Stability Of Nonlinear Differential Systems Research Articles

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.

Sufficient Stability Conditions of Nonlinear Differential Systems Under Impulsive Control With State-Dependent Delay

In this technical note we study the delayed impulsive control of nonlinear differential systems, where the impulsive control involves the delayed state of the system for which the delay is state-dependent. Since the state dependence of the delay makes the impulsive transients dependent on the historical information of the states, which means that it is hard to know exactly a priori how far in the history the information is needed, the main challenge is how to determine the historical states. We resolve this challenge and establish some sufficient conditions for local stability of nonlinear differential systems with state-dependent delayed impulsive control based on impulsive control theory. Two examples are given to show the effectiveness of the proposed approach.

Stability Analysis of Nonlinear Time-Delay Systems Using a Novel Piecewise Positive Systems Method

We propose a method that uses piecewise positive comparison systems to analyze the stability of nonlinear differential systems with time-varying delays. A new stability criterion is provided and two numerical examples are given to demonstrate the advantages of our obtained result.

Stability of nonlinear differential systems with state-dependent delayed impulses

We consider nonlinear differential systems with state-dependent delayed impulses (impulses which involve the delayed state of the system for which the delay is state-dependent). Such systems arise naturally from a number of applications and the stability issue is complex due to the state-dependence of the delay. We establish general and applicable results for uniform stability, uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments. We show how restrictions on the change rates of states and impulses should be imposed to achieve system’s stability, in comparison with general impulsive delay differential systems with state-dependent delay in the nonlinearity, or the differential systems with constant delays. In our approach, the boundedness of the state-dependent delay is not required but derives from the stability result obtained. Examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach.

Novel Criteria for Exponential Stability of Nonlinear Differential Systems With Delay

General nonlinear differential systems with time-varying delay are considered. Several explicit criteria for global exponential stability are presented. Two examples are given to illustrate the obtained results.

Stability of nonlinear differential systems with delay

General nonlinear time-varying differential systems with delay are considered. Several new explicit criteria for exponential stability are given. A discussion of the obtained results and two illustrative examples are presented.

Global exponential stability of certain switched systems with time-varying delays

This paper is focused on global exponential stability of certain switched systems with time-varying delays. By using an average dwell time (ADT) approach that is different from the method in [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213], we establish a new global exponential stability criterion for the switched linear time-delay system under the ADT switching. We also apply this method to a general switched nonlinear time-delay system. A numerical example is given to show the effectiveness of our results.

Several integral inequalities and their applications in nonlinear differential systems

In this paper, several new integral inequalities are presented, which are effective in dealing with the integro-differential inequalities whose variable exponents are greater than one. Compared with existing integral inequalities, those proposed here can be applied to more complicated differential equations. The notions of uniform Lipschitz stability are generalized and the relations between these notions are analyzed. Several sufficient conditions about uniform Lipschitz asymptotic stability of nonlinear systems are established by the proposed integral inequalities. These sufficiently conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature. Finally, an example of uniform Lipschitz asymptotic stability of nonlinear differential systems is shown.

Note on exponential stability of certain nonlinear differential systems with time-varying delays

One of the main results in a recent paper [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213] is extended to a more general nonlinear differential system with time-varying delays. A restrictive condition for global exponential stability in the above paper is also removed.

Effect of leakage time-varying delay on stability of nonlinear differential systems

In this paper, a class of nonlinear differential systems with leakage time-varying delay is considered. Sufficient conditions for the existence-uniqueness and global asymptotic stability of the equilibrium point are derived by using fixed point theorems, Lyapunov–Krasovskii functional and model transformation technique. The stability criterion that depends on the upper bounds of the leakage time-varying delay and its derivative is given in terms of a linear matrix inequality (LMI), which can be efficiently solved via standard numerical software. Even for the case of leakage constant delay, the criterion is shown to be less conservative than a recent publication. Finally, two numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.

On exponential stability of nonlinear differential systems with time-varying delay

Abstract General nonlinear differential systems with time-varying delays are considered. Several explicit criteria for exponential stability are presented. An example is given to illustrate the obtained results. To the best of our knowledge, the results of this note are new.

The Stability of Nonlinear Differential Systems with Random Parameters

The paper deals with nonlinear differential systems with random parameters in a general form. A new method for construction of the Lyapunov functions is proposed and is used to obtain sufficient conditions for L2‐stability of the trivial solution of the considered systems.

On stability of nonlinear differential systems via cone-valued Liapunov function method

The notions of partial, relative and φ 0-stability of ordinary differential equations (ODEs) are introduced in [E.P. Akpan, O. Akinyele, J. Math. Anal. Appl. 164 (1992) 307–324; V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, vol. 1, Academic Press, New York, 1969]. In this paper, we extend these notions to new types of stability namely, φ 0-relative and φ 0-partial stability of ODEs using cone-valued Liapunov function method and comparison technique.

Stability of nonlinear differential systems*

We study a general variationnal stability introduced mainly for non‐autonomous systems. Under integrability conditions, these systems remains stable for general perturbations

On stability of non-linear differential systems

The object of this paper is to study the stability and asymptotic stability of solutions of the non-linear differential equation dx dt = A(t)x + f(t,x) by using the method of equivalent inner products. This method enables one to determine a stability region without the ingenuity in constructing a Lyapunov function. It shows also that for an unstable linear system it is possible to choose a non-linear function so that the non-linear system is stable or asymptotically stable. Both global and regional stability are discussed.