A comparison type theorem for linear neutral fractional systems with distributed delays (original) (raw)
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Mathematics
In the present paper, sufficient conditions are obtained under which the Cauchy problem for a nonlinearly perturbed nonautonomous neutral fractional system with distributed delays and Caputo type derivatives has a unique solution in the case of initial functions with first-kind discontinuities. For this system, by applying a formula for the integral presentation of the solution of the nonhomogeneous linear neutral fractional system, we found some additional natural conditions to ensure that from the global asymptotically stability of the zero solution of the linear part of the nonlinearly perturbed system, global asymptotic stability of the zero solution of the whole nonlinearly perturbed system follows.
Asymptotic stability of nonlinear perturbed neutral linear fractional system with distributed delay
THERMOPHYSICAL BASIS OF ENERGY TECHNOLOGIES (TBET 2020), 2021
In the present paper we study the stability properties of nonlinear perturbed neutral fractional autonomous linear differential systems with distributed delay. It is shown that if the zero solution of the linear part of the nonlinear perturbed system is globally asymptotically stable, then the zero solution of the perturbed nonlinear system is globally asymptotically stable too. The results are based on a formula for the integral presentation of the general solution of a linear autonomous neutral system with distributed delay which was proved by the authors in a previous work.
Advances in Difference Equations, 2019
In this paper, the investigation of the asymptotic stability of Riemann–Liouville fractional neutral systems with variable delays has been presented. The advantage of the Lyapunov functional was used to achieve the desired results. The stability criteria obtained for zero solution of the system were formulated as linear matrix inequalities (LMIs) which can be easily solved. The advantage of the considered method is that the integer-order derivatives of the Lyapunov functionals can be directly calculated. Finally, three numerical examples have been evaluated to illustrate that the proposed method is flexible and efficient in terms of computation and to demonstrate the feasibility of established assumptions by MATLAB-Simulink.
Axioms, 2021
In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.
Fractal and Fractional, 2021
In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix C(t,s) continuous in t,t∈[a,∞),a∈R. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained.
$H_\infty$-Stability Analysis of Fractional Delay Systems of Neutral Type
SIAM Journal on Control and Optimization, 2016
In this paper we consider linear fractional systems of commensurate orders and with commensurate delays, whose characteristic equation is a polynomial in the two variables s α (0 < α < 1) and e −sτ (τ > 0). These systems may have single or multiple chains of poles asymptotic to the imaginary axis. Location of poles of large modulus belonging to these chains are determined by approximation and simple necessary and sufficient H∞-stability conditions are derived.
Mathematics, 2020
The aim of this work is to obtain an integral representation formula for the solutions of initial value problems for autonomous linear fractional neutral systems with Caputo type derivatives and distributed delays. The results obtained improve and extend the corresponding results in the particular case of fractional systems with constant delays and will be a useful tool for studying different kinds of stability properties. The proposed results coincide with the corresponding ones for first order neutral linear differential systems with integer order derivatives.
Uniqueness and Stability Results of Fractional Neutral Differential Equations with Infinite Delay
Int. J. Math. And Appl., 2018
In this paper, some uniqueness and Ulam-Hyers stability results for neutral functional differential equations with fractional order and infinite delay are proved. The problem that is discussed includes the Caputo fractional derivative operator. The technique used is a variety of tools of fractional calculus properties and Banach's fixed point theorem. An example of the obtained results is given. MSC: 34A08, 34K37, 35B35, 47H10.