Linear Functional Systems Research Articles

Constructing Lyapunov–Krasovskii functionals for linear positive continuous time difference systems

Abstract In this paper, a linear positive functional difference system with discrete and distributed delays is studied. It is proved that if the considered system is exponentially stable, then it admits both linear and diagonal quadratic Lyapunov–Krasovskii functionals. In addition, new exponential stability conditions for switched linear positive functional difference systems are obtained on the basis of a constructive approach to the design of common Lyapunov–Krasovskii functionals for the associated families of subsystems.

A multivariate Riesz basis of ReLU neural networks

We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of L2([0,1]) based on the Gershgorin theorem. We also generalize this system to higher dimensions d>1 by a construction, which avoids using (tensor) products. As a consequence, the functions from the new Riesz basis of L2([0,1]d) can be easily represented by neural networks. Moreover, the Riesz constants of this system are independent of d, making it an attractive building block regarding future multivariate analysis of neural networks.

Two Extensions of the Quadratic Nonuniform B-Spline Curve with Local Shape Parameter Series

Two extensions of the quadratic nonuniform B-spline curve with local shape parameter series, called the W3D3C1P2 spline curve and the W3D4C2P1 spline curve, are introduced in the paper. The new extensions not only inherit most excellent properties of the quadratic nonuniform B-spline curve but also can move locally toward or against the fixed control polygon by varying the shape parameter series. They are C1 and C2 continuous separately. Furthermore, the W3D3C1P2 spline curve includes the quadratic nonuniform B-spline curve as a special case. Two applications, the interpolation of the position and the corresponding tangent direction and the interpolation of a line segment, are discussed without solving a system of linear functions. Several numerical examples indicated that the new extensions are valid and can easily be applied.

Fundamental Results on Determining Matrices for a Certain Class of Hereditary Systems

Three major tools are required to investigate the controllability of control systems, namely, determining matrices, index of control systems and controllability Grammian. Determining matrices are the preferred choice for autonomous control systems due to the fact that they are devoid of integral operators in their computations. This article developed the structure of certain parameter-ordered determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for Euclidean controllability of the system. Expressions for the relevant determining matrices were formulated and it was established that the determining matrices for double time-delay linear autonomous functional differential control systems do not exist if one of the time-delays is not an integer multiple of the other paving the way for the investigation of the Euclidean controllability of generic double time-delay control systems.

Global Structure of Determining Matrices for a Class of Differential Control Systems

This paper developed and established unprecedented global results on the structure of determining matrices of generic double time-delay linear autonomous functional differential control systems, with a view to obtaining the controllability matrix associated with the rank condition for the Euclidean controllability of the system. The computational process and implementation of the controllability matrix were demonstrated on the MATLAB platform to determine the controllability disposition of a small-problem instance. Finally, the work examined the computing complexity of the determining matrices.

Some Economic Dynamics Problems for Hybrid Models with Aftereffect

In this paper, we consider a class of economic dynamics models in the form of linear functional differential systems with continuous and discrete times (hybrid models) that covers many kinds of dynamic models with aftereffect. The focus of attention is periodic boundary value problems with deviating argument, control problems with respect to general on-target vector-functional and questions of stability to solutions. For boundary value problems, some sharp sufficient conditions of the unique solvability are obtained. The attainability of on-target values is under study as applied to control problems with polyhedral constraints with respect to control, some estimates of the attainability set as well as estimates to a number of switch-points of programming control are presented. For a class of hybrid systems, a description of asymptotic properties of solutions is given.

On a class of linear continuous-discrete systems with discrete memory

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

Abstract In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

Organic contaminant sorption parameters should only be compared across a consistent system of linear functions

Modeling contaminant sorption data using a linear model is very common; however, the rationale for whether the y-intercept should be constrained or not remains a subject of debate. This article justifies constraining the y-intercept in the linear model to zero. By doing so, one imposes consistency on the system of linear equations, allowing for direct comparison of the sorption coefficients.

К оценке значений линейных функционалов на решениях систем с последействием

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.

Prediction of the wind speed change function by linear regression method

In the article the approximation of the function of wind speed changes by linear functions based on Walsh functions and the prediction of function values by linear regression method is made. It is shown that under the condition of a linear change of the internal resistance of the wind generator over time, it is advisable to introduce the wind speed change function with linear approximation. The system of orthonormal linear functions based on Walsh functions is given. As an example, the approximation of the linear-increasing function with a system of 4, 8 and 16 linear functions based on the Walsh functions is given. The result of the approximation of the wind speed change function with a system of 8 linear functions based on Walsh functions is shown. Decomposition coefficients, mean-square and average relative approximation errors for such approximation are calculated. In order to find the parameters of multiple linear regression the method of least squares is applied. The regression equation in matrix form is given. The example of application of the prediction method of linear regression to simple functions is shown. The restoration result for wind speed change function is shown. Decomposition coefficients, mean-square and average relative approximation errors for restoration of wind speed change function with linear regression method are calculated.

On a Class of Optimal Control Problems for Functional Differential Systems

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components

The structure of the Cauchy operator to a linear continuous-discrete functional differential system with aftereffect and some properties of its components

Linear Functions Based on Walsh Functions

In this paper, the method of constructing an orthonormal system of linear functions for approximating the function of changing the wind speed by linear approximations is considered in order to optimize the operation of the wind generator and to maximize energy recovery from renewable sources. In the construction of this system, the system of orthonormal Walsh functions was taken as the basis. In the application of the Gram-Schmidt process to the system of the basic system of functions an analytic expression for linear functions based on Walsh functions was obtained. The developed system of functions possesses all properties, such as the system of continuous Walsh functions. The discrete version of the developed functions has all the parameters of the base system of functions, and also has the same graphs as the system of discrete Walsh functions. Most common ways of organizing functions within the system were considered. It is established that the error of the approximation of the function of changing the speed of the wind does not depend on the way of the ordering of functions. An example of an approximation of the function of change in wind speed during the day is given with the help of linear functions based on Walsh functions with a dimension of 8, as well as errors of the approximation of the wind speed change function are calculated. The mean square error of approximation of linear processes does not exceed 88%, and the average relative error of approximation in knots is 40%. This system is subjected to the Gibbs effect. That is, the reduced accuracy of approximation at the ends of the approximation interval. Given the Gibbs effect, the developed system allows for the approximation of linear processes with a mean square error that does not exceed 33%, the relative error of approximation in units of approximation decreases to 23%. The choice of the dimension of the system affects the frequency spectrum of the approximating signal, which in turn affects the accuracy of the approximation. Namely, with the increase in the number of functions in the spectrum of the approximating function, the content of high-frequency harmonics increases. Therefore, when increasing the signal frequency, which is subject to approximation, it is expedient to increase the dimension of the system. In turn, for the low-frequency signal approximation, it is expedient to use as little as possible the dimensions of the system. Thus, by choosing the appropriate dimension of the system, we can minimize the error of approximation. Ref. 11, fig. 2, add. 1.

Equivalence of weighted and partial optimality of experimental designs

The recently introduced weighted optimality criteria for experimental designs allow one to place various emphasis on different parameters or functions of parameters of interest. However, various emphasis on parameter functions can also be expressed by considering the well-developed optimality criteria for estimating a parameter system of interest (the partial optimality criteria). We prove that the approaches of weighted optimality and of partial optimality are in fact equivalent for any eigenvalue-based optimality criterion. This opens up the possibility to use the large body of existing theoretical and computational results for the partial optimality to derive theorems and numerical algorithms for the weighted optimality of experimental designs. We demonstrate the applicability of the proven equivalence on a few examples. We also propose a slight generalization of the weighted optimality so that it can represent the experimental objective consisting of any system of linear estimable functions.

Use of Felder and Silverman learning style model for online course design

Learning Management Systems are used in millions of higher education courses, across various countries and disciplines. Teachers build courses reflecting their individual teaching methods, which may not always fit students’ different learning styles. However, limited information is known about how well these courses support the learners. The study aims to explore the use of Felder and Silverman learning style for online course design. The study has used linear transfer function system models to develop fundamentals of feedback by a course analyzer tool. This interactive tool allows teachers to determine a course’s support level for specific learning styles, based on the Felder and Silverman learning style model. The Felder and Silverman learning style model in this study is used to visualize the fit between course and learning style to help teachers improve their course’s support for diverse learning styles. The results of a pilot study successfully validated the course analyzer tool, as it has potential to improve the design of the course in future and allow more insight into overall student performance. The findings suggest that a course designed with certain learning styles in mind can improve learning of the students with those specific learning styles.

ДОСТИЖИМЫЕ ЗНАЧЕНИЯ ЦЕЛЕВЫХ ФУНКЦИОНАЛОВ ДЛЯ ФУНКЦИОНАЛЬНО-ДИФФЕРЕНЦИАЛЬНОЙ СИСТЕМЫ С ИМПУЛЬСНЫМ ВОЗДЕЙСТВИЕМ

For a linear functional differential system with aftereffect and impulses, a description of the attainability set is given. The attainability is considered in the term of a given system of on-target functionals in the case of polyhedral constraints with respect to control and impulses.

Reduction of Linear Functional Systems using Fuhrmann's Equivalence

Functional systems arise in the treatment of systems of partial differential equations, delay-differential equations, multidimensional equations, etc. The problem of reducing a linear functional system to a system containing fewer equations and unknowns was first studied by Serre. Finding an equivalent presentation of a linear functional system containing fewer equations and fewer unknowns can generally simplify both the study of the structural properties of the linear functional system and of different numerical analysis issues, and it can sometimes help in solving the linear functional system. In this paper, Fuhrmann's equivalence is used to present a constructive result on the reduction of under-determined linear functional systems to a single equation involving a single unknown. This equivalence transformation has been studied by a number of authors and has been shown to play an important role in the theory of linear functional systems.

Refocusing distance of a standard plenoptic camera.

Recent developments in computational photography enabled variation of the optical focus of a plenoptic camera after image exposure, also known as refocusing. Existing ray models in the field simplify the camera's complexity for the purpose of image and depth map enhancement, but fail to satisfyingly predict the distance to which a photograph is refocused. By treating a pair of light rays as a system of linear functions, it will be shown in this paper that its solution yields an intersection indicating the distance to a refocused object plane. Experimental work is conducted with different lenses and focus settings while comparing distance estimates with a stack of refocused photographs for which a blur metric has been devised. Quantitative assessments over a 24 m distance range suggest that predictions deviate by less than 0.35 % in comparison to an optical design software. The proposed refocusing estimator assists in predicting object distances just as in the prototyping stage of plenoptic cameras and will be an essential feature in applications demanding high precision in synthetic focus or where depth map recovery is done by analyzing a stack of refocused photographs.