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

On the approximate controllability of some semilinear partial functional integrodifferential equations with unbonded delay

This work concerns the study of the approximate controllability for some nonlinear partial functional integrodifferential equation with infinite delay arising in the modelling of materials with memory, in the framework of Hilbert spaces. We give sufficient conditions that ensure the approximate controllability of the system by supposing that its linear undelayed is part approximately controllable, admits a resolvent operator in the sense of Grimmer, and by making use of the measure of noncompactness and the Mönch fixed-point Theorem. As a result, we obtain a generalization of several important results in the literature, without assuming the compactness of the resolvent operator. An example of applications is given for illustration

Global existence and stability for second order functional evolution equations with infinite delay

In this article, the authors give sufficient conditions for existence and attractivity of mild solutions for second order semi-linear functional evolution equation in Banach spaces using Schauder's fixed point theorem. An example is provided to illustrate the result

Impulsive differential inclusions involving evolution operators in separable Banach spaces

We present some results on the existence of mild solutions and study the topological structures of the sets of solutions for the following first-order impulsive semilinear differential inclusions with initial and boundary conditions: where J=R+, 0 = t 0 < t 1 < … < t m <…, m∈N, lim k→∞ t k = ∞, A(t) is the infinitesimal generator of a family of evolution operators U(t, s) in a separable Banach space E and F is a set-valued mapping. The functions I k characterize the jumps of solutions at the impulse points t k , k = 1, ….The mapping L: PC b →E is a bounded linear operator. We also investigate the compactness of the set of solutions, some regularity properties of the operator solutions, and the absolute retract.Наведено деякi результати про iснування м’яких розв’язкiв та вивчено топологiчну будову множин розв’язкiв для наступних iмпульсних напiвлiнiйних диференцiальних включень першого порядку з початковими та граничними умовами: де J=IR+,0=t0<t1<...<tm<...;(m∈N),limk→∞tk=∞,A(t) — iнфiнiтезимальний генератор сiм’ї операторiв еволюцiї U(t,s) на сепарабельному банаховому просторi E та F — багатозначне вiдображення. Функцiї Ik характеризують стрибки розв’язкiв в точках iмпульсної дiї tk,k=1,... . Вiдображення L:PCb→E є обмеженим лiнiйним оператором. Також дослiджено компактнiсть множини розв’язкiв, деякi властивостi регулярностi операторних розв’язкiв та абсолютну ретрактнiсть

Approximate controllability of impulsive integrodifferential equations with state-dependent delay

This paper considers the approximate controllability of mild solutions for impulsive semilinear integrodifferential equations with statedependent delay in Hilbert spaces. We obtain our significant findings using Grimmer’s resolvent operator theory and Schauder’s fixed point theorem. We give an example at the end to ensure the compatibility of the results

Some Comments on: Existence of Solutions of Abstract Nonlinear Second-Order Neutral Functional Integrodifferential Equations

We establish the existence of mild solutions for a class of abstract second-order partial neutral functional integro-differential equations with infinite delay in a Banach space using the theory of cosine families of bounded linear operators and Schaefer\u27s fixed-point theorem