Lyapunov functions and a control problem

1967 ◽  
Vol 63 (2) ◽  
pp. 435-438 ◽  
Author(s):  
A. A. Kayande ◽  
D. B. Muley
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Control Problem ◽  
Lyapunov Functions ◽  
Differential Inequality ◽  
Differential Systems ◽  
Systems Of Differential Equations ◽  
Linear Differential Systems ◽  
Lyapunov’S Second Method ◽  
Linear Differential

1. One of the most important techniques in the study of non-linear differential systems is the Lyapunov's second method and its extensions. One of the extensions of the method depends upon the fact that the function satisfying a differential inequality can be majorized by the maximal solution of the corresponding differential equation. This method was used extensively by V. Lakshmikantham and others for obtaining results, in a unified way, on stability and boundedness of systems of differential equations.

scholarly journals Analytical Solutions of Fuzzy Linear Differential Equations in the Conformable Setting

2021 ◽  
Vol 2 (2) ◽  
pp. 13-30
Author(s):  
Awais Younus ◽  
Muhammad Asif ◽  
Usama Atta ◽  
Tehmina Bashir ◽  
Thabet Abdeljawad
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Analytical Solutions ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Conformable Derivative ◽  
Systems Of Differential Equations ◽  
Two Systems ◽  
Linear Differential

In this paper, we provide the generalization of two predefined concepts under the name fuzzy conformable differential equations. We solve the fuzzy conformable ordinary differential equations under the strongly generalized conformable derivative. For the order $\Psi$, we use two methods. The first technique is to resolve a fuzzy conformable differential equation into two systems of differential equations according to the two types of derivatives. The second method solves fuzzy conformable differential equations of order $\Psi$ by a variation of the constant formula. Moreover, we generalize our results to solve fuzzy conformable ordinary differential equations of a higher order. Further, we provide some examples in each section for the sake of demonstration of our results.

Generalized systems of linear differential equations

1981 ◽  
Vol 89 (1-2) ◽  
pp. 1-14 ◽  
Author(s):  
Rainer Pfaff
Keyword(s):  
Differential Equations ◽  
Bounded Variation ◽  
Higher Order ◽  
Linear Differential Equations ◽  
Differential Systems ◽  
First Order ◽  
Linear Differential Systems ◽  
Generalized Systems ◽  
Linear Differential ◽  
Derivatives Of

SynopsisWe consider ordinary linear differential systems of first order with distributional coefficients and distributional nonhomogeneous terms. Firstly the coefficients are assumed to be functions, secondly to be first order distributions (i.e. first derivatives of functions which are integrable or of bounded variation), and thirdly to be distributions of higher order.

scholarly journals Algebraic Criteria of the Integral Separation for Solving Certain Classes of Ordinary Differential Equations Systems

2020 ◽  
pp. 1-14
Author(s):  
A. A. Akhrem ◽  
A. P. Nosov
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Systems ◽  
Qualitative Theory ◽  
Small Perturbations ◽  
Differential Systems ◽  
Fundamental Properties ◽  
Linear Differential Systems ◽  
Integral Separation ◽  
Linear Differential

One of the important directions of the qualitative theory of ordinary differential equations is to study the properties of linear systems that satisfy the condition of integral separation. Anyway, integral separation becomes apparent in all studies concerning the asymptotic behavior of the solutions for the linear systems under the action of small perturbations.The papers of V.M. Millionschikov, B.F. Bylov, N.A. Izobov, I.N. Sergeev et al. proved that the available integral separation is the main reason for the rough stability of the characteristic Lyapunov exponents, the rough stability of the highest Lyapunov exponent, and the rough diagonalizability of systems by Lyapunov transformations, and other fundamental properties of linear differential systems.The paper presents the basic properties of the set of linear systems with constant, periodic, reducible coefficients and proves the algebraic criteria for their property of integral separation of solutions to be available.The results can be used in modeling dynamic processes.

Multiplicity and bifurcation of periodic solutions in ordinary differential equations

1979 ◽  
Vol 82 (3-4) ◽  
pp. 225-232 ◽  
Author(s):  
B. Laloux
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Diagonal Matrix ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Linear Differential

SynopsisOne considers the linear differential systems where is a (not necessarily diagonal) matrix and one relates the computation of a general multiplicity defined from this system to the corresponding multiplicity of some eigenvalues of . Then applying these conclusions, one gives simple conditions ensuring the existence of odd or even periodic solutions for systems having the form .

V.—On the Connexion between Linear Differential Systems and Integral Equations

1923 ◽  
Vol 42 ◽  
pp. 43-53 ◽  
Author(s):  
E. L. Ince
Keyword(s):  
Integral Equation ◽  
Differential Equations ◽  
Integral Equations ◽  
Green's Function ◽  
Linear Differential Equations ◽  
Differential Systems ◽  
Definite Integrals ◽  
Linear Differential Systems ◽  
Linear Differential ◽  
The Relationship

This paper summarises the results of an attempt to extend the theory upon which the relationship between linear differential equations and integral equations is based. The case in which the nucleus K(x, s) of the integral equation arises as a Green's function is well known; the nucleus is there characterised by its having discontinuous derivates when x = s. The method here dealt with is virtually an extension of Laplace's and analogous methods for solving linear differential equations by definite integrals, and leads to nuclei which are continuous and have continuous derivates for x = s.

scholarly journals Nonautonomous Differential Equations in Banach Space and Nonrectifiable Attractivity in Two-Dimensional Linear Differential Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Siniša Miličić ◽  
Mervan Pašić
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Finite Interval ◽  
Zero Solution ◽  
Two Dimensional ◽  
Matrix Elements ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Nonautonomous Differential Equations ◽  
Linear Differential

We study the asymptotic behaviour on a finite interval of a class of linear nonautonomous singular differential equations in Banach space by the nonintegrability of the first derivative of its solutions. According to these results, the nonrectifiable attractivity on a finite interval of the zero solution of the two-dimensional linear integrable differential systems with singular matrix-elements is characterized.

scholarly journals The Composition of Linear Differential Systems

1932 ◽  
Vol 3 (1) ◽  
pp. 10-15 ◽  
Author(s):  
J. M. Whittaker
Keyword(s):  
Differential Equation ◽  
Differential Operator ◽  
Lower Order ◽  
Galois Theory ◽  
Algebraic Equations ◽  
Differential Systems ◽  
Linear Differential Systems ◽  
Analogous Process ◽  
Linear Differential ◽  
Image Position

A common method of solving a linear differential equation consists in expressing the differential operator as a product of factors. The possibility of doing so has been studied extensively by Vessiot, following the work of Picard and Drach, on the lines of the Galois theory of algebraic equations. The analogous process of resolving a, linear differential system, consisting of an equation together with boundary conditions, into two or more systems of lower order does not seem to have been investigated. Such a resolution is not always possible, even in cases where the differential equation can be factorised. Thus the systemis equivalent to the systemsbut the systemcannot be so resolved.

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