Fractional Differential Inequality Research Articles

Synchronization of fractional-order quaternion-valued neural networks with image encryption via event-triggered impulsive control

The finite-time synchronization and impulsive synchronization of fractional-order quaternion-valued neural networks (FOQVNNs) with delay are investigated in this research. To begin, instead of employing a decomposition method, a Lyapunov direct approach is proposed to achieve the finite-time synchronization of FOQVNNs using event-triggered control based on state errors. The settling time is then estimated using a unique fractional differential inequality. The event-triggered impulsive controller, which updates only during impulsive instants, is proposed to be more inexpensive and efficient. Then, utilizing fractional Lyapunov theory and the comparison principle, several novel quaternion-valued linear matrix inequalities criteria are defined to assure the impulsive synchronization of the delayed FOQVNNs. Furthermore, the positive lower bound of the inter-event time is obtained to exclude the Zeno behavior. As proof of the feasibility and validity of these theoretical findings obtained, numerical examples are provided. At last, the use of the event-triggered control technique for FOQVNNs in image encryption and decryption is demonstrated and validated.

New existence and stability results of mild solutions for fuzzy fractional differential variational inequalities

The aim of this work is to study the stability for a class of fuzzy fractional differential variational inequalities modeled by variational inequalities coupled with fuzzy fractional differential inclusions. By introducing the parametric problem with the help of two parameters in the involved mapping and constraint set, we first establish an existence result for the parametric fuzzy fractional differential variational inequality (PFFDVI), and we also establish the compactness and continuous dependence on the initial value for the PFFDVI. Furthermore, when the parameters are perturbed, we obtain the closedness of solution mapping for the PFFDVI. Then we further investigate the upper semicontinuity of solution mapping for the PFFDVI under some mild conditions. Finally, in order to illustrate the obtained results, we give a numerical example at the end of this paper.

Fuzzy fractional delay differential inclusions driven by hemivariational inequalities in Banach spaces

This paper investigates a novel class of nonlinear dynamical fuzzy systems referred to as fuzzy fractional delay differential hemivariational inequalities (FFDDHVIs) in Banach spaces. These systems integrate fractional fuzzy differential inclusions with delay and hemivariational inequalities. The existence theorem for the HVIs is established based on the KKM theorem. Moreover, specific propositions are proved, covering the superpositional measurability and the upper semicontinuity for the HVIs. Next, by using the fixed points theorem, we establish the existence and compactness of mild solution sets for the FFDDHVIs under certain mild conditions. Finally, As an illustrative application, we investigate a frictional quasistatic contact problem for viscoelastic materials, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals.

On the critical exponents for a fractional diffusion-wave equation with a nonlinear memory term in a bounded domain

In this paper, we prove sharp blow-up and global existence results for a time fractional diffusion-wave equation with a nonlinear memory term in a bounded domain, where the fractional derivative in time is taken in the sense of the Caputo type. Moreover, we also give a result for nonexistence of global solutions to a wave equation with a nonlinear memory term in a bounded domain. The proof of blow-up results is based on the eigenfunction method and the asymptotic properties of solutions for an ordinary fractional differential inequality.

Quasi-Synchronization of Fractional Multiweighted Coupled Neural Networks via Aperiodic Intermittent Control.

This article investigates the quasi-synchronization for fractional multiweighted coupled neural networks (FMCNNs) with discontinuous activation functions and mismatched parameters. First, under the generalized Caputo fractional-order derivative operator, a novel piecewise fractional differential inequality is established to study the convergence of fractional systems, which significantly extends some related published results. Subsequently, by exploiting the new inequality and Lyapunov stability theory, some sufficient quasi-synchronization conditions of FMCNNs are presented by aperiodic intermittent control. Meanwhile, the exponential convergence rate and synchronization error's bound are given explicitly. Finally, the validity of theoretical analysis is confirmed by numerical examples and simulations.

Optimal control results for fractional differential hemivariational inequalities of order r (1, 2)

ABSTRACT In this paper, we investigate the optimal control results for fractional differential hemivariational inequalities of order r ∈ ( 1 , 2 ) . Multivalued maps, fractional calculus, and the fixed point theorem are being applied to evaluate the main findings of this study. To begin, a well-known fixed point theorems of multivalued maps and the properties of generalized Clarke subdifferential are used to establish and prove the existence of mild solutions. Furthermore, we obtained the conclusion that the presented control system has optimal control under some mild conditions. Finally, an example is given to illustrate the theory.

On Cauchy-type problems with weighted R-L fractional derivatives of a function with respect to another function and comparison theorems

<abstract><p>The main aim of this paper is to study the Cauchy problem for nonlinear differential equations of fractional order containing the weighted Riemann-Liouville fractional derivative of a function with respect to another function. The equivalence of this problem and a nonlinear Volterra-type integral equation of the second kind have been presented. In addition, the existence and uniqueness of the solution to the considered Cauchy problem are proved using Banach's fixed point theorem and the method of successive approximations. Finally, we obtain a new estimate of the weighted Riemann-Liouville fractional derivative of a function with respect to functions at their extreme points. With the assistance of the estimate obtained, we develop the comparison theorems of fractional differential inequalities, strict as well as nonstrict, involving weighted Riemann-Liouville differential operators of a function with respect to functions of order $ \delta $, $ 0 < \delta < 1 $.</p></abstract>

A stochastic fractional differential variational inequality with Lévy jump and its application

This paper investigates a stochastic fractional differential variational inequality with Lévy jump (SFDVI with Lévy jump), which comprises a stochastic fractional differential equation with Lévy jump and a stochastic variational inequality. By employing the successive approximation method as well as the projection technique, we establish unique existence of the solution for the SFDVI with Lévy jump under some mild conditions. Moreover, the main results are applied to obtain the unique existence of the solution for the spatial price equilibrium problem in stochastic environments.

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

A class of Hilfer fractional differential evolution hemivariational inequalities with history-dependent operators

Optimal control problem of fractional evolution inclusions with Clarke subdifferential driven by quasi-hemivariational inequalities

In this paper, we introduce and study an optimal control problem for a new class of fractional differential quasi-hemivariational inequalities with nonlocal initial conditions and Clarke subdifferentials. The main tools in our study are the theory of fractional calculus, the fixed point argument for condensing multimaps and Balder’s theorem. We prove the solvability of abstract nonlocal fractional differential quasi-hemivariational inequalities and then show the existence of solutions to the associated optimal control problems.

Fractional partial differential variational inequality

In this present paper, we introduce and study a dynamical systems involving fractional derivative operator and nonlocal condition, which is constituted of a fractional evolution equation and a time-dependent variational inequality, and is named as fractional partial differential variational inequality (FPDVI, for short). By employing the estimates involving the one-and two-parameter Mittag-Leffler functions, fixed-point theory for set-value mappings, and non-compactness measure theory, we develop a general framework to establish the existence of smooth solutions to (FPDVI).

Adaptive Control Design and Fractional Differential Inequality Approach to Complete Synchronization Analysis on Caputo Delayed Quaternion-valued Network Systems

Adaptive Control Design and Fractional Differential Inequality Approach to Complete Synchronization Analysis on Caputo Delayed Quaternion-valued Network Systems

Quasi-synchronization of fractional-order complex-value neural networks with discontinuous activations

In this paper, under the framework of Filippov solutions, quasi-synchronization issue of fractional-order complex-valued neural networks (FCNNs) with discontinuous activations and uncertainties is investigated. Firstly, using Laplace transform and the property of Mittag-Leffler function, a novel fractional differential inequality is derived. Then combining the newly constructed inequality with delay feedback control scheme, several quasi-synchronization conditions are obtained for the considered FCNNs by means of non-decomposable method. Eventually, a numerical example is provided to substantiate validation of the proposed results.

Finite-time adaptive synchronization of fractional-order delayed quaternion-valued fuzzy neural networks

Based on direct quaternion method, this paper explores the finite-time adaptive synchronization (FAS) of fractional-order delayed quaternion-valued fuzzy neural networks (FODQVFNNs). Firstly, a useful fractional differential inequality is created, which offers an effective way to investigate FAS. Then two novel quaternion-valued adaptive control strategies are designed. By means of our newly proposed inequality, the basic knowledge about fractional calculus, reduction to absurdity as well as several inequality techniques of quaternion and fuzzy logic, several sufficient FAS criteria are derived for FODQVFNNs. Moreover, the settling time of FAS is estimated, which is in connection with the order and initial values of considered systems as well as the controller parameters. Ultimately, the validity of obtained FAS criteria is corroborated by numerical simulations.

Asymptotically periodic solutions for fractional differential variational inequalities

Asymptotically periodic solutions for fractional differential variational inequalities

Laplace transform and nonlinear control design for quasi‐projective synchronization for Caputo inertial delayed neural networks

The quasi‐projective synchronization problems for Caputo inertial neural networks (INNs) with delays are analyzed in this article. The considered model is transformed into two subsystems by selecting a suitable variable substitution. To realize synchronization, a novel fractional differential inequality is established, and the new lemma is derived via Laplace transform. Applying the given lemmas and inequality techniques, the synchronization conditions are inferred through different nonlinear feedback controllers. From the derivation of the theorems, it can be seen that the designed controllers are more extensive and efficacious. Furthermore, the error bound is validly estimated, and the validity of obtained theorems is illustrated by the numerical simulations.

Nonexistence results for a sequential fractional differential problem

In this paper, we study a class of sequential fractional differential inequalities involving Caputo fractional derivatives with different orders. The nonexistence of nontrivial global solutions is investigated in a suitable space via the test function technique and some properties of fractional integrals. Our results are supported by numerical examples.

Synchronization of fractional-order delayed neural networks with reaction–diffusion terms: Distributed delayed impulsive control

In this study, a new fractional Halanay-like impulsive differential inequality, which considers the system delay and impulse delay, is proposed for the first time. And accordingly, the quasi-synchronization problem of delayed fractional-order coupled reaction–diffusion neural networks (CRDNNs) is investigated. It is important to know that there are still relatively few studies on the impulsive synchronization control of fractional-order systems with time delay, and this study is an effective complement to this relevant field. To achieve quasi-synchronization, the distributed delayed impulsive control strategy is applied, which utilizes the local state information and the weighted average delayed information of other available neurons. Then, based on Wirtinger inequality, Lyapunov stability theory, and the properties of Mittag-Leffler function, some quasi-synchronization criteria are derived by solving some linear matrix inequalities (LMIs). Finally, a numerical simulation is provided to demonstrate the effectiveness of theoretical results, and reveal the effects of impulse delay and fractional order on the system synchronization rate.

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

Adaptive quasi-synchronization analysis for Caputo delayed Cohen–Grossberg neural networks

The quasi-synchronization (QS) issues for Caputo delayed Cohen–Grossberg neural networks (CGNNs) are discussed in this article. To begin with, a novel lemma is established by constructing suitable fractional differential inequality. Due to the advantages of adaptive control schemes with reducing control cost and having high tracking accuracy, two different adaptive controllers are designed, respectively. Applying the proposed lemma, inequality techniques and Lagrange’s mean value theorem, the conditions of QS are obtained by selecting appropriate Lyapunov functions. Finally, two numerical examples in different dimensions are shown to test the correctness of the gained theorems.