Mittag-Leffler Matrix Function Research Articles

Relatively exact controllability of fractional stochastic neutral system with two incommensurate constant delays

This paper is devoted to analyzing a kind of fractional stochastic neutral system (FSNS). Firstly, by introducing the notion of newly defined two‐parameter Mittag–Leffler matrix function, we derive the solution of the corresponding linear stochastic system. Subsequently, for the linear case, by virtue of the Grammian matrix, we give a suffcient and necessary condition to guarantee the relatively exact controllability for the addressed case. Furthermore, for the nonlinear one, the relatively exact controllability is obtained by fixed point and explore it via Banach contraction principle. Finally, two examples are provided to intensify our theoretical conclusions.

A Grammian matrix and controllability study of fractional delay integro-differential Langevin systems

<abstract><p>This study focused on introducing a fresh model of fractional operators incorporating multiple delays, termed fractional integro-differential Langevin equations with multiple delays. Additionally, the research evaluated the relative controllability of this model within finite-dimensional spaces. Employing fixed-point theory yields the desired outcomes, with the controllability assessment facilitated by Schauder's fixed point and the Grammian matrix defined through the Mittag-Leffler matrix function. Validation of the results was conducted through an application.</p></abstract>

New results on controllability analysis of nonlinear fractional order integrodifferential Langevin system with multiple delays

Nowadays, the study of fractional differential equations (FDEs) is one of the most interesting research topics. In this article, we study a novel nonlinear fractional order Langevin system involving delays. The primary focus of this article is to analyze the relative controllability of a nonlinear integrodifferential fractional Langevin dynamical system with multiple delays in control for finite-dimensional spaces. Fundamentals of the fixed-point theory are used to achieve the desired results. Using Schauder’s fixed theorem, sufficient conditions for the controllability are established and the Mittag-Leffler matrix function defines the controllability Grammian matrix. Additionally, the derived results are validated through an application.

Controllability of damped dynamical systems modelled by Hilfer fractional derivatives

In this article, we investigate the controllability of damped dynamical system modelled by Hilfer derivative of fractional order and . The Mittag-Leffler matrix function of two parameters has been used to represent the solution of Hilfer fractional problem of damped dynamical system. With the use of the Schawder fixed point theorem and the contraction mapping theorem, the existence and uniqueness of the linear and non-linear system of Hilfer dynamical control system are determined. More specifically, fractional calculus are used to provide sufficient criteria for the controllability results. Finally, examples are given to show how the key results might be justified.

Controllability of Impulsive Neutral Fractional Stochastic Systems

The study of dynamic systems appears in various aspects of dynamical structures such as decomposition, decoupling, observability, and controllability. In the present research, we study the controllability of fractional stochastic systems (FSF) and examine the Poisson jumps in finite dimensional space where the fractional impulsive neutral stochastic system is controllable. Sufficient conditions are demonstrated with the aid of fixed point theory. The Mittag-Leffler (ML) matrix function defines the controllability of the Grammian matrix (GM). The relation to symmetry is clear since the controllability Grammian is a hermitian matrix (since the integrand in its definition is hermitian) and this is the complex version of a symmetric matrix. In fact, such a Grammian becomes a symmetric matrix in the specific scenario where the controllability Grammian is a real matrix. Some examples are provided to demonstrate the feasibility of the present theory.

On the matrix function $_pR_q(A, B; z)$ and its fractional calculus properties

The main objective of the present paper is to introduce and study the function $_pR_q(A, B; z)$ with matrix parameters and investigate the convergence of this matrix function. The contiguous matrix function relations, differential formulas and the integral representation for the matrix function $_pR_q(A, B; z)$ are derived. Certain properties of the matrix function $_pR_q(A, B; z)$ have also been studied from fractional calculus point of view. Finally, we emphasize on the special cases namely the generalized matrix $M$-series, the Mittag-Leffler matrix function and its generalizations and some matrix polynomials.

Extended Beta and Gamma Matrix Functions via 2-Parameter Mittag-Leffler Matrix Function

The main aim of this article is to study an extension of the Beta and Gamma matrix functions by using a two-parameter Mittag-Leffler matrix function. In particular, we investigate certain properties of these extended matrix functions such as symmetric relation, integral representations, summation relations, generating relation and functional relation.

Controllability analysis of multiple fractional order integro-differential damping systems with impulsive interpretation

We study the controllability criteria for a class of fractional integro-differential damped systems with impulsive perturbations. The solution representation is derived for both linear and non-linear damped systems, and the introduced formulations were constructed by employing Laplace transformation with Mittag-Leffler matrix function. We present necessary and sufficient conditions for the indicated systems to be controllable in finite dimensional spaces. Here, controllability conditions are proposed by using Grammian matrix as a powerful tool. Two numerical examples are also given at end to demonstrate the obtained theory.

Finite-time stability of Hadamard fractional differential equations in weighted Banach spaces

The main purpose of this paper is to investigate the finite-time stability of Hadamard fractional differential equations (HFDEs). Firstly, the standard definitions of finite-time stability of HFDEs in compatible Banach spaces are proposed. In light of the method of successive approximation and Beesack inequality with weakly singular kernel, the criteria of finite-time stability for linear and nonlinear HFDEs are established, respectively. Then with regard to linear HFDEs with pure delay, a novel fractional delayed matrix function (also called delayed Mittag-Leffler matrix function) is given. Specific to nonlinear HFDEs with constant time delay, both Beesack inequality and Hölder inequality are utilized in the framework of the generalized Lipschitz condition. Finally, several indispensable simulations are implemented to verify the effectiveness and practicability of the main results.

Finite time stability of fractional delay difference systems: A discrete delayed Mittag-Leffler matrix function approach

Abstract A discrete delayed Mittag-Leffler matrix function is developed in this paper. Based on this function, an explicit formula of the solution of fractional delay difference system (FDDS) is derived. Furthermore, a criterion on finite time stability (FTS) of FDDS with constant coefficients is obtained by use of this formula. However, it can’t be directly used to investigate the FTS of FDDS with variable coefficients. To overcome this difficulty, a comparison theorem of FDDS is established to obtain a criterion of the FTS of FDDS with variable coefficients. Finally, a numerical example is given to show the effectiveness of the proposed results.

Numerical approach to the controllability of fractional order impulsive differential equations

AbstractIn this manuscript, a numerical approach for the stronger concept of exact controllability (total controllability) is provided. The proposed control problem is a nonlinear fractional differential equation of order\alpha \in (1,2]with non-instantaneous impulses in finite-dimensional spaces. Furthermore, the numerical controllability of an integro-differential equation is briefly discussed. The tool for studying includes the Laplace transform, the Mittag-Leffler matrix function and the iterative scheme. Finally, a few numerical illustrations are provided through MATLAB graphs.

Stability and controllability analysis of fractional damped differential system with non-instantaneous impulses

In this paper, we prove the existence, stability and controllability results for fractional damped differential system with non-instantaneous impulses. The results are obtained by using Banach fixed-point theorem, nonlinear functional analysis, Mittag-Leffler matrix function and controllability Grammian matrix. At last, some numerical examples are given to illustrate the obtained theory.

Total controllability of neutral fractional differential equation with non-instantaneous impulsive effects

In this article, we establish some sufficient conditions for total controllability of a neutral fractional differential system with impulsive conditions in the finite-dimensional spaces. This type of controllability concerns the controllability problem not only at the final time but also at the impulse time. We use Mittag-Leffler matrix function, nonlinear functional analysis, controllability Grammian type matrices and fixed point theorem due to Krasnoselskii’s to establish these results. Also, we extend these results for the studied problem with the integro term. At last, some simulated examples are given to validate the obtained results.

The controllability of damped fractional differential system with impulses and state delay

We discuss the controllability for a damped fractional differential system with impulses and state delay, which involves Caputo fractional derivatives. Deriving the condition based on the Gramian matrix defined by the Mittag-Leffler matrix function and Laplace transformation, we establish necessary and sufficient conditions of controllability criteria. Finally, we construct two numerical examples to support the result.

Controllability of fractional-order damped systems with time-varying delays in control

In this study, we focus on the controllability of fractional-order damped systems in linear and nonlinear cases with multiple time-varying delays in control. For the linear system based on the Mittag-Leffler matrix function, we define a controllability Gramian matrix, which is useful in judging whether the system is controllable or not. Furthermore, in two special cases, we present serval equivalent controllable conditions which are easy to verify. For the nonlinear system, under the controllability of its corresponding linear system, we obtain a sufficient condition on the nonlinear term to ensure that the system is controllable. Finally, two examples are given to illustrate the theory.

Synchronization of Butterfly Fractional Order Chaotic System

In this paper, we study the synchronization of a nonlinear fractional system, and analyze its time response and chaotic behaviors. We represent a solution for considered system by employing the Mittag-Leffler matrix function and Jacobian matrix. Thereafter, we prove synchronization of error system between drive-response systems using stability theory and linear feedback control methods. Finally, numerical simulations are presented to show the effectiveness of the theoretical results.

Relative Controllability of Nonlinear Fractional Langevin Systems with Delays in Control

This paper is concerned with the relative controllability of fractional Langevin dynamical systems with both multiple delays and distributed delays in control for finite dimensional spaces. Sufficient conditions for the controllability are obtained using Schauder’s fixed point theorem and the controllability Grammian matrix which is defined by the Mittag–Leffler matrix function. Examples are provided to illustrate the theory.

分数阶常微分方程的改进精细积分法

Based on the definition of the Mittag-Leffler function, the precise iteration computation scheme for the Mittag-Leffler matrix function was constructed. Compared with the normal iteration scheme for exponential functions, the constructed scheme has additional correction items. The expression of the correction item is related to the order of the fractional derivative. For dynamic fractional ordinary differential equation D^(α)v=Hv with the Caputo fractional definition, the solution function value at the endpoint of the time phase can be obtained with the precise iteration method. The numerical examples demonstrated effectiveness and efficiency of the presented method.

Generalized Mittag-Leffler Matrix Function and Associated Matrix Polynomials

The Mittag-Leffler function has been found useful in solving certain problems in Science and Engineering. On the other hand, noticing the occurrence of certain matrix functions in Special functions’ theory in general and in Statistics and Lie group theory in particular, we introduce here a matrix analogue of a recently generalized form of Mittag-Leffler function. This function yields the matrix analogues of the Saxena-Nishimoto’s function, Bessel-Maitland function, Dotsenko function and the Elliptic Function. We obtain matrix differential equation and eigen matrix function property for the proposed matrix function. Also, a generalized Konhauser matrix polynomial is deduced and its inverse series relations and generating function are derived.

A novel controllability analysis of impulsive fractional linear time invariant systems with state delay and distributed delays in control

In this paper, we investigate the controllability of impulsive fractional linear time invariant systems with state delay and distributed delays in control. By using the controllability Grammian matrix which is defined by the Mittag-Leffler matrix function , a new set of ufficient conditions are obtained for the considered system to be controllable. Finally, two numerical examples are given to demonstrate the validity of the obtained theoretical results.