Conditions for stability and stabilization of systems of neutral type with nonmonotone Lyapunov functionals

We suggest new tests for the stability and uniform asymptotic stability of an equilibrium in systems of neutral type. By using these tests, we prove conditions for optimal stabilization and derive new estimates for perturbations that can be countered by a system closed by an optimal control. We show that, by using nonmonotone sign-indefinite functionals as Lyapunov functionals, one can obtain conditions for uniform asymptotic stability that do not contain the a priori requirement of stability of the difference operator and do not imply the boundedness of the right-hand side of the system. When studying the action of perturbations on the stabilized systems, these conditions permit one to obtain new estimates of perturbations preserving the stabilizing properties of optimal controls. The obtained estimates do not imply any constraint on the value of perturbations in some domains of the phase space that are defined when constructing an optimal stabilizing control. Some examples are considered.     

Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.     

On the pth moment exponential stability criteria of neutral stochastic functional differential equations

The paper discusses the pth moment exponential stability for a general class of neutral stochastic functional differential equations of the Ito type. This investigation can be very complicated, even in many special cases, by using usual methods based on Lyapunov functionals. In this paper we present criteria which are relatively easy to verify the pth moment exponential stability of the solutions of such equations.     

Functionals with sign-definite derivative for the stabilization of delay systems

We present new conditions for the uniform asymptotic stability of equilibria in delay systems. These conditions are based on Lyapunov functionals that have negative definite derivatives along the trajectories of the system only on some part of the phase space.By using these conditions, we establish new optimal stabilization tests, which admit the use of a performance functional whose weight functional is not negative definite on the entire phase space.We introduce a new notion of smallness for perturbations in delay systems and present stabilization tests by the first approximation.     

Asymptotic behaviors for functionals of random dynamical systems

In this article, we consider asymptotic behaviors for functionals of dynamical systems with small random perturbations. First, we present a deviation inequality for Gaussian approximation of dynamical systems with small random perturbations under Hölder norms and establish the moderate deviation principle and the central limit theorem for the dynamical systems by the deviation inequality. Then, applying these results to forward-backward stochastic differential equations and diffusions in small time intervals, combining the delta method in large deviations, we give a moderate deviation principle for solutions of forward-backward stochastic differential equations with small random perturbations, and obtain the central limit theorem, the moderate deviation principle and the iterated logarithm law for functionals of diffusions in small time intervals.     

On the Robust Stability of Solutions to Periodic Systems of Neutral Type

Under consideration is some class of linear systems of neutral type with periodic coefficients. We obtain the conditions on perturbations of the coefficients which preserve the exponential stability of the zero solution. Using a special Lyapunov–Krasovskii functional, we establish some estimates that characterize the rate of exponential decay at infinity of the solutions of the perturbed systems.     

On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.     

Dynamics of a hysteretic neuron model

A mathematical model of a single isolated artificial neuron with hysterisis is formulated by means of a neutral delay differential equation. The asymptotic and exponential stability of such a model are investigated. Sufficient conditions for the exponential stability of a linear integral difference inequality are obtained. In the absence of hysterisis effect, our model reduces to a known model of a single neuron. Usually asymptotic stability of neutral delay differential equations is studied by means of degenerate Lyapunov–Kravsovskii functionals. In this article, perhaps for the first time exponential stability of a class of neutral differential equations are studied by means of the exponential stability of an affiliated difference inequality. While generalization to Hopfield type hysteretic neural networks is possible, such a generalization is not considered in this article.     

On the behavior of positive solutions of semilinear elliptic equations in asymptotically cylindrical domains

The goal of this note is to study the asymptotic behavior of positive solutions for a class of semilinear elliptic equations which can be realized as minimizers of their energy functionals. This class includes the Fisher-KPP and Allen–Cahn nonlinearities. We consider the asymptotic behavior in domains becoming infinite in some directions. We are in particular able to establish an exponential rate of convergence for this kind of problems.     

Estimates of solutions to the linear differential equations of neutral type with several delays of the argument

Under study is a systemof linear differential equations of neutral type with several delays of the argument. We obtain the conditions on the matrix coefficients of the system under which all solutions decrease with an exponential rate at infinity. Using some functionals of Lyapunov-Krasovskii type, the uniform estimates of solutions are established.     

Linear Stability of Viscous Roll Waves

The purpose of this paper is to study the linear stability of “viscous” roll waves. These are periodic continuous traveling waves solutions of viscous perturbations of inhomogeneous hyperbolic systems. We first study the scalar case for the Burgers equation and for an inhomogeneous hyperbolic equation. Then we analyze the stability of roll waves, solutions of the shallow water equations with a real viscosity. In both cases, we first analyze the Evans function and compute an asymptotic expansion in the low frequency regime. Under a strong spectral stability condition, we prove the linear stability of viscous roll waves, solutions of the Saint Venant equations, with pointwise estimates on the Green functions.     

An improved stability criterion for a class of neutral differential equations

This work gives an improved criterion for asymptotical stability of a class of neutral differential equations. By introducing a new Lyapunov functional, we avoid the use of the stability assumption on the main operators and derive a novel stability criterion given in terms of a LMI, which is less restricted than that given by Park [J.H. Park, Delay-dependent criterion for asymptotic stability of a class of neutral equations, Appl. Math. Lett. 17 (2004) 1203–1206] and Sun et al. [Y.G. Sun, L. Wang, Note on asymptotic stability of a class of neutral differential equations, Appl. Math. Lett. 19 (2006) 949–953].     

一类线性多步法关于变延迟非线性中立型微分方程的渐近稳定性

1引言中立型微分方程广泛出现于生物学、物理学及工程技术等诸多领域.数值求解中立型微分方程时,数值方法的稳定性研究具有无容置疑的重要性,其中渐近稳定性的研究是其重要组成部分.对于线性中立型延迟微分方程,渐近稳定性研究已有许多重要结果,如文献[1,2,3,4,5,6]等.对于非线性中立型变延迟微分方程,数值方法的稳定性研究近几年才有进展.2000年,Bellen等在文献[7]中讨论了Runge-Kutta法求解一类特殊的中立型延迟微分     

Positivity and stability of linear functional differential equations with infinite delay

General linear functional differential equations with infinite delay are considered. We first give an explicit criterion for positivity of the solution semigroup of linear functional differential equations with infinite delay and then a Perron‐Frobenius type theorem for positive equations. Next, a novel criterion for the exponential asymptotic stability of positive equations is presented. Furthermore, two sufficient conditions for the exponential asymptotic stability of positive equations subjected to structured perturbations and affine perturbations are provided. Finally, we applied the obtained results to problems of the exponential asymptotic stability of Volterra integrodifferential equations. To the best of our knowledge, most of the results of this paper are new.     

Decay characteristics for differential equations without linear terms

The asymptotic stability of isolated critical points of differential systems without linear terms is related to the signum of the type numbers introduced by Coleman. This leads to asymptotic estimates for solutions of a class of differential equations with homogeneous first approximation.     

Asymptotic stability of ground states in 2D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equations in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time-dependent, Hamiltonian, linearized dynamics around a carefully chosen one-parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

STABILITY OF SOLUTIONS OF FIRST ORDER NONLINEAR NEUTRAL DIFFERENTIAL EQUATIONS WITH OSCILLATORY COEFFICIENTS

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Stability of solutions of Volterra integrodifferential equations

Systems with past memory (or after-effect), the state of which is given by nonlinear Volterra- type integrodifferential equations with small perturbations, are investigated. The equations depend on functionals in integral form and, in particular, on analytic functionals represented by Fréchet series. The integral kernels can allow for singularities with Abel’s kernel. The stability under persistent disturbances, and the structure of the general solution, are investigated in the neighborhood of zero for an equation with holomorphic nonlinearity assuming asymptotic stability of the trivial solution of the linearized unperturbed equation. Stability in the critical cases (in Lyapunov’s sense) of a single zero root and of pairs of pure imaginary roots for the unperturbed equation is analyzed.     

Instability conditions for solutions of delay systems and localization of limit sets

We obtain new tests for the instability of the trivial solutions of equations with deviating argument. In contrast to earlier-known results, these tests use nonmonotone Lyapunov functionals. The class of such functionals contains Lyapunov-Krasovskii functionals as well as Lyapunov-Razumikhin functions as special cases. By localizing the limit sets of solutions, in a number of instability tests, we have been able to drop the requirement that the derivative of the Lyapunov functional according to the system be negative definite.