Stability Of Functional Differential Equations Research Articles

Exponential stabilization of fixed and random time impulsive delay differential system with applications.

In this work, we study the problem of p-th moment global exponential stability for functional differential equations and scalar chaotic delayed equations under random impulsive effects. Meanwhile, the p-th moment global exponential synchronization for the proposed equations is also discussed, whereas the main results are proved by using Lyapunov function and Razumikhin technique. Furthermore, the impact of fixed and random time impulses are presented by applying the results to Mackey Glass blood cell production model and Ikeda bistable resonator model. Finally, the effectiveness of fixed and random impulses are depicted via graphical representations.

Scalar criteria for exponential stability of functional differential equations

Using a novel approach, we present some scalar criteria for the exponential stability of general functional differential equations. Both delay-independent and delay-dependent stability criteria are provided. A discussion of the obtained results is given.

Further results on exponential stability of functional differential equations

ABSTRACTGeneral non-linear functional differential equations are considered. New explicit criteria for the exponential stability are presented. The stability criteria given in this paper include many existing results as particular cases. In particular, they unify, generalise and improve some ones published recently in [Ngoc, P. H. A. (2012). On exponential stability of non-linear differential systems with time-varying delay. Applied Mathematics Letters, 25(9), 1208–1213 and Ngoc, P. H. A. (2013b). Novel criteria for exponential stability of functional differential equations. Proceedings of the American Mathematical Society, 141(9), 3083–3091]. Two examples are given to show the effectiveness and advantage of the obtained results.

Stability for Functional Differential Equations with Delay in Banach Spaces

In this paper, using Sadovskii’s fixed point theorem and properties of the measure of noncompactness, we obtain asymptotic stability results for solutions of nonlinear differential equation with variable delay. Results presented in this paper extend some previous results due to Burton [1], Becker and Burton [2], Ardjouni and Djoudi [3], and Jin and Luo [4].

Persistent impulsive effects on stability of functional differential equations with finite or infinite delay

In this paper, the stability problem of impulsive functional differential equations (IFDEs) is considered. Several criteria ensuring the uniform stability of IFDEs with finite or infinite delay are derived by establishing some new Razumikhin conditions. Different with the current existing ones, our development results can be applied to delay systems with persistent impulsive effects. Finally, an example and its computer simulation is given to show the effectiveness and advantages of the proposed method.

Razumikhin method for impulsive functional differential equations of neutral type

Although the well-known Razumikhin method has been well developed for the stability of functional differential equations with or without impulses and it is very useful in applications, so far there is almost no result of Razumikhin type on stability of impulsive functional differential equations of neutral type. The purpose of this paper is to close this gap and establish some Razumikhin-based stability results for impulsive functional differential equations of neutral type. A kind of auxiliary function N(t) that has great randomicity is introduced to Razumikhin condition. Some examples are given to show the effectiveness and advantages of the developed method.

Hyers–Ulam stability of linear functional differential equations

In this paper, the stability of some classes of linear functional differential equations was discussed by direct method, iteration method, fixed point method and open mapping theorem. It is shown that the Hyers–Ulam stability holds true for y(n)=g(t)y(t−τ)+h(t). The stability of functional differential equations with multiple delays of first order and general delay differential equations also have been discussed.

Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching

This work develops moment exponential stability of functional differential equations (FDEs) with Markovian switching, in which a two-time-scale (real time $t$ and fast time $t/\epsilon$ with a small parameter $\epsilon>0$) continuous-time and finite-state Markov chain is used to represent the switching process. The essence is that there is a time-scale separation, which is motivated by the consideration of networked control systems and manufacturing systems. Under suitable conditions, we establish a Razumikhin-type theorem on the $p$th moment exponential $\epsilon$-stability for the small parameter $\epsilon$. By virtue of the Razumikhin-type theorem, we further deduce mean-square exponential stability results for delay differential equations (DDEs) and ordinary differential equations (ODEs) with two-time-scale Markovian switching. These stability results show that the overall system may be stabilized by the Markov switching even when some of the underlying subsystems are unstable. It is noted that in the presence of the Markovian switching, the stationary distribution of the fast changing part of the Markov chain plays an important role. Explicit conditions for the mean-square exponential stability of linear equations are derived and illustrative examples are provided to demonstrate our results.

On exponential stability of functional differential equations with variable impulse perturbations

We consider a class of functional differential equations subject to perturbations, which vary in time, and we study the exponential stability of solutions of these equations using the theory of generalized ordinary differential equations and Lyapunov functionals. We introduce the concept of variational exponential stability for generalized ordinary differential equations and we develop the theory in this direction by establishing conditions for the trivial solutions of generalized ordinary differential equations to be exponentially stable. Then, we apply the results to get corresponding ones for impulsive functional differential equations. We also present an example of a delay differential equation with Perron integrable right-hand side where we apply our result.

Finite-time uniform stability of functional differential equations with applications in network synchronization control

In this paper, we investigate finite-time uniform stability of functional differential equations with applications in network synchronization control. First, a Razumikhin-type theorem is derived to ensure finite-time uniform stability of functional differential equations. Based on the theoretical results, finite-time uniform synchronization is proposed for a class of delayed neural networks and delayed complex dynamical networks by designing nontrivial and simple control strategies and some novel criteria are established. Especially, a feasible region of the control parameters for each neuron is derived for the realization of finite-time uniform synchronization of the addressed neural networks, which provide a great convenience for the application of the theoretical results. Finally, two numerical examples with numerical simulations are provided to show the effectiveness and feasibility of the theoretical results.

Novel criteria for exponential stability of functional differential equations

We first give explicit criteria for exponential stability of general linear nonautonomous functional differential equations. Then the obtained results are extended to nonlinear functional differential equations. Two examples are given to illustrate the results. To the best of our knowledge, the results of this note are new.

Stability of functional differential equations with variable impulsive perturbations via generalized ordinary differential equations

We consider a class of functional differential equations with variable impulses and we establish new stability results. We discuss the variational stability and variational asymptotic stability of the zero solution of a class of generalized ordinary differential equations where our impulsive functional differential equations can be embedded and we apply that theory to obtain our results, also using Lyapunov functionals.

On the spectral stability of functional-differential equations

A boundary value problem for an elliptic functional-differential equation with contraction and dilatation of the arguments of the desired function in the leading part is considered in a starshaped bounded domain. Estimates for the modification of eigenvalues of the operator of the problem under internal deformations of the domain are obtained.

Stability of functional differential equations with oscillating coefficients and distributed delays

We consider the scalar equation ˙ x(t)+ m ∑ j=1 aj(t) h 0 x(t −s)drj(s )= 0 (h = const > 0, ˙ x = dx/dt), where rj(s) are nondecreasing functions. Besides, we do not require that aj(t) are positive for all t 0. So the function z+ m ∑=1 aj(t) h 0 e −zs drj(s) can have zeros in the right-hand plane for some t 0. It is proved that the considered equation is exponentially stable, provided aj(t )= bj +cj(t) ,w herebj are positive constants, such that all the zeros of the function z+∑ m j=1bj h 0 e −zs drj(s) are in the open left-hand plane, and the integrals t 0 cj(s)ds (j = 1,...,m) are sufficiently small for all t > 0.

Stochastic suppression and stabilization of functional differential equations

Without the linear growth condition or the one-sided linear growth condition, this paper discusses whether or not stochastic noise feedback can stabilize a given unstable nonlinear functional system x ̇ ( t ) = f ( x t , t ) . Since f may defy the linear growth condition or the one-sided linear growth condition, this system may explode in a finite time. To stabilize this system, this paper stochastically perturbs this system into the stochastic functional differential system d x ( t ) = f ( x t , t ) d t + q x ( t ) d w 1 ( t ) + σ | x ( t ) | β x ( t ) d w 2 ( t ) by two independent Brownian motions w 1 ( t ) and w 2 ( t ) . This paper shows that the Brownian motion w 2 ( t ) may suppress the potential explosion of the solution for appropriate β . Moreover, for sufficiently large q , this stochastic functional system is exponentially stable. These results can be used to examine stochastic stabilization.

Stability analysis for a class of functional differential equations and applications

The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov–Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems.

On the stability problem for functional differential equations with infinite delay

We study the stability of functional differential equations with infinite delay, using the Lyapunov functional of constant sign with a derivative of constant sign. Limit equations are constructed in a special phase space. We establish a theorem on localization of a positive limit set and theorems on the stability and the asymptotic stability. The results are illustrated by examples.

Delayed feedback control near Hopf bifurcation

The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean.

Stability in functional differential equations established using fixed point theory

In this paper we consider a nonlinear scalar delay differential equation with variable delays and give some new conditions for the boundedness and stability by means of Krasnoselskii’s fixed point theory. A stability theorem with a necessary and sufficient condition is proved. The results in [T.A. Burton, Stability by fixed point theory or Liapunov’s theory: A comparison, Fixed Point Theory 4 (2003) 15–32; T.A. Burton, T. Furumochi, Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dynamic Systems and Applications 11 (2002) 499–519; B. Zhang, Fixed points and stability in differential equations with variable delays, Nonlinear Analysis 63 (2005) e233–e242] are improved and generalized. Some examples are given to illustrate our theory.

Asymptotic stability for impulsive functional differential equation

In this paper, sufficient conditions are given for asymptotic stability and uniformly asymptotic stability of impulsive functional differential equation of the form { x ′ ( t ) = f ( t , x t ) , t ⩾ t 0 , Δ x = I k ( t , x ( t − ) ) , t = t k , k ∈ Z + , and the conditions in theorems extend or improve the corresponding ones in [J. Yan, J. Shen, Impulsive stabilization of functional differential equation by Liapunov–Razumikhin functions, Nonlinear Anal. 37 (1999) 245–255; J. Shen, Z. Luo, Impulsive stabilization of functional differential equations via Liapunov functionals, J. Math. Anal. Appl. 240 (1999) 1–15; L. Hatvani, On the asymptotic stability for nonautonomous functional differential equations by Liapunov functionals, Trans. Amer. Math. Soc. 354 (2002) 3555–3571].