Fractional Delay Systems Research Articles

Finite‐time stability and numerical approximations of fractional neutral delay systems involving proportional Caputo derivative

This paper deals with a class of fractional neutral delay systems involving proportional Caputo derivative. Maintaining the finite‐time stability of fractional‐order systems is a major challenge, as their capacity to mimic complex dynamics draws more attention to them. Therefore, the paper presents a novel finite‐time stability criterion based on the Banach fixed‐point theorem. A decomposition formula for proportional Caputo derivative is provided. This formula allows us to derive a new numerical technique for efficiently solving the proposed problem. Finally, extensive numerical results are performed to illustrate and validate the proposed theoretical results.

Relative controllability for [formula omitted]Caputo fractional delay control system

This paper resolves around relative controllability of ψ−fractional delayed differential equations in finite dimensional space. The Mittag Leffler type of ψ−delayed perturbation matrix function with two parameters exhibits the Grammian matrix of fractional delay system. Based on this Grammian matrix we derived the necessary and sufficient conditions for the linear system which is relatively controllable. The fixed point approach is applied for obtaining a controllability result for semilinear system. This concept can be applied to some examples in order to illustrate the efficacy of our results.

Controllability of switched integro‐neutral fractional delay systems over arbitrary time domain

In this paper, we examine various qualitative properties for some class of fractional impulsive switched integro‐neutral dynamic systems over arbitrary time domain. We establish the existence, stability, and controllability of the scrutinized models. This article is divided into three parts: The first part is concerned with the existence and uniqueness, the second part is dedicated to the Hyers–Ulam stability, and in the third part, we examine the controllability analysis. We establish our results via the Banach and Leray–Schauder fixed‐point approaches along with the Arzela–Ascoli theorem. To show the implications and effectiveness of established results, we present a simulated numerical example at the end.

T-controllability of higher-order fractional stochastic delay system via integral contractor

The existence, uniqueness and trajectory (T)-controllability results for mild solutions to the fractional neutral stochastic integro-delay differential system are presented in this article. To demonstrate the results, the concept of bounded integral contractors is combined with the stochastic result and sequencing technique. In contrast to previous publications, we do not need to specify the induced inverse of the controllability operator to prove the controllability results, and the relevant non-linear function does not have to meet the Lipschitz condition. Furthermore, T-controllability results for neutral integro-delay differential systems with non-local conditions have been established. Finally, an application to demonstrate the acquired results is discussed.

An analytical criterion for stability testing of delayed Bode’s transfer function with unity feedback

The unity feedback where delayed Bode’s transfer function is placed in forward path is used as reference transfer function in controller design for fractional time delay systems. In this study, bounded input-bounded output stability of this reference transfer function is investigated. This investigation is based on a theorem for fractional order time delay systems using Hassard’s techniques. Sufficient and necessary conditions are obtained for testing the stability of this reference transfer function. Moreover, an explicit formula is given for calculating the number of unstable roots in case of instability. The illustrative examples show that this stability criterion works efficiently.

Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

Well-Posedness and Hyers–Ulam Stability of Fractional Stochastic Delay Systems Governed by the Rosenblatt Process

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays ( since we are giving more degrees of freedom via the fractional delays ) and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform ( TFT ) of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm ( CSA ) are observed to be superior when compared with other techniques such as real-coded genetic algorithm ( RGA ), particle swarm optimization ( PSO ) and gradient-search ( GS ) methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Optimal nonlinear system identification using fractional delay second-order volterra system

The aim of this work is to design a fractional delay second order Volterra filter that takes a discrete time sequence as input and its output is as close as possible to the output of a given nonlinear unknown system which may have higher degree nonlinearities in the least square sense. The basic reason for such a design is that rather than including higher than second degree nonlinearities in the designed system, we use the fractional delay degrees of freedom to approximate the given system. The advantage is in terms of obtaining a better approximation of the given nonlinear system than is possible by using only integer delays U+0028 since we are giving more degrees of freedom via the fractional delays U+0029 and simultaneously it does not require to incorporate higher degree nonlinearities than two. This work hinges around the fact that if the input signal is a decimated version of another signal by a factor of M, then fractional delays can be regarded as delays by integers less than M. Using the well known formula for calculating the discrete time Fourier transform U+0028 DTFT U+0029 of a decimated signal, we then arrive at an expression for the DTFT of the output of a fractional delay system in terms of the unknown first and second order Volterra system coefficients and the fractional delays. The final energy function to be minimized is the norm square of the difference between the DTFT of the given output and the DTFT of the output of the fractional delay system. Minimization over the filter coefficients is a linear problem and thus the final problem is to minimize a highly nonlinear function of the fractional delays which is accomplished using search techniques like the gradient-search and nature inspired optimization algorithms. The effectiveness of the proposed method is demonstrated using two nonlinear benchmark systems tested with five different input signals. The accuracy of the stated models using the globally convergent metaheuristic, cuckoo-search algorithm U+0028 CSA U+0029 are observed to be superior when compared with other techniques such as real-coded genetic algorithm U+0028 RGA U+0029, particle swarm optimization U+0028 PSO U+0029 and gradient-search U+0028 GS U+0029 methods. Finally, statistical analysis affirms the potential of the proposed designs for its successful implementation.

Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1<α<2 with delay in Banach spaces. Firstly, we study the existence of the PC-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results.

Fractional Vieta-Fibonacci wavelets: application for systems of fractionaldelay differential equations

In this study, the fractional Vieta-Fibonacci wavelets are introduced. These wavelets are applied to construct a numerical method to solve a class of fractional delay systems of differential equations. To this end, some relationships regarding fractional integration and derivative of these wavelets are extracted at the first. Then, all of the unknown functions in the system under consideration are approximated by these wavelets. Next, by substituting these approximations into the system and applying the collocation method, a system of algebraic equations is obtained. Finally, by solving the extracted system and can determine the unknown coefficients, a solution is obtained for the main system. Moreover, the upper bound of error for approximation with the fractional Vieta-Fibonacci wavelets and the convergence analysis of presented approach is derived. The proposed method is evaluated with solving several examples.

Exponential Stability of Fractional Large-Scale Neutral Stochastic Delay Systems with Fractional Brownian Motion

Mathematics plays an important role in many fields of finance. In particular, it presents theories and tools widely used in all areas of finance. Moreover, fractional Brownian motion (fBm) and related stochastic systems have been used to model stock prices and other phenomena in finance due to the long memory property of such systems. This manuscript provides the exponential stability of fractional-order Large-Scale neutral stochastic delay systems with fBm. Based on fractional calculus (FC), Rn stochastic space and Banach fixed point theory, sufficiently useful conditions are derived for the existence of solution and exponential stability results. In this study, we tackle the nonlinear terms of the considered systems by applying local assumptions. Finally, to verify the theoretical results, a numerical simulation is provided.

Relatively exact controllability of fractional stochastic delay system driven by Lévy noise

In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. First, we derive the solution of linear FSDS via delayed matrix functions of Mittag–Leffler (M‐L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen's inequality, Hölder's inequality, and Itô's isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle. Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through examples.

Global fractional Halanay inequalities approach to finite-time stability of nonlinear fractional order delay systems

In this paper, we investigate the robust finite time stability of fractional order systems with time varying delay and nonlinear perturbation. We improve the sufficient condition for general fractional delay systems by utilizing the special structure of a singular Gronwall inequality. For stable fractional delay systems, our approach bases on a global Halanay type inequality in differential and integral forms. A sharper delay dependent sufficient condition for robust finite time stability of such systems is formulated in terms of the Mittag-Leffler functions and the delayed size. The connection between the new sufficient condition and the previous results is compared and discussed thoroughly.

Optimal Control of Fractional Integro-differential Evolution Equation with Infinite Delay of order (1; 2)

In this paper, we will examine the facts in the image of infinite time delay on the mild solution’s solvability and fractional integro-differential Evolution Equation of Banach space’s optimal control choosing order ? ? (1,2). For investigation of mild solution of the system, continuation, uniqueness and the existence of the solution with the help of grown-wall inequality will be discussed shortly. After constructing the lagrange problem, the demonstration of actuality of optimal control of fractional integro delay system is being proceed. And at the end we will discuss the regarding result by an adequate example.

Design of Optimal Sliding Mode Control based on Linear Matrix Inequality for Fractional Time-Varying Delay Systems

Design of Optimal Sliding Mode Control based on Linear Matrix Inequality for Fractional Time-Varying Delay Systems

Controllability of Fractional Stochastic Delay Systems Driven by the Rosenblatt Process

In this work, we consider linear and nonlinear fractional stochastic delay systems driven by the Rosenblatt process. With the aid of the delayed Mittag-Leffler matrix functions and the representation of solutions of these systems, we derive the controllability results as an application. By introducing a fractional delayed Gramian matrix, we provide sufficient and necessary criteria for the controllability of linear fractional stochastic delay systems. Furthermore, by employing Krasnoselskii’s fixed point theorem, we establish sufficient conditions for the controllability of nonlinear fractional stochastic delay systems. Finally, an example is given to illustrate the main results.

Controllability and Hyers–Ulam Stability of Fractional Systems with Pure Delay

Linear and nonlinear fractional-delay systems are studied. As an application, we derive the controllability and Hyers–Ulam stability results using the representation of solutions of these systems with the help of their delayed Mittag–Leffler matrix functions. We provide some sufficient and necessary conditions for the controllability of linear fractional-delay systems by introducing a fractional delay Gramian matrix. Furthermore, we establish some sufficient conditions of controllability and Hyers–Ulam stability of nonlinear fractional-delay systems by applying Krasnoselskii’s fixed-point theorem. Our results improve, extend, and complement some existing ones. Finally, numerical examples of linear and nonlinear fractional-delay systems are presented to demonstrate the theoretical results.

Analytical Solution of the Fractional Linear Time-Delay Systems and their Ulam-Hyers Stability

We introduce the delayed Mittag-Leffler type matrix functions, delayed fractional cosine, and delayed fractional sine and use the Laplace transform to obtain an analytical solution to the IVP for a Hilfer type fractional linear time-delay systemD0,tμ,νzt+Azt+Ωzt−h=ftof order1<μ<2and type0≤ν≤1, with nonpermutable matricesAandΩ. Moreover, we study Ulam-Hyers stability of the Hilfer type fractional linear time-delay system. Obtained results extend those for Caputo and Riemann-Liouville type fractional linear time-delay systems with permutable matrices and new even for these fractional delay systems.

Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

This research studies optimal control of a new subclass of variable-order fractional delay systems whose its order is a piecewise constant function. This category of systems has not been discussed in the literature yet. An effective methodology based on a generalization of the fractional-order Chebyshev functions is offered for providing a solution with high level of precision. A detailed consideration regarding the convergence of the new framework is furnished. Moreover, two important estimates connected to the best approximation of the mentioned fractional basis in the Sobolev space and Hilbert space are achieved. Because direct implementation of the Riemann-Liouville integral operator (RLIP) leads to probably some serious drawbacks, such as numerical challenges, instability and unexpected oscillatory behaviour of the system under examination, a key integral operator connected to the basis under consideration is attained. The capacity and capability of the suggested numerical scheme are illustrated and verified through our numerical findings.

Optimal control of nonlinear fractional order delay systems governed by Fredholm integral equations based on a new fractional derivative operator

We aim to develop a direct transcription approach for solving a notable category of optimal control problems governed by nonlinear fractional Fredholm integral equations having delays in both input and output signals. The foundation of the new methodology is based on a multi domains decomposition scheme by utilizing the fractional-order Legendre functions. A new fractional derivative operator associated with the fractional basis is introduced by using the Caputo fractional derivative operator. With the use of derivative and delay operators, one can transform the dynamical system related to the fractional control problem into a new system containing algebraic equations. A wide variety of challenging test problems are studied to provide a detailed explanation of the designed approach.