Solutions Of Cauchy Research Articles

On optimal control for linear distributed time-delay systems

This article presents the solution to the infinite horizon optimal control problem for linear distributed time-delay systems. It is based on the use of the Cauchy solution for distributed time-delay systems. In contrast to previous results for systems with concentrated delays, the form of the functional and its properties are formally justified. An important property is demonstrated: the Bellman functional admits upper and lower bounds; thus, it is proved to be positive definite. Additionally, some experimental results of a temperature process are presented.

A novel insight into the interface fracture between magnesium phosphate cement mortar and cement concrete

ABSTRACT This paper focused on fracture problem about interfacial transition zone (ITZ) formed after repairing between magnesium phosphate cement (MPC) mortar and old Portland cement concrete (PCC) pavements. Firstly, the mode Ⅱ cracks on the ITZ were hypothesized and the interface fracture model with dislocation theory was proposed. The stress field at the crack tip was represented by Airy stress potential function and Fourier Transform, and the structural characteristics of the kernel solution of Cauchy were retained. Then, combined with the oscillation term, the Dirac function was replaced by the normal distribution function to obtain stable fracture parameters. Besides, the ϵ about 8 ‰ of the crack length in model was analyzed. Secondly, the elastic moduli of ITZs were obtained by nanoindentation experiments. Results showed the elastic modulus of ITZ between MPC mortar and ITZ-1 (the interface in concrete) was lowest. Such weakest position in repair interface was further verified by finite element analysis (FEA). Finally, a twodimensional model was developed to validate the fracture model. Simulation results were found consistent with the theoretical solution of interface fracture model. The developed interface fracture model can provide insights for evaluating the shear behavior of the interface between MPC mortar and concrete.

Continuous dependence of fixed points on parameters, remetrization theorems, and an application to the initial value problem

We show that the recent result of Bełdziński et al. on continuous dependence of fixed points on parameters, which extends a classical parametrized version of the Banach fixed point theorem, may be treated as a particular case of the contraction principle for a Nemytskiĭ operator on a space of continuous functions with values in a metric space. Next, we establish a generalization of the result of Bełdziński et al. involving an equicontinuous family of mappings having contractive th iterates for some . As an application, we obtain a theorem on continuous dependence of solutions of Cauchy's problem on both the initial values and the parameters. Finally, we establish two remetrization theorems and we discuss a possibility of deriving our main result from the Banach contraction principle by using a remetrization of the space.

The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers

The non-existence of a Hamel-basis and the general solution of Cauchy's functional equation for nonnegative numbers

SUBSTANTIATION OF AMPLITUDE-FREQUENCY CHARACTERISTICS AND DESIGN PARAMETERS OF THE SEPARATOR WITH A VIBRATION DRIVE

The main effects of the broken design of the vibrating separator are: increased crushing force in the process of separating the slurry mass in this robot, which was moved by a path on the working container with a sieve of the conical shape of the vibrating ruh; a plethora of minds passing through a lot of products through the opening that they reached for the rahunka of the sieve surface of the bulk coliving; Changes in energy consumption and increase in the minds of robotic support units during the operation of the projected vibrating separator, which have been achieved by the addition of additional spring elements between the separator housing and bearing units of the vertical drive shaft of the vibration separator. The application of the proponated vibrating screen to the working organs of the volumetric kolivalny roar allows to increase the productivity and capacity of the process of distributing solid-state husky masses. To determine the rational parameters in the process of vibrating sieving, we took away the ratio of the rot of the victorious organs at the end of the sieve surface when victorious to the Lagrange method of the 2nd genus. When zastosuvanni Cauchy solutions for linear non-homogeneous differential lines, the remainder of the rozvnіz was eliminated. Removing the fallowness of the amplitude of the crushing, the vibratory characteristics and the vibration acceleration, the intensity of the crushing movement made it possible to carry out a mathematical analysis of the power and energy characteristics of the vibrodrive of the finished separator. The expansion of the conical sieve surface allows you to create expanses of gyration or colossal progressive movement, which allows you to realize the importance of a large separation of dry masses. The results of the analytical investigation made it possible to prime the optimal cut to the working sieve surface. On the basis of the analysis carried out, the grounding and clarification of the design parameters of the vibratory system were presented and the design of this technical system was presented.

Linear and nonlinear solvers for simulating multiphase flow within large-scale engineered subsurface systems

Multiphase flow simulation is well-known to be computationally demanding, and modeling large-scale engineered subsurface systems entails significant additional numerical challenges. These challenges arise from: (a) the presence of small-scale discrete features like shafts, tunnels, waste packages, and barriers; (b) the need to accurately represent both the waste form processes at the small spatial scale of the repository and the large-scale transport processes throughout heterogeneous geological formations; (c) the strong contrast in material properties such as porosity and permeability, as well as the nonlinear constitutive relations for multiphase flow. Numerical solution entails discretization of the coupled system of nonlinear governing equations and solving a linear system of equations at each Newton–Raphson iteration. Practical problems require a very large number of unknowns that must be solved efficiently using iterative methods in parallel on high-performance computers. The unique challenges noted above can lead to an ill-conditioned Jacobian matrix and non-convergence with Newton’s method due to discontinuous nonlinearity in constitutive models. Moreover, practical applications can require numerous Monte-Carlo simulations to explore uncertainly in material properties, geological heterogeneity, failure scenarios, or other factors; governmental regulatory agencies can mandate these as part of Performance Assessments. Hence there is a need for flexible, robust, and computationally efficient methods for multiphase flow in large-scale engineered subsurface systems.We apply the open-source simulator PFLOTRAN to the practical problem of performance assessment of the US DOE Waste Isolation Pilot Plant (WIPP) site. The simulator employs a finite volume discretization and uses the PETSc parallel framework. We evaluate the performance of several preconditioners for the iterative solution of the linearized Jacobian system; these range from stabilized-biconjugate-gradient with block-Jacobi preconditioning (BCGS) to methods adopted from reservoir modeling, such as the constrained pressure residual (CPR) two-stage preconditioner and flexible generalized residual solver (FGMRES). We also implement within PETSc the general-purpose nonlinear solver, Newton trust-region dogleg Cauchy (NTRDC), which truncates the Newton update or modifies the update with a Cauchy solution that is within the quadratic model trust-region of the objective function. We demonstrate the effectiveness of each method for a series of test problems with increasing difficulty. We find that the NTRDC and FGMRES-CPR-ABF (FCA) preconditioners generally perform best for the test problem having the extreme nonlinear processes, achieving a 50x speed-up compared with BCGS. The most ill-conditioned and extreme nonlinear simulations do not converge with BCGS (as one may expect), but they do complete the simulation with NTRDC and FCA. We also investigate the strong scalability of each method and demonstrate the impact of node-packing upon parallel performance on modern processor architectures.

Simulation of the movement of the supporting leg of an exoskeleton with two links of variable length in 3D

A two-link model of exoskeleton with variable-length links for supporting the lower limbs of the human musculoskeletal system is proposed in the article. The researched model differs from the existing ones by the variable-length links, and by the angle calculation method. While in the existing models, the angles are calculated from the regular direction – from vertical, or from horizontal, – in the proposed research they are calculated between the links. As for practical exoskeleton implementation, the proposed method of angle calculation is appropriate to the actual working conditions of the electrical motors with the reduction gears installed in the hinges, which change the angles between the links. The construction of a variable-length exoskeleton link consists of two absolutely solid weighty sections located at both ends of the link and one weightless section between them in the center of the link. In the weightless section, there is a drive that creates a control longitudinal force, which realizes the increase or decrease in the length of the link in the required manner and provides the necessary maintenance of the length of the link when the person moves in the exoskeleton. The links are connected to each other using spherical hinges. Drives are installed in each hinge, creating control torques, which provide a relative rotational movement of the links. The jointly controlling longitudinal forces and moments realize the maintenance of the posture or the movement of the link in the required manner and, in relation to the exoskeleton, the repetition of the basic biomechanical properties of the human musculoskeletal system. The mathematical model in the form of the system of Lagrange differential equations of the second kind is obtained. The obtained mathematical model is examined for existence and uniqueness of the Cauchy solution. The kinematic trajectory of the link motion has been synthesized, which simulates the anthropomorphic movement of the supporting leg during the single-support phase of movement, and the control actions required for its implementation has been found. The significance of the results obtained in the process of modeling lies in the ability to create active exoskeletons, prostheses in medicine, anthropomorphic robots, and spacesuits that take into account the biomechanical features of the functioning of the human musculoskeletal system.

A Generalized Gronwall Inequality for Caputo Fractional Dynamic Delta Operator

In this paper we obtain generalized Gronwall type inequality using Caputo Fractional delta operator. Also we have obtained the existence of solution of Cauchy's Type problem on fractional dynamic equations using dynamic delta operator. Applying the obtained inequality we study the properties of solution on fractional dynamic equations.

Local Proportional-Integral Boundary Feedback Stabilization for Quasilinear Hyperbolic Systems of Balance Laws

The proportional-integral (PI) boundary stabilization of nonlinear hyperbolic systems of balance laws is investigated for the $H^2$-norm, in which the control and output measurements are all located at the boundaries. The boundary conditions of the system are subject to unknown constant disturbances. The induced closed-loop system is proven to be locally exponentially stable with respect to the steady states. To this end, a set of matrix inequalities is given by constructing a new Lyapunov function as a weighted $H^2$-norm of the classical Cauchy solution and the integral of boundary output. Furthermore, the traffic flow dynamics of a freeway section are modeled with the Aw--Rascle--Zhang model. To stabilize the oscillations of traffic demand, a local PI boundary feedback controller is designed with integration of on-ramp metering and variable speed limit control. The exponential convergence of the nonlinear traffic flow dynamics in an $H^2$ sense is achieved and validated with simulations.

On higher differentiability of solutions of parabolic systems with discontinuous coefficients and (p,q)-growth

AbstractWe consider weak solutions$u:\Omega _T\to {\open R}^N$to parabolic systems of the type$$u_t-{\rm div}\;a(x,t,Du) = 0\quad {\rm in}\;\Omega _T = \Omega \times (0,T),$$where the functiona(x,t,ξ) satisfies (p,q)-growth conditions. We give an a priori estimate for weak solutions in the case of possibly discontinuous coefficients. More precisely, the partial maps$x\mapsto a(x,t,\xi )$under consideration may not be continuous, but may only possess a Sobolev-type regularity. In a certain sense, our assumption means that the weak derivatives$D_xa(\cdot ,\cdot ,\xi )$are contained in the class$L^\alpha (0,T;L^\beta (\Omega ))$, where the integrability exponents$\alpha ,\beta $are coupled by$$\displaystyle{{p(n + 2)-2n} \over {2\alpha }} + \displaystyle{n \over \beta } = 1-\kappa $$for someκ∈ (0,1). For the gap between the two growth exponents we assume$$2 \les p < q \les p + \displaystyle{{2\kappa } \over {n + 2}}.$$Under further assumptions on the integrability of the spatial gradient, we prove a result on higher differentiability in space as well as the existence of a weak time derivative$u_t\in L^{p/(q-1)}_{{\rm loc}}(\Omega _T)$. We use the corresponding a priori estimate to deduce the existence of solutions of Cauchy–Dirichlet problems with the mentioned higher differentiability property.

Global large smooth solutions for 3-D Hall-magnetohydrodynamics

In this paper, the global smooth solution of Cauchy's problem of incompressible, resistive, viscous Hall-magnetohydrodynamics (Hall-MHD) is studied. By exploring the nonlinear structure of Hall-MHD equations, a class of large initial data is constructed, which can be arbitrarily large in $H^3(\mathbb{R}^3)$. Our result may also be considered as the extension of work of Lei-Lin-Zhou from the second-order semi-linear equations to the second-order quasilinear equations, because the Hall term elevates the Hall-MHD system to the quasilinear level.

Dry friction distributed over a contact patch between a rigid body and a visco-elastic plane

We consider the dynamics of an absolutely rigid body moving along a rough horizontal plane. We assume that the plane deforms during the motion so that the contact patch is non-planar and has non-zero, but comparatively small area. In the contact patch, the vertical reactions are proportional to the vertical deformations and their rates, that is, we consider Kelvin–Voigt model. The tangent forces are assumed to be classic Coulomb dry friction in sliding regime (no stiction in contact patch). Due to the viscous part of Kelvin–Voigt law, the contact patch, the plane’s normal reaction, friction force and torque depend on the position, orientation, velocity of the center of mass and the angular velocity of the body. The model does not involve any additional dynamic parameter and gives the friction force and torque directly for any given state of the body. To show the advances of the model, we recall the analytical solution of Cauchy’s initial problem with general initial conditions of ODE governing the dynamics of the homogeneous sphere on a horizontal plane (results of Zobova and Treschev in Proc. Steklov Inst. Math. 281:91–118, 2013). Here we generalize the model for arbitrary convex bodies and study its main properties. The model is compared to the continuous velocity-base friction model for pure sliding (Brown and McPhee in ASME J. Comput. Nonlinear Dyn. 11(5):054502, 2016) and to the combined dry friction (Awrejcewicz and Kudra in Multibody Syst. Dyn., 2018, https://doi.org/10.1007/s11044-018-9624-9 ) in case of sliding with spinning. We illustrate the model considering the results of numerical integration of the Cauchy problem for a controlled differential-drive vehicle on a horizontal plane.

Refined Meshless Local Strong Form solution of Cauchy–Navier equation on an irregular domain

This paper considers a numerical solution of a linear elasticity problem, namely the Cauchy–Navier equation, using a strong form method based on a local Weighted Least Squares (WLS) approximation. The main advantage of the employed numerical approach, also referred to as a Meshless Local Strong Form method, is its generality in terms of approximation setup and positions of computational nodes. In this paper, flexibility regarding the nodal position is demonstrated through two numerical examples, i.e. a drilled cantilever beam, where an irregular domain is treated with a relatively simple nodal positioning algorithm, and a Hertzian contact problem, where again, a relatively simple h-refinement algorithm is used to extensively refine discretization under the contact area. The results are presented in terms of accuracy and convergence rates, using different approximations and refinement setups, namely Gaussian and monomial based approximations, and a comparison of execution time for each block of the solution procedure.

Analysis of Lavrentiev-finite element methods for data completion problems

The variational finite element solution of Cauchy’s problem, expressed in the Steklov–Poincare framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in Azaiez et al. (Inverse Probl Sci Eng 18:1063–1086, 2011). The present work concentrates on the numerical analysis of the semi-discrete problem. We perform the mathematical study of the error to rigorously establish the convergence of the global bias-variance error.

Identifying the stored energy of a hyperelastic structure by using an attenuated Landweber method

We consider the nonlinear inverse problem of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field as well as from surface sensor measurements. The displacement field is represented as a solution of Cauchy’s equation of motion, which is a nonlinear elastic wave equation. Hyperelasticity means that the first Piola-Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that a dictionary of suitable functions is available. The aim is to recover the stored energy with respect to this dictionary. The considered inverse problem is of vital interest for the development of structural health monitoring systems which are constructed to detect defects in elastic materials from boundary measurements of the displacement field, since the stored energy encodes the mechanical properties of the underlying structure. In this article we develop a numerical solver using the attenuated Landweber method. We show that the parameter-to-solution map satisfies the local tangential cone condition. This result can be used to prove local convergence of the attenuated Landweber method in the case that the full displacement field is measured. In our numerical experiments we demonstrate how to construct an appropriate dictionary and show that our method is well suited to localize damages in various situations.

Local Exponential Stabilization of Semi-Linear Hyperbolic Systems by Means of a Boundary Feedback Control

This letter investigates the boundary feedback control for a class of semi-linear hyperbolic partial differential equations with nonlinear relaxation, which is local Lipschitz continuous with a stable matrix structure. A sufficient condition in terms of linear inequalities is developed for the existence of global Cauchy solutions and the exponential stability by seeking a balance between the relaxation term and the boundary condition. These results are illustrated with an application to the boundary feedback control for a class of hyperbolic Lotka–Volterra models.

Radial solutions to the Cauchy problem for □1+3U = 0 as limits of exterior solutions

We consider the strategy of realizing the solution of a Cauchy problem (CP) with radial data as a limit of radial solutions to initial-boundary value problems posed on the exterior of vanishing balls centered at the origin. The goal is to gauge the effectiveness of this approach in a simple, concrete setting: the three-dimensional (3d), linear wave equation [Formula: see text] with radial Cauchy data [Formula: see text], [Formula: see text]. We are primarily interested in this as a model situation for other, possibly nonlinear, equations where neither formulae nor abstract existence results are available for the radial symmetric CP. In treating the 3d wave equation, we therefore insist on robust arguments based on energy methods and strong convergence. (In particular, this work does not address what can be established via solution formulae.) Our findings for the 3d wave equation show that while one can obtain existence of radial Cauchy solutions via exterior solutions, one should not expect such results to be optimal. The standard existence result for the linear wave equation guarantees a unique solution in [Formula: see text] whenever [Formula: see text]. However, within the constrained framework outlined above, we obtain strictly lower regularity for solutions obtained as limits of exterior solutions. We also show that external Neumann solutions yield better regularity than external Dirichlet solutions. Specifically, for Cauchy data in [Formula: see text], we obtain [Formula: see text]-solutions via exterior Neumann solutions, and only [Formula: see text]-solutions via exterior Dirichlet solutions.

Elementary volume and measurability properties of additive functions

Abstract The paper is concerned with some aspects of the theory of elementary volume from the measure-theoretical standpoint. It is shown that there exists a nontrivial solution of Cauchy's functional equation, nonmeasurable with respect to every translation invariant measure on the real line, extending the one-dimensional Lebesgue measure.

Cauchy Solution of the Kinematic Wave Shallow-Water Equations Using Square-Grid Finite-Element Method

AbstractThe kinematic wave system of shallow-water equations is posed as an initial value problem, or Cauchy problem, to investigate error analysis using the square-grid finite-element method. The consistent, lumped, and upwind models of the square finite-element method are analyzed using the Fourier (or von Neumann) stability method as separate initial value problems for two-dimensional shallow-water equations. The small difference between the Cauchy solution and integer (or half) multiples of the eigenvalue solution of amplification factors using the eigenvalue method indicates that the Cauchy solution is a possible stable numerical solution of the kinematic wave shallow-water equations within the reasonable limits of solution accuracy. The nodal amplification factors are less than or equal to unity for implicit finite-element schemes for all of the three formulations—consistent, lumped, and upwind for all of the wave numbers, implying unconditional stability. For explicit and semi-implicit finite-eleme...

Global solvability of Cauchy–Dirichlet problem for fully nonlinear parabolic systems

We show existence and uniqueness theorems of global strong solutions of Cauchy–Dirichlet problem, for second order fully nonlinear parabolic systems, that satisfy the Campanato's condition of ellipticity. We use the Campanato's near operators theory.