Uniform Stability of a Class of Fractional-Order Nonautonomous Systems with Multiple Time Delays

In mathematics, to a large extent, control theory addresses the stability of solutions of differential equations, which can describe the behavior of dynamic systems. In this paper, a class of fractional-order nonautonomous systems with multiple time delays modeled by differential equations is considered. A sufficient condition is established for the existence and uniqueness of solutions for such systems involving Caputo fractional derivative, and the uniform stability of solution is studied. At last, two examples are given to demonstrate the applicability of our results

Stability and stabilization of fractional order time delay systems

U ovom radu predstavljeni su neki osnovni rezultati koji se odnose na kriterijume stabilnosti sistema necelobrojnog reda sa kašnjenjem kao i za sisteme necelobrojnog reda bez kašnjenja.Takođe, dobijeni su i predstavljeni dovoljni uslovi za konačnom vremenskom stabilnost i stabilizacija za (ne)linearne (ne)homogene kao i za perturbovane sisteme necelobrojnog reda sa vremenskim kašnjenjem. Nekoliko kriterijuma stabilnosti za ovu klasu sistema necelobrojnog reda je predloženo korišćenjem nedavno dobijene generalizovane Gronval nejednakosti, kao i 'klasične' Belman-Gronval nejednakosti. Neki zaključci koji se odnose na stabilnost sistema necelobrojnog reda su slični onima koji se odnose na klasične sisteme celobrojnog reda. Na kraju, numerički primer je dat u cilju ilustracije značaja predloženog postupka.In this paper, some basic results of the stability criteria of fractional order system with time delay as well as free delay are presented. Also, we obtained and presented sufficient conditions for finite time stability and stabilization for (non)linear (non)homogeneous as well as perturbed fractional order time delay systems. Several stability criteria for this class of fractional order systems are proposed using a recently suggested generalized Gronwall inequality as well as 'classical' Bellman-Gronwall inequality. Some conclusions for stability are similar to those of classical integerorder differential equations. Finally, a numerical example is given to illustrate the validity of the proposed procedure

Non-Lyapunov stability of the fractional-order time-varying delay systems

U ovom radu, kriterijumi stabilnosti na konačnom vremenskom intervalu su prošireni na nelinearne nehomogene perturbovane sisteme necelobrojnog reda koji sadrže višestruka vremenski promenljiva kašnjenja. Dobijeni su dovoljni uslovi stabilnosti za sisteme necelog reda sa višestrukim vremenskim kašnjenjima korišćenjem generalizovanog i klasičnog Gronwallovog pristupa. Numerički primer je dat u cilju ilustracije značaja dobijenog rezultata.In this paper, the finite-time stability criteria are extended to nonlinear nonhomogeneous perturbed fractional-order systems including multiple time-varying delays. The sufficient conditions of a stability for the fractional systems with multiple time delays are obtained by using the generalized and classical Gronwall's approach. A numerical example is presented to illustrate the validity of the obtained result

Non-Lyapunov stability of the fractional-order time-varying delay systems

U ovom radu, kriterijumi stabilnosti na konačnom vremenskom intervalu su prošireni na nelinearne nehomogene perturbovane sisteme necelobrojnog reda koji sadrže višestruka vremenski promenljiva kašnjenja. Dobijeni su dovoljni uslovi stabilnosti za sisteme necelog reda sa višestrukim vremenskim kašnjenjima korišćenjem generalizovanog i klasičnog Gronwallovog pristupa. Numerički primer je dat u cilju ilustracije značaja dobijenog rezultata.In this paper, the finite-time stability criteria are extended to nonlinear nonhomogeneous perturbed fractional-order systems including multiple time-varying delays. The sufficient conditions of a stability for the fractional systems with multiple time delays are obtained by using the generalized and classical Gronwall's approach. A numerical example is presented to illustrate the validity of the obtained result

Stability of fractional order time delay systems

In this paper, some basic results of the stability criteria of fractional order system with time delay as well as free delay are presented. Also, they are obtained and presented sufficient conditions for finite time stability for (non)linear (non)homogeneous as well as perturbed fractional order time delay systems. Several stability criteria for this class of fractional order systems are proposed using a recently suggested generalized Gronwall inequality as well as “classical” Bellman-Gronwall inequality. Some conclusions for stability are similar to that of classical integer-order differential equations. Last, a numerical example is given to illustrate the validity of the proposed procedure

New Trends in Differential and Difference Equations and Applications

This is a reprint of articles from the Special Issue published online in the open-access journal Axioms (ISSN 2075-1680) from 2018 to 2019 (available at https://www.mdpi.com/journal/axioms/special issues/differential difference equations)

Asymptotic behaviour of general nonautonomous Nicholson equations with mixed monotonicities

A general nonautonomous Nicholson equation with multiple pairs of delays in {\it mixed monotone} nonlinear terms is studied. Sufficient conditions for permanence are given, with explicit lower and upper uniform bounds for all positive solutions. Imposing an additional condition on the size of some of the delays, and by using an adequate difference equation of the form $x_{n+1}=h(x_n)$, we show that all positive solutions are globally attractive. In the case of a periodic equation, a criterion for existence of a globally attractive positive solution is provided. The results here constitute a significant improvement of recent literature, in view of the generality of the equation under study and of sharper criteria obtained for situations covered in recent works. Several examples illustrate the results.Comment: 23 page

On the Stability of Some Discrete Fractional Nonautonomous Systems

Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed