Controllability of Fractional Differential Equation of Order α∈(1,2] With Non‐Instantaneous Impulses

The objective of this paper is to establish the sufficient condition for the exact controllability of a control system governed by the fractional differential equation of order α∈(1,2] with non‐instantaneous impulses in a Hilbert space X. The results are obtained using the α‐order cosine family and Banach fixed point theorem. Also, the exact controllability of an integrodifferential problem is studied. Finally, a few examples are given to illustrate the applications of these abstract results.     

On The Exact Controllability of Impulsive Fractional Semilinear Functional Differential Inclusions

In this paper, we deal with the exact controllability for impulsive fractional semilinear functional differential inclusions in Hilbert spaces. Under suitable assumptions, the exact controllability of the control systems are formulated and proved. The main tools in our study rely on the fixed point theorem for multivalued maps due to Dhage associated with evolution systems.     

Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay

We prove the approximate controllability of control systems governed by a class of partial neutral functional differential systems of fractional order with state-dependent delay in an abstract space. Sufficient conditions for approximate controllability of the control systems are established provided the approximate controllability of the corresponding linear control systems. The results are obtained by using the Krasnoselskii–Schaefer type fixed point theorem with the fractional power of operators. An example is provided to illustrate the main results.     

Approximate controllability of fractional nonlocal delay semilinear systems in Hilbert spaces

We study the existence and approximate controllability of a class of fractional nonlocal delay semilinear differential systems in a Hilbert space. The results are obtained by using semigroup theory, fractional calculus and Schauder’s fixed point theorem. Multi-delay controls and a fractional nonlocal condition are introduced. Furthermore, we present an appropriate set of sufficient conditions for the considered fractional nonlocal multi-delay control system to be approximately controllable. An example to illustrate the abstract results is given.     

Approximate controllability results for non-densely defined fractional neutral differential inclusions with Hille–Yosida operators

In this paper, we mainly focused the approximate controllability results for a class of non-densely defined fractional neutral differential control systems. First, we establish a set of sufficient conditions for the approximate controllability for a class of fractional differential inclusions with infinite delay where the linear part is non-densely defined and satisfies the Hille–Yosida condition. The main techniques rely on Bohnenblust– Karlin's fixed point theorem, operator semigroups and fractional calculus. Further, we extend the result to study the approximate controllability concept with nonlocal conditions. Finally, an example is also given for the illustration of the obtained theoretical results.     

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations (FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.      

Pinning Synchronization Between Two General Fractional Complex Dynamical Networks With External Disturbances

In this paper, the pinning synchronization between two fractional complex dynamical networks with nonlinear coupling, time delays and external disturbances is investigated. A Lyapunov-like theorem for the fractional system with time delays is obtained. A class of novel controllers is designed for the pinning synchronization of fractional complex networks with disturbances. By using this technique, fractional calculus theory and linear matrix inequalities, all nodes of the fractional complex networks reach complete synchronization. In the above framework, the coupling-configuration matrix and the innercoupling matrix are not necessarily symmetric. All involved numerical simulations verify the effectiveness of the proposed scheme.      

Relative controllability of nonlinear switched fractional systems

This article investigates relative controllability of nonlinear switched fractional systems (SFSs). With the aid of Mittag-Leffler functions and invariant subspace theory, two sufficient and necessary conditions for corresponding linear SFSs are first established. Then, piecewise continuous control functions and a novel nonlinear operator are constructed to overcome the difficulties arising from switching rules, nonlinearity, and fractional derivatives. Under different constraints on nonlinear functions, two controllability conditions depending on system parameters for nonlinear SFSs are proposed by applying Schauder's and Banach's fixed point theorems, respectively. The obtained criteria may well show the influence of coefficient matrices, fractional derivatives, and switching rules on relative controllability. In addition, our proposed method is also applicable for integer-order switched systems. Finally, two simulated examples are worked out to verify the theoretical results.     

Variational Calculus With Conformable Fractional Derivatives

Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of invariance are obtained. As particular cases, we prove fractional versions of Noether's symmetry theorem. Invariant conditions for fractional optimal control problems, using the Hamiltonian formalism, are also investigated. As an example of potential application in Physics, we show that with conformable derivatives it is possible to formulate an Action Principle for particles under frictional forces that is far simpler than the one obtained with classical fractional derivatives.      

Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation

This paper mainly discusses fractional differential approach to detecting textural features of digital image and its fractional differential filter. Firstly, both the geo- metric meaning and the kinetic physical meaning of fractional differential are clearly explained in view of information theory and kinetics, respectively. Secondly, it puts forward and discusses the definitions and theories of fractional stationary point, fractional equilibrium coefficient, fractional stable coefficient, and fractional grayscale co-occurrence matrix. At the same time, it particularly discusses frac- tional grayscale co-occurrence matrix approach to detecting textural features of digital image. Thirdly, it discusses in detail the structures and parameters of nxn any order fractional differential mask on negative x-coordinate, positive x-coordi- nate, negative y-coordinate, positive y-coordinate, left downward diagonal, left upward diagonal, right downward diagonal, and right upward diagonal, respectively. Furthermore, it discusses the numerical implementation algorithms of fractional differential mask for digital image. Lastly, based on the above-mentioned discus- sion, it puts forward and discusses the theory and implementation of fractional differential filter for digital image. Experiments show that the fractional differential-based image operator has excellent feedback for enhancing the textural details of rich-grained digital images.     

Controllability of stochastic integrodifferential systems

In this paper sufficient conditions for the complete controllability of a stochastic semilinear integrodifferential system in finite dimensional spaces are established using the resolvent matrix and the Banach fixed point theorem. An example is provided to illustrate the result.     

Impulsive boundary value problem for nonlinear differential equations of fractional order

In this paper, we prove the existence and uniqueness of solutions for the boundary value problem of nonlinear impulsive differential equations of fractional order q∈(1,2]. Our results are based on Altman’s fixed point theorem and Leray-Schauder’s fixed point theorem.     

Fractional variable order discrete-time systems,their solutions and properties

In this paper, fractional variable order discrete state-space systems based on different definitions of fractional variable order difference are investigated.The general solution of these systems is derived. Moreover, necessary and sufficient conditions for reachability and observability are given and proven. The sufficient conditions for controllability are proposed too.     

Approximate controllability for fractional semilinear parabolic equations

In this paper, we deal with the control systems described by a large class of fractional semilinear parabolic equations. Firstly, we reformulate the fractional parabolic equations into abstract fractional differential equations associated with a semigroup on an appropriate Banach space. Secondly, we introduce a suitable concept on a mild solution for this kind of fractional parabolic equations and present the existence and uniqueness of mild solution by utilizing the theory of semigroup of linear operator, nonlinear analysis method and fixed point theorem. Then, the approximate controllability of the fractional semilinear parabolic equations is formulated and proved. At the end of the paper, an example is given to illustrate our main results.     

Fuzzy Adaptive Control of a Fractional Order Chaotic System With Unknown Control Gain Sign Using a Fractional Order Nussbaum Gain

In this paper we propose an improved fuzzy adaptive control strategy, for a class of nonlinear chaotic fractional order (SISO) systems with unknown control gain sign. The online control algorithm uses fuzzy logic sets for the identification of the fractional order chaotic system, whereas the lack of a priori knowledge on the control directions is solved by introducing a fractional order Nussbaum gain. Based on Lyapunov stability theorem, stability analysis is performed for the proposed control method for an acceptable synchronization error level. In this work, the Grünwald-Letnikov method is used for numerical approximation of the fractional order systems. A simulation example is given to illustrate the effectiveness of the proposed control scheme.      

Existence of Optimal Mild Solutions and Controllability of Fractional Impulsive Stochastic Partial Integro‐Differential Equations with Infinite Delay

In this paper, we investigate a new class of fractional impulsive stochastic partial integro‐differential equations with infinite delay in Hilbert spaces. By using the stochastic analysis theory, fractional calculus, analytic α‐resolvent operator and the fixed point technique combined with fractional powers of closed operators, we firstly give the existence of of mild solutions and optimal mild solutions for the these equations. Next, the controllability of the controlled fractional impulsive stochastic partial integro‐differential systems with not instantaneous impulses is presented. Finally, examples are also given to illustrate our results.     

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

This paper introduces a “spectral observability condition” for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates, and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one-dimensional output systems from Müntz–Szász theorem on the closed span of sets of power functions.     

基于四元数分数阶方向微分的图像增强

根据已有的四元数和分数阶微分的知识,以及它们在现代信号处理中的应用,推导出一种新知识:四元数分数阶方向微分,并将其应用于图像增强.该方法首先将一幅彩色图像用一个四元数函数表示,给出四元数函数的分数阶方向微分的定义和计算方法,继而推导出沿八个方向的四元数分数阶方向导数的数值计算模板,并根据八个方向的分数阶导数求出图像平面...     

Controllability analysis for a class of linear quadratic conformable fractional game-based control systems

Controllability analysis for a class of linear quadratic conformable fractional game-based control systems (CFGBCSs) is investigated in this article. Supported by the definition and property of conformable fractional, the payoff function for linear quadratic CFGBCSs is proposed, and Nash equilibrium solution is obtained via variational method for Hamilton function correspondingly. Subsequently, the state transition matrix and controllability matrix are presented to deduce the sufficient and necessary conditions for the controllability of Nash equilibrium solution of time-varying linear quadratic CFGBCSs and time-invariant linear quadratic CFGBCSs, respectively. Finally, two numerical examples are furnished to justify the validity of the matrix full rank condition.     

Fractional partial differential equation denoising models for texture image

In this paper,a set of fractional partial differential equations based on fractional total variation and fractional steepest descent approach are proposed to address the problem of traditional drawbacks of PM and ROF multi-scale denoising for texture image.By extending Green,Gauss,Stokes and Euler-Lagrange formulas to fractional field,we can find that the integer formulas are just their special case of fractional ones.In order to improve the denoising capability,we proposed 4 fractional partial differential equation based multiscale denoising models,and then discussed their stabilities and convergence rate.Theoretic deduction and experimental evaluation demonstrate the stability and astringency of fractional steepest descent approach,and fractional nonlinearly multi-scale denoising capability and best value of parameters are discussed also.The experiments results prove that the ability for preserving high-frequency edge and complex texture information of the proposed denoising models are obviously superior to traditional integral based algorithms,especially for texture detail rich images.