A general method for stability controllability in the theory of fractional-order differential systems

线性分数脉冲演化方程在Hilbert空间中的近似可控性

Approximate Controllability of Linear Fractional Impulsive Evolution Equations in Hilbert Spaces

本文研究了Hilbert空间中线性分数脉冲演化方程的近似可控性，通过分数解算子与脉冲算子结合推导mild解，并基于残差算子的收敛性条件提出近似可控性判据。

时间分数阶薛定谔方程中随机量子扩散率的平均可控性

Averaged Controllability of Time-Fractional Schrödinger Equations with Random Quantum Diffusivity

本文研究了时间分数阶薛定谔方程在随机量子扩散率下的平均可控性问题，证明了绝对连续扩散率无法实现同时零可控性，并提出了一种双参数分数特征函数以实现任意时间的零平均可控性。

基于预解算子技术的二阶脉冲中立型微分方程的可控性

Controllability of a second-order impulsive neutral differential equation via resolvent operator technique

本文利用预解算子技术研究Banach空间中二阶非自治中立型脉冲积微分方程的温和解存在性与近似可控性，并借助半群理论、图范数和Krasnoselskii不动点定理给出证明，最后以实例验证结论。

阶数为 $q\in ]1,2[$ 的非局部条件分数阶微分系统的近似能控性

Approximate Controllability of Fractional Differential Systems with Nonlocal Conditions of Order $q\in ]1,2[$

本文研究 Banach 空间中阶数为 $1<q<2$ 的非局部条件分数阶非线性微分方程的近似能控性，基于预解算子理论、分数阶微积分和 Krasnoselskii 不动点定理推导出充分条件，改进了现有成果并通过实例验证应用。

希尔伯特空间中二阶发展微分包含的近似能控性

Approximate controllability of second-order evolution differential inclusions in Hilbert spaces

本文考虑了希尔伯特空间中一类二阶发展微分包含.研究了一类二阶控制系统的近似能控性。首先，我们在希尔伯特空间中建立了一类二阶发展微分包含的近似能控性的充分条件。我们利用Bohnenblust-Karlin的不动点定理证明了我们的主要结果。进一步，我们将结果推广到研究了非局部条件下的近似能控性概念，并将结果推广到研究了非局部条件下脉冲控制系统的近似能控性。最后给出一个例子来说明我们的主要结果。

Linearized Controllability Analysis of Semilinear Partial Differential Equations

It is well-known that the controllability of finite-dimensional nonlinear systems can be established by showing the controllability of the linearized system. However, this classical result does not generalize to infinite-dimensional nonlinear systems. In this paper, we restrict ourselves to semilinear infinite-dimensional systems, and show that the exact controllability of the linearized system implies exact controllability of the nonlinear system. The restrictions concerning the nonlinear operator are similar to those that can be found in the literature about the linearized stability analysis…

On the stability and null-controllability of an infinite system of linear differential equations

In this work, the null controllability problem for a linear system in $\ell^2$ is considered, where the matrix of a linear operator describing the system is an infinite matrix with $λ\in \mathbb R$ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if $λ\le-1$, which shows the fine difference between the finite and the infinite-dimensional systems. When $λ\le-1$ we also show that the system is null controllable in large. We also show a dependence of the stability on the norm i.e. the same system considered in $\ell^\infty$ is not asymptotically s…

一类具有脉冲的1<α<2阶非局部分数阶中立型随机积分微分包含的研究

A study of nonlocal fractional neutral stochastic integrodifferential inclusions of order $1<α<2$ with impulses

本文研究Hilbert空间中一类带脉冲的非局部分数阶中立型随机积分微分包含，结合不动点定理、分数阶余弦族与半群理论证明了凸值和非凸值情形下温和解的存在性，并建立了系统近似可控性的充分条件。

分数阶微分方程混沌系统的同步与保密通信

Synchronization and secure communication using some chaotic systems of fractional differential equations

利用Caputo分数阶导数$α,$ $α\in (0,1),$，研究了一类分数阶微分方程的混沌系统.我们将证明，他们可以同步和反同步使用适当的非线性控制函数。分数阶微分方程的同步或反同步误差系统用于保密通信。

分数阶微分方程的线性化渐近稳定性

Linearized Asymptotic Stability for Fractional Differential Equations

证明了分数阶微分方程的线性化渐近稳定性定理。更精确地说，我们证明了非线性Caputo分数阶微分方程的平衡点是渐近稳定的，如果它在平衡点的线性化是渐近稳定的。因此，我们扩展李雅普诺夫第一方法的分数阶微分方程证明，如果频谱的线性化包含在部门$\{λ\in \C : |\arg λ| > \frac{απ}{2}\}$，其中$α> 0$表示阶的分数阶微分方程，然后平衡的非线性分数阶微分方程是渐近稳定的。

On the Nonlinear $Ψ$-Hilfer Hybrid Fractional Differential Equations

In this paper, we initially derive the equivalent fractional integral equation to $Ψ$-Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The primary objective of the paper is to obtain estimates on $Ψ$-Hilfer derivative and utilize it to derive the hybrid fractional differential inequalities involving $Ψ$-Hilfer derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions, comparison theorems and uniqueness of the solution.

Generation of nonlocal fractional dynamical systems by fractional differential equations

We show that any two trajectories of solutions of a one-dimensional fractional differential equation (FDE) either coincide or do not intersect each other. In contrary, in the higher dimensional case, two different trajectories can meet. Furthermore, one-dimensional FDEs and triangular systems of FDEs generate nonlocal fractional dynamical systems, whereas a higher dimensional FDE does, in general, not generate a nonlocal dynamical system.

高维空间中分数阶微分方程的稳定流形

On stable manifolds for fractional differential equations in high dimensional spaces

本文的目的是建立任意有限维空间中分数阶微分方程双曲平衡点附近的稳定流形。

Mittag-Leffler Dichotomy and Roughness of Conformable Fractional Differential Equations

The solutions of traditional fractional differential equations neither satisfy group property nor generate dynamical systems, so the study on hyperbolicity is in blank. Relying on the new proposed conformable fractional derivative, we investigate dichotomy of conformable fractional equations, including structure of solutions of linear systems, Mittag-Leffler dichotomy and stability, roughness and nonuniform dichotomy.

Port-Hamiltonian descriptor systems are relative generically controllable and stabilizable

The present work is a successor of [Ilchmann, Kirchhoff 2022] on generic controllability and of [Ilchmann, Kirchhoff 2023] on relative generic controllability of linear differential-algebraic equations. We extend the result from general, unstructured differential-algebraic equations to differential-algebraic equations of port-Hamiltonian type. We derive new results on relative genericity. These findings are the basis for characterizing relative generic controllability of port-Hamiltonian systems in terms of dimensions. A similar result is proved for relative generic stabilizability.

Stability analysis of multi-term fractional-differential equations with three fractional derivatives

Necessary and sufficient stability and instability conditions are obtained for multi-term homogeneous linear fractional differential equations with three Caputo derivatives and constant coefficients. In both cases, fractional-order-dependent as well as fractional-order-independent characterisations of stability and instability properties are obtained, in terms of the coefficients of the multi-term fractional differential equation. The theoretical results are exemplified for the particular cases of the Basset and Bagley-Torvik equations, as well as for a multi-term fractional differential equat…

On Controllability of Driftless Control System on Symmetric Spaces

Symmetric spaces arise in wide variety of problems in Mathematics and Physics. They are mostly studied in Representation theory, Harmonic analysis and Differential geometry. As many physical systems have symmetric spaces as their configuration spaces, the study of controllability on symmetric space is quite interesting. In this paper, a driftless control system of type \dot{x}= \sum_{i=1}^m u_if_i(x) is considered on a symmetric space. For this we have established global controllability condition which is illustrated by few examples of exponential submanifolds of SE(3) and random matrix ensemb…

On Nonlinear Hybrid Fractional Differential Equations with Atangana-Baleanu-Caputo Derivative

In this paper, we develop the theory of nonlinear hybrid fractional differential equations involving Atangana--Baleanu--Caputo (ABC) fractional derivative. We construct the equivalent fractional integral equation and establish the existence results through it. Further, we build up the theory of inequalities for ABC--hybrid fractional differential equations and use it to examine the uniqueness, existence of a maximal and minimal solution and the comparison results.

分数阶布朗运动驱动的随机微分方程的生存性

Viability for stochastic differential equations driven by fractional Brownian motion

本文利用路径方法证明了Hurst参数为1/2 < H < 1的分数阶布朗运动驱动的多维不定常随机微分方程解的一个生存性结果.该充分条件也是带状态约束的分数阶微分方程解的另一个整体存在性结果。

Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations

The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the…