Stability Analysis for a Nonautonomous, Inhomogeneous System with Application to Tethered Satellite Formations

linear periodic impulsive differential equations are considered. The impulse effect instants are assumed to satisfy the average dwell-time condition (the ADT condition). The stability problem is reduced to studying the stability of an auxiliary abstract impulsive differential equation. This is a perturbed periodic impulsive differential equation, which considerably simplifies the construction of a Lyapunov function. Sufficient conditions for the asymptotic stability of abstract linear periodic impulsive differential equations are obtained. It is shown that the ADT conditions lead to less conservative dwell-time estimates guaranteeing asymptotic stability. Bibliography: 24 titles.

A system of linear interval equations is called solvable if each system of linear equations contained therein is solvable. In the main result of this paper it is proved that solvability of a general rectangular system of linear interval equations can be characterized in terms of nonnegative solvability of a finite number of systems of linear equations which, however, is exponential in matrix size; the problem is proved to be NP-hard. It is shown that three earlier published results are consequences of the main theorem, which is compared with its counterpart valid for linear interval inequalities that turn out to be much less difficult to solve.

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

Iterative refinement of solution with biparameter for solving ill-conditioned systems of linear algebraic equations

A method for solving systems of linear interval equations applied to the Leontief input–output model of economics

Dynamics of a flexible tethered satellite system utilising various materials for coplanar and non-coplanar models

In this paper, a novel approach based on the Floquet theory is applied for the stability analysis of a mass–spring system with switchable stiffness. The Reid model is used to describe the dynamics of this semi-active vibration control problem. The semi-active control is achieved by a spring which commutes between a maximum and minimum stiffness according to a prescribed state-dependent rule and its performance is characterized by a system parameter, which relates to the extreme values of the stiffness. In order to apply the Floquet theorem, the Reid model is written as a linear periodic differential equation by converting the state-dependent rule into a time-periodic control law. The application of the theory allows us to obtain the Floquet multipliers and exponents in terms of the system parameter. The multipliers lie inside the unitary circle showing asymptotic stability, while the exponents are used to solve an optimization problem by applying a sensitivity analysis. Our results are validated by analyzing the Reid model using nonlinear analysis techniques. According to our findings, the present approach provides a useful tool to analyze the vibration control of linear systems with switchable stiffness in a natural and straightforward way, which also gives mathematical tractability for optimization purposes. In addition, this approach can be extended to study the cases of multi-degree-of-freedom systems and forced systems.

In previous chapters, we have learned to solve a wide range of differential equations. Primarily, our focus has been on linear differential equations: first-order linear equations, higher-order linear equations with constant coefficients, and systems of linear equations with constant coefficients. Indeed, we have learned through a variety of techniques that under the proviso that a differential equation or system is linear, we can almost always find a solution.

In 1968–1969, A. V. Jategaonkar published his famous constructions of left but not right noetherian rings that provided counterexamples to several important conjectures of that era. These examples, and others like them, seemed to indicate that, in general, the task of completely understanding the structure of indecomposable injective modules over one-sided noetherian rings was hopeless. In this paper I show how to deduce by natural methods, directly from the known description of these rings and their properties, explicit computational descriptions of the indecomposable injective left modules over Jategaonkar's rings. I use these explicit descriptions to answer some simple structural questions about the indecomposables. The structure of indecomposable injective modules over a commutative noetherian ring has been well understood since the work of Matlis[10] in the 1950's. There is a natural correspondence between the indecomposable injectives and the localizations of the ring, and the structure of ...

We consider homogeneous and inhomogeneous systems of linear difference equations with coefficients that are-periodic functions of discrete time. For homogeneous systems, sufficient conditions for the existence of periodic and almost periodic solutions are obtained. For inhomogeneous systems, it is shown that a necessary and sufficient condition for the existence of an N-periodic solution is the existence of a bounded solution. Necessary and sufficient conditions for theN orthogonality of the fundamental matrix of the homogeneous system are established. Illustrative examples are given.

Electrodynamic tethers that make use of electricity-conducting tethers in space could provide a propellant-free propulsion system in low Earth orbit. In 2010, deployment of a bare electrodynamic tape tether was attempted in a S-520-25 sounding rocket experiment by an international team of Japan, Europe, the USA and Australia. A new scheme, the inverse-origami method, is employed for the tape tether deployment and the space deployment of tape tether was the first success in the world. Understanding of deployment dynamics of tape tethers is prerequisite to establishing a highly reliable deployment technique of tape tethers in space. This paper introduces results of tether deployment experiments on the ground after the space demonstration. A high-speed camera is used to capture the dynamic behaviour of the tape tether under the deployment regime. The analysis of the experimental results shows the deployment drag induced by the friction force between the tape tether and the exit of the storage box as well as the deformation of the tape tether.

Weighted pseudo-almost periodic solutions of a class of abstract differential equations

Another constructive proof is presented for the fact that a system of linear equations with integer coefficients in bounded integer variables is equivalent to a single equation, which is a linear combination of the original ones. The equation is obtained in a number of steps; in each step two equations are replaced by a single one. This replacement is performed subject to the condition that the remaining equations hold and a final equation with relatively small coefficients is obtained. It may be inefficient however to calculate small coefficients, as the original coefficients can be used to represent the final ones in a suitably chosen number system.

The first equations of the 1940 pantograph were considered in the works of Mahler. He used functional differential equations with compression arguments in number theory. He used functional differential equations with compression arguments in number theory. In 1971, J. Ockendon used functional differential equations with transformed arguments of the form to describe the motion of an electric locomotive pantograph. Recently, pantograph type equations have been widely used. For example, a configurable (pantograph) phone device holder, a retractable (pantograph) microphone device, etc. In this article, on the segment discusses a two-point boundary value problem for systems of integro-differential equations of the pantograph type. To solve this problem, the parameterization method proposed by Professor D. Dzhumabaev is used. To do this, we divide the segment in question into parts. Let's denote the value of the desired function at the starting point of each segment through the parameters and replace it in the intervals . Then the problem under consideration is formally divided into two parts, i.e., a system of linear algebraic equations with respect to the introduced parameters and to the Cauchy problem for a system of integro-differential equations of the pantograph type. Thus, the problem is reduced to a closed system for determining the solution of a special Cauchy problem for systems of integro–differential equations of the pantograph type and a system of linear equations. Based on this, an algorithm for determining the solution of the initial problem is proposed.

We present two new parallel algorithms for exact (error-free) solution of a system of linear equations on a distributed-memory multiprocessor. The exact solution is obtained using the congruence technique which consists of two steps: First, the system of linear equations is converted to systems of linear congruence equations with respect to several prime moduli, and each of these systems is solved on a separate processor. Then, these solutions are combined using the mixed-radix conversion algorithm to obtain the exact solution. The first step is completely (embarrassingly) parallel with no communication requirements among the processors. We improve our previous work and describe two efficient parallel algorithms for the second step. We present the results of our experiments on an Intel iPSC/860 with 8 processors. A linear system of dimension 128 with integer entries as large as 10577 is solved in about 195 seconds on 8 processors with an efficiency of 99.76%.