Dnipro National Universuty) / Журнал з оптимізації, диференціальних рівнянь та їх застосувань
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Discrete Processes and Chaos in Systems of Ordinary Differential Equations
Get PDFA method for constructing a one-dimensional discrete mapping describing a certain periodic process in a general system of ordinary autonomous differential equations is proposed. The resulting discrete mapping is then used to prove the existence of chaos in the original system of differential equations
Existence and Uniqueness Solutions of Fuzzy Fractional Integral Equation of Volterra-Stieltjes Type
Get PDFIn this paper, we establish the existence and uniqueness results to the Cauchy problem posed for a fuzzy fractional Volterra-Stieltjes integrodifferential equation. The method of successive approximations is used to prove the existence, whereas the contraction theory is applied to prove the uniqueness of the solution to the problem
On Increasing of Resolution of Satellite Images via Their Fusion with Imagery at Higher Resolution
Get PDFIn 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.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
Solutions to a Simplified Initial Boundary Value Problem for 1D Hyperbolic Equation with Interior Degeneracy
Get PDFA 1-parameter initial boundary value problem (IBVP) for a linear homogeneousdegenerate wave equation (JODEA, 28(1), 1) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function.A 1-parameter initial boundary value problem (IBVP) for a linear homogeneousdegenerate wave equation (JODEA, 28(1), 1 â“ 42) in a space-time rectangle is considered. The origin of degeneracy is the power law coefficient function with respect to the spatial distance to the symmetry line of the rectangle, the exponent being the only parameter of the problem, ranging in (0,1) and (1,2) and producing the weak and strong degeneracy respectively. In the case of weak degeneracy separation of variables is used in the rectangle to obtain the unique bounded continuous solution to the IBVP, having the continuous flux. In the case of strong degeneracy the IBVP splits into the two derived IBVPs posed respectively in left and right half-rectangles and solved separately using separation of variables. Continuous matching of the obtained left and right families of bounded solutions to the IBVPs results in a linear integro-differential equation of convolution type. The Laplace transformation is used to solve the equation and obtain a family of bounded solutions to the IBVP, having the continuous flux and depending on one undetermined function
Space-Time-Domain Decomposition for Optimal Control Problems Governed by Linear Hyperbolic Systems
Get PDFIn this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work [31, 32] and answer a long standing open question as to whether the combination of time- and space-domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by P. L. Lions [25] for elliptic problems.In this article, we combine a domain decomposition method in space and time for optimal control problems with PDE-constraints described in [2] to a simultaneous space-time decomposition applied to optimal control problems for systems of linear hyperbolic equations with distributed control. We thereby extend the recent work [31, 32] and answer a long standing open question as to whether the combination of time- and space-domain decomposition for the method under consideration can be put into one single convergent iteration procedure. The algorithm is designed for a semi-elliptic system of equations obtained from the hyperbolic optimality system by the way of reduction to the adjoint state. The focus is on the relation to the classical procedure introduced by P. L. Lions [25] for elliptic problems
Nodal Stabilization of the Flow in a Network with a Cycle
Get PDFIn this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through the nodes of the network. The analysis of the decay of the Riemann invariants requires to follow backwards all the characteristics that enter such a node and contribute to the evolution. This means that with each nodal reflection/crossing the number of characteristics that contribute to the evolution increases. We show how for simple networks with a sufficient number of damping nodal controlers it is possible to keep track of this family of characteristics and use this approach to analyze the exponential stability of the system. The analysis is based on an adapted version of Gronwall’s lemma that allows us to take into account the possible increase of the Riemann invariants when the characteristic curves cross a node of the network. Our example is motivated by applications in the control of gas pipeline flow, where the graphs of the networks often contain many cycles.In this paper we discuss an approach to the stability analysis for classical solutions of closed loop systems that is based upon the tracing of the evolution of the Riemann invariants along the characteristics. We consider a network where several edges are coupled through node conditions that govern the evolution of the Riemann invariants through the nodes of the network. The analysis of the decay of the Riemann invariants requires to follow backwards all the characteristics that enter such a node and contribute to the evolution. This means that with each nodal reflection/crossing the number of characteristics that contribute to the evolution increases. We show how for simple networks with a sufficient number of damping nodal controlers it is possible to keep track of this family of characteristics and use this approach to analyze the exponential stability of the system. The analysis is based on an adapted version of Gronwall’s lemma that allows us to take into account the possible increase of the Riemann invariants when the characteristic curves cross a node of the network. Our example is motivated by applications in the control of gas pipeline flow, where the graphs of the networks often contain many cycles
Computer Simulation of the Stress-Strain State of the Plate with Circular Hole and Functionally Graded Inclusion
Get PDFComputer simulation of the behavior of a thin elastic rectangular plate with a circular hole and an annular inclusion made of functionally graded material has been carried out. Using the finite element method, the influence of the geometric and mechanical parameters of the inclusion on the concentration of stresses around the hole is investigated and various laws of the change in the modulus of elasticity of a functionally graded material are specified. A comparative analysis of the results has been carried out. The recommendations for reducing stress concentration are given.Computer simulation of the behavior of a thin elastic rectangular plate with a circular hole and an annular inclusion made of functionally graded material has been carried out. Using the finite element method, the influence of the geometric and mechanical parameters of the inclusion on the concentration of stresses around the hole is investigated and various laws of the change in the modulus of elasticity of a functionally graded material are specified. A comparative analysis of the results has been carried out. The recommendations for reducing stress concentration are given
Finding the Zeros of a High-Degree Polynomial Sequence
Get PDFA 1-parameter initial-boundary value problem for a linear spatially 1-dimensionalhomogeneous degenerate wave equation, posed in a space-time rectangle, in case of strong degeneracy, was reduced to a linear integro-differential equation of convolution type (JODEA, 29(1) (2021), pp. 1–31). The former was then solved by applying the Laplace transformation, and the solution formula was inverted in the form of the Neumann series. The current study deals with an other approach to the inversion of the solution formula, based on invoking the Bromwich integral and the Cauchy residue theorem for the integrand. The denominator of the integrand being an infinite series with respect to rational functions of the complex variable, converges quite rapidly and can be approximated with finite series of m terms. Therefore finding the zeros of the approximated denominator reduces to finding the zeroes of a polynomial of degree 2m. For the resulting polynomial sequence some numerical approaches have been applied.A 1-parameter initial-boundary value problem for a linear spatially 1-dimensionalhomogeneous degenerate wave equation, posed in a space-time rectangle, in case of strong degeneracy, was reduced to a linear integro-differential equation of convolution type (JODEA, 29(1) (2021), pp. 1–31). The former was then solved by applying the Laplace transformation, and the solution formula was inverted in the form of the Neumann series. The current study deals with an other approach to the inversion of the solution formula, based on invoking the Bromwich integral and the Cauchy residue theorem for the integrand. The denominator of the integrand being an infinite series with respect to rational functions of the complex variable, converges quite rapidly and can be approximated with finite series of m terms. Therefore finding the zeros of the approximated denominator reduces to finding the zeroes of a polynomial of degree 2m. For the resulting polynomial sequence some numerical approaches have been applied.
On the Equivalence of Real Dynamic Process and Its Neural Network Quadratic Models
Get PDFA dynamic process defined by its own time series is considered. Using the methods of qualitative recurrent analysis, the dimension of the embedding
space and the optimal time delay of the specified series are determined. Using these characteristics, a neural network with a quadratic activation
function is modeled. The simulation result is presented in the form of a system of neural ODEs. After that, the Lyapunov exponents of the real
dynamic system and its neural network model are calculated. Then the closeness of these exponents for a real system and its model makes it
possible to judge the adequacy (equivalence) of both dynamic processes. Examples are given.A dynamic process defined by its own time series is considered. Using the methods of qualitative recurrent analysis, the dimension of the embedding space and the optimal time delay of the specified series are determined. Using these characteristics, a neural network with a quadratic activation function is modeled. The simulation result is presented in the form of a system of neural ODEs. After that, the Lyapunov exponents of the real dynamic system and its neural network model are calculated. Then the closeness of these exponents for a real system and its model makes it possible to judge the adequacy (equivalence) of both dynamic processes. Examples are given
On Boundary Null Controllability of Strongly Degenerate Hyperbolic Systems on Star-Shaped Planar Network
Get PDFIn this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.In this paper we discuss the problem of boundary exact null controllability for weakly and strongly degenerate linear wave equation defined on star-shaped planar network. The network is represented by a singular measure in a bounded planar domain. The novelty of this article lies in the degeneration of the leading coefficient representative of the material properties at the common node of network. We discuss the existence of weak and strong solutions to the degenerate hyperbolic problem and establish the corresponding controllability properties.