On asymptotic properties of solutions to fractional differential equations

Cong, Nguyen Dinh; Tuan, H. T.; Trinh, Hieu

doi:10.1016/j.jmaa.2019.123759

Cited by 46 publications

(37 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We will show a characterization of the Mittag-Leffler stability through fractional Lyapunov functions. [3,7,9,15]. The novelty relies in the bounded case (Theorem 3.1(ii)) and the underscore that the stability is respect to the initial time of the derivative ( t = 0 ).…”

Section: Asymptotic Stabilitymentioning

confidence: 99%

“…Such X is called a positively invariant set of (2.1). A sufficient condition to have X = R n is that f be continuous and Lipschitz continuous in its first argument [3,5]. Note that for the uniqueness, x 0 must be specified at the initial time of the fractional derivative ( t = 0 in our case).…”

Section: Stability Preliminariesmentioning

confidence: 99%

“…(ii) After sending this article, we have taken note of an alternative version of Theorem 2.4 segregated in the results of [3,Theorem 6, Lemma 5, Remark 10, Remark 7], where no differentiability but just a local Lipschitz condition is required on f .…”

Section: Remark 25 (I) As a Consequence Fractional Systems Of Derivation Order Lesser Than One Cannot Be Exponentiallymentioning

confidence: 99%

See 2 more Smart Citations

Converse theorems in Lyapunov’s second method and applications for fractional order systems

Gallegos¹,

Duarte‐Mermoud²

2019

Turk J Math

View full text Add to dashboard Cite

We establish a characterization of the Lyapunov and Mittag-Leffler stability through (fractional) Lyapunov functions, by proving converse theorems for Caputo fractional order systems. A hierarchy for the Mittag-Leffler order convergence is also proved which shows, in particular, that fractional differential equation with derivation order lesser than one cannot be exponentially stable. The converse results are then applied to show that if an integer order system is (exponentially) stable, then its corresponding fractional system, obtained from changing its differentiation order, is (Mittag-Leffler) stable. Hence, available integer order control techniques can be disposed to control nonlinear fractional systems. Finally, we provide examples showing how our results improve recent advances published in the specialized literature.

show abstract

Section: Asymptotic Stabilitymentioning

confidence: 99%

Section: Stability Preliminariesmentioning

confidence: 99%

Section: Remark 25 (I) As a Consequence Fractional Systems Of Derivation Order Lesser Than One Cannot Be Exponentiallymentioning

confidence: 99%

See 1 more Smart Citation

Converse theorems in Lyapunov’s second method and applications for fractional order systems

Gallegos¹,

Duarte‐Mermoud²

2019

Turk J Math

View full text Add to dashboard Cite

show abstract

“…In order to find out the essential performance of the established equations, the existence and stability of the solutions of the equations is the first prerequisite. In the last few years, several results on this topic were presented including asymptotic stability [1,4,15,24]), exponential stability [2,25] and Mittag-Leffler stability [5,19,[27][28][29][30]. The general method for analyzing the stability is based on the first method of Lyapunov, the second method of Lyapunov and other mathematical techniques.…”

Section: Introductionmentioning

confidence: 99%

Stability analysis of fractional-order systems with randomly time-varying parameters

Wang

Ding

Nieto

2021

NAMC

View full text Add to dashboard Cite

This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.

show abstract

“…A rather detailed account of diverse recent theoretical advances and applications of fractional calculus in the various fields can be found in the books of Sabatier et al [14], Hilfer [6] and Atanackovic et al [1]. Works such as [2,3,7,8,11,15,16,[22][23][24][25][26][27][28][29] and the monographs [4,13] analyzed qualitative and quantitative aspects of the solution of the fractional-order differential equation. The methods employed in the aforementioned literature include the sequential technique of successive approximation as well as the classical fixed-point approaches of Banach and Schauder.…”

Section: Introductionmentioning

confidence: 99%

On the concept of a conformable fractional differential equation

Shaw¹,

Othman²

2021

J., eng., them. sci.

View full text Add to dashboard Cite

A new simple well-behaved definition of the fractional derivative termed as conformable fractional derivative and introducing a geometrical approach of fractional derivatives, non-integral order initial value problems are an attempt to solve in this article. Based on the geometrical interpretation of the fractional derivatives, the solution curve is approximated numerically. Two special phenomena are employed for concave upward and downward curves. In order to obtain the solution of fractional order differential equation (FDE) with the integer-order initial condition, some new criteria on fractional derivatives are proposed.

show abstract

On asymptotic properties of solutions to fractional differential equations

Cited by 46 publications

References 32 publications

Converse theorems in Lyapunov’s second method and applications for fractional order systems

Converse theorems in Lyapunov’s second method and applications for fractional order systems

Stability analysis of fractional-order systems with randomly time-varying parameters

On the concept of a conformable fractional differential equation

Contact Info

Product

Resources

About