Infinite Delay Research Articles

Stochastic fractional diffusion equations containing finite and infinite delays with multiplicative noise

In this work, we investigate stochastic fractional diffusion equations with Caputo–Fabrizio fractional derivatives and multiplicative noise, involving finite and infinite delays. Initially, the existence and uniqueness of mild solution in the spaces C p ( [ − a , b ] ; L q ( Ω , H ˙ r ) ) ) and C δ ( ( − ∞ , b ] ; L q ( Ω , H ˙ r ) ) ) are established. Next, besides investigating the regularity properties, we show the continuity of mild solutions with respect to the initial functions and the order of the fractional derivative for both cases of delay separately.

On the qualitative analyses solutions of new mathematical models of integro‐differential equations with infinite delay

This paper deals with the uniformly stability (US), globally uniformly asymptotically stability (GUAS) and instability of zero solution as well as integrability and boundedness of nonzero solutions of certain nonlinear integro‐differential equations (IDEs). We prove five new theorems, which include sufficient conditions related to these fundamental qualitative properties of solutions to the IDEs considered. The main tools used in the proof are two new and suitable Lyapunov–Krasovskiiˇ functionals (LKFs). In particular cases, two numerical examples are given and solved via the fourth order Runge–Kutta method (RKM) in MATLAB to illustrate the theoretical results of this paper. Compared with the existing results on the fundamental qualitative behaviors of scalar IDEs, our results are new, original, more effective and convenient for tests and applications.

Lyapunov Functionals in Integral Equations

Lyapunov functions/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it is therefore further explored in this paper. In this manuscript, we utilize Lyapunov functionals combined with Laplace transform to qualitatively analyze the solutions of the integral equation In addition, we extend our method to nonlinear integral equations, integral equations with infinite delay, and integral equations with several kernels. We mention that Laplace transform has been used to solve integral equations of convolution types but has never been applied directly to integral equations that are not of the convolution type. In addition, our method allows us to find the upper estimates, and our necessary conditions are easy to verify.

An Analysis on the Optimal Control for Fractional Stochastic Delay Integrodifferential Systems of Order 1 < γ < 2

The purpose of this paper is to investigate the optimal control for fractional stochastic integrodifferential systems of order 1 < γ < 2. To ensure the existence and uniqueness of mild solutions, we first gather a novel list of requirements. Further, the existence of optimal control for the stated issue is given by applying Balder’s theorem. Additionally, we extend our existence outcomes with infinite delay. The outcomes are obtained via fractional calculus, Hölder’s inequality, the cosine family, stochastic analysis techniques, and the fixed point approach. The theory is shown by an illustration, as well.

Existence Results for Functional Perturbed Differential Equations of Fractional Order with State-Dependent Delay in Banach Spaces

In this paper, we provide sufficient conditions for the existence of solutions of initial value problem, for perturbed partial functional hyperbolic differential equations of fractional order involving Caputo fractional derivative with state-dependent delay by reducing the research to the~search of the existence and the uniqueness of fixed points of appropriate operators. Our main result for this problem is based on a nonlinear alternative fixed point theorem for the~sum of a completely continuous operator and a contraction one in Banach spaces due to Burton and Kirk and a fractional version of Gronwall's inequality. We should observe the structure of the space and the properties of the~operators to obtain existence results. To our knowledge, there are very few papers devoted to fractional differential equations with finite and/or infinite constant delay on bounded domains. Many other questions and issues can be investigated regarding the existence in the space of weighted continuous functions, the uniqueness, the structure of the~solutions set and also whether or not the condition satisfied by the operators are optimal. This paper can be considered as a contribution in this setting case. Examples are given to illustrate this work.

Ulam–Hyers stability for an impulsive Caputo–Hadamard fractional neutral stochastic differential equations with infinite delay

This paper addresses existence and Ulam–Hyers stability (UHS) problems for an impulsive Caputo–Hadamard fractional neutral functional stochastic differential equation with infinite delay (FNFSDEwID). We first prove the existence and uniqueness of the solution using Banach fixed point theorem and standard stochastic analysis techniques. We then tackle the UHS under a Lipschitz condition on a bounded and closed interval. We end up with an illustrative example that corroborates our theoretical findings.

Existence and controllability results for stochastic impulsive integro-differential equations with infinite delay

Existence and controllability results for stochastic impulsive integro-differential equations with infinite delay

Periodic solutions of second‐order degenerate differential equations with infinite delay in Banach spaces

AbstractWe consider the well‐posedness of the second‐order degenerate differential equations with infinite delay on [0, 2π] in Lebesgue–Bochner spaces and periodic Besov spaces , where , and M are closed linear operators in a Banach space X satisfying and the kernels . Using known operator‐valued Fourier multiplier theorems, we are able to give necessary and sufficient conditions for the well‐posedness of (P) in and . These results are applied to examine some concrete examples.

Controllability results to non-instantaneous impulsive with infinite delay for generalized fractional differential equations

This paper discusses controllability results for active types with infinite-time delay of non-instantaneous impulsive fractional differential equations. The model is constructed based on the generalized Caputo (Caputo-Katugampola) fractional derivative and the control function with non-local Katugampola fractional integral as a boundary condition. Our principal results are established by giving some sufficient hypotheses, utilizing well-known fractional calculus truths and using Krasnoselskii’s fixed point theorem. The infinite time delay has been treated with the abstract phase space techniques and fulfilling the ensuing axioms due to Hale and Kato. It turns out that under some sufficient conditions, the problem has at least one controllable solution. An implementation of our theoretical results is demonstrated by a numerical example.

Stabilization of linear systems cascaded with a chain of infinite-delayed integrators

This paper considers stabilization of linear systems cascaded with a chain of infinite-delayed integrators. Distributed infinite input delay is also taken into consideration. The systems under consideration can be seen as a special class of linear strict-feedback systems with infinite delays. A novel predictor feedback controller is proposed via an n-step backstepping procedure. The Lyapunov theorem developed for infinite-delayed systems in our previous work is applied. It is shown that the proposed controller can globally stabilize the concerned systems. To our best knowledge, it is a first work on control of systems with simultaneous distributed infinite state and input delays. A simulation example is provided to illustrate the effectiveness of our controller.

A note on approximate controllability of second-order impulsive stochastic Volterra-Fredholm integrodifferential system with infinite delay

This manuscript examines the approximate controllability of a second-order stochastic Volterra-Fredholm integrodifferential system including delay and impulses. Primarily, by utilizing stochastic theory, the cosine family of operators, and the fixed point approach, we verify the existence of mild solutions for the given system. In particular, we establish a new set of sufficient requirements for the approximate controllability of the system. On the condition that the associated linear system is approximately controllable, the outcome is derived. In addition, we extend our system with nonlocal conditions. To demonstrate the theory of the primary outcomes, an example is shown.

Optimal control results for impulsive fractional delay integrodifferential equations of order 1 < r < 2 via sectorial operator

This research investigates the existence of nonlocal impulsive fractional integrodifferential equations of order 1 < r < 2 with infinite delay. To begin with, we discuss the existence of a mild solution for the fractional derivatives by using the sectorial operators, the nonlinear alternative of the Leray–Schauder fixed point theorem, mixed Volterra–Fredholm integrodifferential types, and impulsive systems. Furthermore, we develop the optimal control results for the given system. The application of our findings is demonstrated with the help of an example.

Controllability for Fractional Evolution Equations with Infinite Time-Delay and Non-Local Conditions in Compact and Noncompact Cases

The goal of this dissertation is to explore a system of fractional evolution equations with infinitesimal generator operators and an infinite time delay with non-local conditions. It turns out that there are two ways to regulate the solution. To demonstrate the presence of the controllability of mild solutions, it is usual practice to apply Krasnoselskii’s theorem in the compactness case and the Sadvskii and Kuratowski measure of noncompactness. A fractional Caputo approach of order between 1 and 2 was used to construct our model. The families of linear operators cosine and sine, which are strongly continuous and uniformly bounded, are used to achieve the mild solution. To make our results seem to be applicable, a numerical example is provided.

Stability of neutral Volterra integro-differential equations with infinite delay

Stability of neutral Volterra integro-differential equations with infinite delay

A note concerning to approximate controllability of Atangana–Baleanu fractional neutral stochastic integro‐differential system with infinite delay

This article is primarily targeting the approximate controllability outcomes for the Atangana–Baleanu fractional neutral stochastic integro‐differential equation with infinite delay. First, by using stochastic analysis, fractional calculus, and Krasnoselskii fixed point techniques, we demonstrate the existence of mild solutions for the fractional stochastic evolution equations. Then we present a sufficient condition that ensures the approximate controllability of the stochastic evolution equations. Our findings are then applied to nonlocal conditions. At last, an example is provided to define our primary results.

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

Stabilisation with general decay rate by delay feedback control for nonlinear neutral stochastic functional differential equations with infinite delay

A dual-model semi-supervised self-organizing fuzzy inference system for data stream classification

Semi-supervised learning from data streams is widely considered as a highly challenging task to be further researched. In this paper, a novel dual-model self-organizing fuzzy inference system composed of two recently introduced evolving fuzzy systems (EFSs) is proposed for semi-supervised learning from data streams in infinite delay environments. After being primed with a small amount of labelled data during the warm-up period, the proposed model is able to continuously self-learn and self-expand its knowledge base from unlabelled data on a chunk-by-chunk basis with minimal human expert involvement. Thanks to its dual-model structure, the proposed model combines the merits of the two EFS models such that it can continuously identify new prototypes from new pseudo-labelled data to self-improve its knowledge base whilst keeping the impact of pseudo-labelled errors on its decision-making minimized. Numerical examples based on various benchmark problems demonstrate the efficacy of the proposed method, showing its strong potential in real-world applications by offering higher classification accuracy over the state-of-the-art competitors whilst retaining high computational efficiency.

Existence results for abstract functional differential equations with infinite state-dependent delay and applications

Existence results for abstract functional differential equations with infinite state-dependent delay and applications

Output feedback stabilization of linear time‐varying systems with infinite distributed input delays

AbstractThis article investigates the output feedback stabilization problem of linear time‐varying (LTV) systems with infinite distributed input delays. A novel observer is designed to estimate the system states and then a low gain output feedback controller is developed based on the estimated states. It is shown that the resulting closed‐loop control system is globally exponentially stable under some mild assumptions. Moreover, it should be pointed out that to the best of our knowledge, the output feedback stabilization problem of LTV systems with infinite distributed input delays has not yet been studied in open literature. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed controllers.

Approximate Controllability of Delayed Fractional Stochastic Differential Systems with Mixed Noise and Impulsive Effects

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^∈(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.