Asymptotic Stability Of Zero Solution Research Articles

Synchronization of nonautonomous neural networks with Caputo derivative and time delay

<abstract> <p>The synchronization problem of delayed nonautonomous neural networks with Caputo derivative is studied in this article. Firstly, new neural networks are proposed by introducing variable parameters into known models, and the analytical formula of the synchronous controller is given according to the new neural networks. Secondly, from the drive-response systems corresponding to the above delayed neural networks, their error system is obtained. Thirdly, by constructing the Lyapunov function and utilizing the Razumikhin-type stability theorem, the asymptotic stability of zero solution for the error system is verified, and some sufficient conditions are achieved to ensure the global asymptotic synchronization of studied neural networks. Finally, some numerical simulations are given to show the availability and feasibility of our obtained results.</p> </abstract>

On the asymptotic stability of singular systems with a constant and two time-varying delays

In this paper, asymptotic stability of zero solution of a kind of delay singular system is investigated. Based on the delayed-decomposition approach, two new results are obtained on the asymptotic stability of the considered system by using of some well-known inequalities and Lyapunov- Krasovskiĭ functionals (LKFs). Finally, two examples with numerical simulation are given to illustrate effectiveness of the proposed method by MATLAB-Simulink.

Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach

In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications.

Criterion for robustness of global asymptotic stability to external perturbations of linear time-varying systems

ABSTRACT In this paper, we establish a new criterion for robustness of global asymptotic stability of zero solution of LTV system in the presence of external perturbations (disturbances). To prove the result, logarithmic norm will be used under which the stability becomes a topological notion depending on the chosen vector norm in the state-space

Logarithmic norm-based analysis of robust asymptotic stability of nonlinear dynamical systems

In this paper, we address the issue of robust asymptotic stability of zero solution for the nonlinear dynamical systems x˙=A(t)x+g(x,t). Using the approach based on the logarithmic norm and combining with the variation constant formula we show the persistence of local or global “attractiveness” of the origin for a wide class of perturbations.

New Qualitative Results for Solutions of Functional Differential Equations of Second Order

In this paper, we are concerned with the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second order by the second method of Lyapunov. We obtain sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations considered. We give an example for illustrations by MATLAB-Simulink, which shows the behaviors of the orbits. The findings of this paper extend and improve some results that can be found in the literature.

The Lyapunov Stability of the Cauchy - Dirichlet Problem for the Generalized Hoff Equation

We consider the initial boundary value problem with homogeneous Dirichlet boundary conditions for the generalized Ho equation in a bounded domain. This equation models the dynamics of buckling of a double-tee girder under constant load and belongs to a large class of Sobolev type semilinear equations (We can isolate the linear and non-linear parts of the operator acting on the original function). The paper addresses the stability of zero solution of this problem. There are two methods in the theory of stability: the rst one is the study of stability by linear approximation and the second one is the study of stability by Lyapunov function. We use the second Lyapunov's method adapted to the case of incomplete normed spaces. The main result of this paper is a theorem on the stability and asymptotic stability of zero solution to this problem.

Asymptotic stability of the zero solution for degenerate retarded differential equations

In this paper, we mainly discuss relations between the asymptotic stability of the zero solution for retarded differential equations and real parts of all characteristic roots of characteristic equations. We obtain a necessary and sufficient condition for the asymptotic stability of the zero solution. We point out the mistake of [W.E. Brumley, On the asymptotic behavior of solutions of differential–difference equations of neutral type, Journal of Differential Equations 7 (1) (1970) 175–188], generalize the result of [H. Ren, Z. Zhen, Algebraic criteria of asymptotic stability of zero solution of neutral-type equation x ̇ ( t ) + c x ̇ ( t − τ ) + a x ( t ) + b x ( t − τ ) = 0 , Acta Mathematica Sinica 42 (6) (1999) 1–11] and solve a problem posed in [Z.-X. Zheng, Introduction to Functional Differential Equations, Anhui Educational Press, 1992].

On the stability of solutions to a certain fourth-order delay differential equation

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

On stability of solutions of certain fourth-order delay differential equations

By the use of the Liapunov functional approach, a new result is obtained to ascertain the asymptotic stability of zero solution of a certain fourth-order non-linear differential equation with delay. The established result is less restrictive than those reported in the literature.

Some kinds of liapunov functional in stability theory of RFDE

In this paper, some kinds of Liapunov functionals are constructed. By these constructions we have obtained some criteria for the uniform asymptotic stability of zero solution of several RFDE systems. These criteria are in concise forms, easily checked and applicable.