scholarly journals Solvability and Optimal Controls of Semilinear Riemann-Liouville Fractional Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
Xue Pan ◽  
Xiuwen Li ◽  
Jing Zhao
Keyword(s):  
Differential Equations ◽  
Control Systems ◽  
Fractional Derivatives ◽  
Broad Class ◽  
Mild Solutions ◽  
Optimal Controls ◽  
Infinite Dimensional ◽  
Mild Conditions ◽  
Fractional Differential ◽  
Fractional Control

We consider the control systems governed by semilinear differential equations with Riemann-Liouville fractional derivatives in Banach spaces. Firstly, by applying fixed point strategy, some suitable conditions are established to guarantee the existence and uniqueness of mild solutions for a broad class of fractional infinite dimensional control systems. Then, by using generally mild conditions of cost functional, we extend the existence result of optimal controls to the Riemann-Liouville fractional control systems. Finally, a concrete application is given to illustrate the effectiveness of our main results.

scholarly journals Analysis and Optimal Control of φ-Hilfer Fractional Semilinear Equations Involving Nonlocal Impulsive Conditions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2084
Author(s):  
Sarra Guechi ◽  
Rajesh Dhayal ◽  
Amar Debbouche ◽  
Muslim Malik
Keyword(s):  
Mild Solutions ◽  
Semigroup Theory ◽  
Nonlocal Conditions ◽  
Optimal Controls ◽  
Impulsive Conditions ◽  
Semilinear Equations ◽  
New Class ◽  
Fractional Differential ◽  
The Fixed Point Theorem ◽  
Fractional Control

The goal of this paper is to consider a new class of φ-Hilfer fractional differential equations with impulses and nonlocal conditions. By using fractional calculus, semigroup theory, and with the help of the fixed point theorem, the existence and uniqueness of mild solutions are obtained for the proposed fractional system. Symmetrically, we discuss the existence of optimal controls for the φ-Hilfer fractional control system. Our main results are well supported by an illustrative example.

On fractional differential equations with state-dependent delay via Kuratowski measure of noncompactness

Filomat ◽  
2017 ◽  
Vol 31 (2) ◽  
pp. 451-460 ◽  
Author(s):  
Mohammed Belmekki ◽  
Kheira Mekhalfi
Keyword(s):  
Differential Equations ◽  
Measure Of Noncompactness ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Kuratowski Measure Of Noncompactness ◽  
State Dependent ◽  
State Dependent Delay ◽  
Liouville Fractional Derivative ◽  
Fractional Differential ◽  
Functional Differential

This paper is devoted to study the existence of mild solutions for semilinear functional differential equations with state-dependent delay involving the Riemann-Liouville fractional derivative in a Banach space and resolvent operator. The arguments are based upon M?nch?s fixed point theoremand the technique of measure of noncompactness.

scholarly journals The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Xianzhen Zhang ◽  
Zuohua Liu ◽  
Hui Peng ◽  
Xianmin Zhang ◽  
Shiyong Yang
Keyword(s):  
Differential Equation ◽  
Integral Equation ◽  
General Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Order System ◽  
Impulsive Systems ◽  
Fractional Differential ◽  
Noninstantaneous Impulses

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

scholarly journals Positive solutions for a system of fractional differential equations with p-Laplacian operator and multi-point boundary conditions

2018 ◽  
Vol 23 (5) ◽  
pp. 771-801 ◽  
Author(s):  
Rodica Luca
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Positive Solutions ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Existence Results ◽  
Laplacian Operator ◽  
Point Boundary ◽  
Existence And Nonexistence ◽  
Fractional Differential

>We investigate the existence and nonexistence of positive solutions for a system of nonlinear Riemann–Liouville fractional differential equations with parameters and p-Laplacian operator subject to multi-point boundary conditions, which contain fractional derivatives. The proof of our main existence results is based on the Guo–Krasnosel'skii fixed-point theorem.

scholarly journals EXISTENCE AND STABILITY RESULTS FOR FRACTIONAL DIFFERENTIAL EQUATIONS WITH TWO CAPUTO FRACTIONAL DERIVATIVES

2019 ◽  
pp. 341
Author(s):  
Mohamed Houas ◽  
Mohamed Bezziou
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Contraction Mapping ◽  
Nonlocal Boundary ◽  
Stability Of Solutions ◽  
Caputo Fractional Derivatives ◽  
Existence And Stability ◽  
Fractional Differential ◽  
Stability Results

In this paper, we discuss the existence, uniqueness and stability of solutions for a nonlocal boundary value problem of nonlinear fractional differential equations with two Caputo fractional derivatives. By applying the contraction mapping and O’Regan fixed point theorem, the existence results are obtained. We also derive the Ulam-Hyers stability of solutions. Finally, some examples are given to illustrate our results.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

On the Application of Fractional Derivatives to the Study of Memristor Dynamics

2020 ◽  
pp. 342-363
Author(s):  
Rawid Banchuin
Keyword(s):  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Phase Portraits ◽  
Fractional Differential

In this chapter, the authors report their work on the application of fractional derivative to the study of the memristor dynamic where the effects of the parasitic fractional elements of the memristor have been studied. The fractional differential equations of the memristor and the memristor-based circuits under the effects of the parasitic fractional elements have been formulated and solved both analytically and numerically. Such effects of the parasitic fractional elements have been studied via the simulations based on the obtained solutions where many interesting results have been proposed in the work. For example, it has been found that the parasitic fractional elements cause both charge and flux decay of the memristor and the impasse point breaking of the phase portraits between flux and charge of the memristor-based circuits similarly to the conventional parasitic elements. The effects of the order and the nonlinearity of the parasitic fractional elements have also been reported.

Sign in / Sign up

Export Citation Format

Share Document