Stability Of Linear Differential Equations Research Articles

Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform.

Results on Ulam‐type stability of linear differential equation with integral transform

The main theme of this study is to implement the Sumudu integral transform technique to solve the stability problem of linear differential equations. Another important aspect of this paper is to investigate the Ulam–Hyers and Ulam–Hyers–JRassias stability of linear differential equations by using Sumudu transform method. Further, the results are extended to the Mittag‐Leffler–Ulam–Hyers and Mittag‐Leffler–Ulam–Hyers–JRassias stability of these differential equations. As an application point of view, the Sumudu transform is used to find Ulam stabilities of differential equations arising in field‐controlled DC servo motor with position control.

Characterization of Ulam-Hyers stability of linear differential equations with periodic coefficients

Let X be a Banach space over R or C. We study Ulam–Hyers stability on the time interval R of linear differential equations with continuous coefficients, considering solutions with values in X for higher order equations, and solutions with values in Xn for first-order systems. Roughly speaking, Ulam–Hyers stability of an equation is its property of having an exact solution close to an approximate solution. For a first-order system x′=Ax, we obtain that its Ulam–Hyers stability is equivalent to another important property, namely for each continuous and bounded function g:R→Xn there exists a bounded solution of x′=Ax+g. Using this result, theory of Jordan forms of matrices and Floquet theory, we characterize the Ulam–Hyers stability of a system x′=Ax with periodic coefficients in terms of its characteristic multipliers. Our main result states that the Ulam–Hyers stability of a linear equation with periodic coefficients of order n with unknown x is equivalent to the Ulam–Hyers stability of the corresponding system satisfied by (x,x′,…,x(n−1)).

On Input-to-State Stability of Linear Difference Equations and Its Characterization with a Lyapunov Functional

This paper investigates necessary and sufficient Lyapunov conditions for Input-to-State Stability (ISS) of Linear Difference Equations with pointwise delays and an additive exogenous signal. Grounding on recent works in the literature on necessary conditions for the exponential stability of such Difference Equations, we propose a quadratic Lyapunov functional involving the derivative of the so-called delay Lyapunov matrix of the corresponding homogeneous Difference Equation. We prove that the ISS of Linear Difference Equations is equivalent to the existence of an ISS Lyapunov functional. We apply this result to the stability and ISS analysis of hyperbolic Partial Differential Equations of conservation laws.

On the asymptotic stability of linear difference equations with time-varying coefficients

<abstract><p>Issues of the asymptotic stability for linear difference equations with time-varying coefficients are discussed. It is shown that, in contrast to equations with constant coefficients, the condition of Schur stability of the characteristic polynomial for a linear difference equation with time-varying coefficients is neither necessary nor sufficient for the asymptotic stability of the difference equation. It is proved that the analog of Kharitonov's theorem on robust stability and the edge theorem do not hold for a difference equation if the coefficients of the equation are not constant.</p></abstract>

Ulam type stability of second-order linear differential equations with constant coefficients having damping term by using the Aboodh transform

The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of second order with constant coefficients having damping term using the Aboodh transform method. We also obtain the Hyers-Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.

Application of Fourier Transform to Study Hyers-Ulam Stability of Linear Differential Equations

In this paper, we investigate the generalized Hyers-Ulam stabilities of the fourth-order homogeneous differential equation and the non-homogeneous linear differential equation by applying Fourier transform method.

Mahgoub transform and Hyers-Ulam stability of $ n^{th} $ order linear differential equations

<abstract><p>The main aim of this paper is to investigate various types of Hyers-Ulam stability of linear differential equations of $ n^{th} $ order with constant coefficients using the Mahgoub transform method. We also show the Hyers-Ulam constants of these differential equations and give some main results.</p></abstract>

On stability of linear differential equations with commensurate delayed arguments

The paper studies a class of linear differential equations with several delayed arguments formed by iterates of a given function. The main result of this paper improves the existing stability criteria and formulates an effective necessary and sufficient condition relating stability of the studied differential equations to stability of some auxiliary difference equations. In the case of a two-delay equation, this condition is presented explicitly in terms of the equation’s parameters. As an accompanying result, the asymptotic decay rate of the solutions is described as well.

Hyers\u2013Ulam stability of first-order linear differential equations using Aboodh transform

The main aim of this paper is to investigate various types of Ulam stability and Mittag-Leffler stability of linear differential equations of first order with constant coefficients using the Aboodh transform method. We also obtain the Hyers–Ulam stability constants of these differential equations using the Aboodh transform and some examples to illustrate our main results are given.

Mittag-Leffler\u2013Hyers\u2013Ulam stability of differential equation using Fourier transform

This research paper aims to present the results on the Mittag-Leffler–Hyers–Ulam and Mittag-Leffler–Hyers–Ulam–Rassias stability of linear differential equations of first, second, and nth order by the Fourier transform method. Moreover, the stability constant of such equations is obtained. Some examples are given to illustrate the main results.

Ulam stability of linear differential equations using Fourier transform

The purpose of this paper is to study the Hyers-Ulam stability and generalized Hyers-Ulam stability of general linear differential equations of $n$th order with constant coefficients by using the Fourier transform method. Moreover, the Hyers-Ulam stability constants are obtained for these differential equations.

On strong stability and robust strong stability of linear difference equations with two delays

This note studies strong stability of the linear difference equation x(t)=Ax(t−a)+Bx(t−b) where a>0,b>0 are constants and A,B are n×n square matrices. A necessary and sufficient condition in terms of a linear matrix inequality (LMI), where the coefficients A,B appear as a linear function, is introduced. Such a condition is further used to deal with the robust strong stability problem with norm bounded uncertainty, whose solutions are expressed by LMIs. A time-domain interpretation of the LMI condition in terms of integral Lyapunov–Krasovskii functional is given, which helps to reveal the relationships among the existing methods. A numerical example demonstrates the effectiveness of the proposed method.

The Analysis of the Systems Stability of Linear Differential Equations Based on the Transformation of Difference Schemes

The approach to the analysis of Lyapunov systems stability of linear ordinary differential equations based on multiplicative transformations of difference schemes of numerical integration is presented. As a result of transformations, the stability criteria in the form of necessary and sufficient conditions are formed. The criteria are invariant with respect to the right side of the system and do not require its transformation with respect to the difference scheme, the length of the gap and the step of the solution. A distinctive feature of the criteria is that they do not use the methods of the qualitative theory of differential equations. In particular, for the case of systems with a constant matrix of the coefficients it is not necessary to construct a characteristic polynomial and estimate the values of the characteristic numbers. When analyzing the system stability with variable matrix coefficients, it is not necessary to calculate the characteristic indicators. The varieties of criteria in an additive form are obtained, the stability analysis based on them being equivalent to the stability assessment based on the criteria in a multiplicative form. Under the conditions of a linear system stability (asymptotic stability) of differential equations, the criteria of the systems stability (asymptotic stability) of linear differential equations with a nonlinear additive are obtained. For the systems of nonlinear ordinary differential equations the scheme of stability analysis based on linearization is presented, which is directly related to the solution under study. The scheme is constructed under the assumption that the solution stability of the system of a general form is equivalent to the stability of the linearized system in a sufficiently small neighborhood of the perturbation of the initial data. The matrix form of the criteria allows implementing them in the form of a cyclic program. The computer analysis is performed in real time and allows coming to an unambiguous conclusion about the nature of the system stability under study. On the basis of a numerical experiment, the acceptable range of the step variation of the difference method and the interval length of the difference solution within the boundaries of the reliability of the stability analysis is established. The approach based on the computer analysis of the systems stability of linear differential equations is rendered. Computer testing has shown the feasibility of using this approach in practice.

Hyers–Ulam stability of first order linear differential equations of Carathéodory type and its application

The purpose of this study is to derive Hyers–Ulam stability of the linear differential equation x′=ax+f(t) without the assumption of continuity of f(t). The obtained result is applied to a control system.

The z-transform method for the Ulam stability of linear difference equations with constant coefficients

Applying the z-transform method, we study the Ulam stability of linear difference equations with constant coefficients. To a certain extent, our results can be viewed as an important complement to the existing methods and results.

A General Method for the Ulam Stability of Linear Differential Equations

This paper deals with the Ulam stability of linear differential equations by using the method of variation of parameters, which provides a unified method to study the Ulam stability problem of linear differential equations of n-th order with constant and nonconstant coefficients. As an application of the main results, we also obtain the Hyers–Ulam stability of the Cauchy–Euler differential equations of second order, third order and n-th order. Our results make up to some deficiencies in the relevant literature.

Об экспоненциальной устойчивости линейных разностных уравнений с задержками

The article gives an overview of recent results on the stability of finite-difference equations with delay. All results are compared with known signs of exponential stability of linear difference equations. The results are obtained using the Bohl-Perron theorem and comparing the equation under study with an equation for which the Cauchy function is positive. The Bohl-Perron theorem allows us to reduce the question of the exponential stability of a linear difference equation with delay to the solvability of an operator equation in one of the functional infinite-dimensional spaces. That is, in fact, to an estimate of the norm or the spectral radius of a bounded linear operator in this space. For this estimation, different difference inequalities are used. One way to obtain such inequalities is to evaluate the fundamental solution in the event that this solution is positive.

On the stability of linear differential equations of second order

The aim of this paper is to investigate the Hyers-Ulam stability of the linear differential equation$$y''(x)+alpha y'(x)+beta y(x)=f(x)$$in general case, where $yin C^2[a,b],$ $fin C[a,b]$ and $-infty<a<b<+infty$. The result of this paper improves a result of Li and Shen [textit{Hyers-Ulam stability of linear differential equations of second order,} Appl. Math. Lett. 23 (2010) 306--309].

Invariance of Hyers-Ulam stability of linear differential equations and its applications

We prove that the generalized Hyers-Ulam stability of linear differential equations of nth order (defined on I) is invariant under any monotone one-to-one correspondence $\tau: I \to J$ which is n times continuously differentiable. Moreover, using this result, we investigate the generalized Hyers-Ulam stability of the linear differential equation of second order and the Cauchy-Euler equation.