Controllability Problem of Fractional Neutral Systems: A Survey

The following article presents recent results of controllability problem of dynamical systems in infinite-dimensional space. Generally speaking, we describe selected controllability problems of fractional order systems, including approximate controllability of fractional impulsive partial neutral integrodifferential inclusions with infinite delay in Hilbert spaces, controllability of nonlinear neutral fractional impulsive differential inclusions in Banach space, controllability for a class of fractional neutral integrodifferential equations with unbounded delay, controllability of neutral fractional functional equations with impulses and infinite delay, and controllability for a class of fractional order neutral evolution control systems

Filippov's theorem for impulsive differential inclusions with fractional order

In this paper, we present an impulsive version of Filippov's Theorem for fractional differential inclusions of the form: $$\begin{array}{rlll} D^{\alpha}_*y(t) &\in& F(t,y(t)), &\hbox{ a.e. } \, t\in J\backslash \{t_{1},\ldots,t_{m}\},\ \alpha\in(1,2],\\ y(t^+_{k})-y(t^-_k)&=&I_{k}(y(t_{k}^{-})), &k=1,\ldots,m,\\ y'(t^+_{k})-y'(t^-_k)&=&\overline{I}_{k}(y'(t_{k}^{-})), &k=1,\ldots,m,\\ y(0)&=&a,\ y'(0)=c,\ \end{array}$$ where $J=[0,b],$ $D^{\alpha}_*$ denotes the Caputo fractional derivative and $F$ is a set-valued map. The functions $I_k,\overline{I}_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m$)

Impulsive stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay

In this paper, we prove the existence of mild solutions for the following first-order impulsive semilinear stochastic functional differential inclusions driven by a fractional Brownian motion with infinite delay in the case where the right hand side is convex or nonconvex-valued. The results are obtained by using two fixed point theorems for multivalued mappings.Ministerio de Economía y Competitividad (España) MTM2011-22411Junta de Andalucía. Consejería de Innovación, Ciencia y Empresa 2010/FQM314Junta de Andalucía P12-FQM-149

Existence of Solutions for Fractional-Order Neutral Differential Inclusions with Impulsive and Nonlocal Conditions

This paper investigates the existence of solutions for fractional-order neutral impulsive differential inclusions with nonlocal conditions. Utilizing the fractional calculus and fixed point theorem for multivalued maps, new sufficient conditions are derived for ensuring the existence of solutions. The obtained results improve and generalize some existed results. Finally, an illustrative example is given to show the effectiveness of theoretical results