Exponential stability of nonlinear differential equations in the presence of perturbations or delays

The exponential stability of nonlinear nonautonomous nonsmooth differential equations in finite dimensional linear spaces is investigated. It turns out that exponential stability is preserved under sufficiently small perturbations or delays. Explicit estimates for the relation between the exponential decay rates and the size of the perturbation or delay are presented. The results are valid for differential equations given by Lipschitz continuous vector fields.     

Exponential dichotomies of linear dynamic equations on measure chains under slowly varying coefficients

Unifying ordinary differential and difference equations, we consider linear dynamic equations on measure chains or time scales, which possess an exponential dichotomy uniformly in a parameter, and show that this dichotomy is robust, if the mentioned parameter changes slowly in time. Here, the equations can be infinite dimensional and are not assumed to be invertible.     

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.     

Application of finite difference method of lines on the heat equation

In this article, we apply the method of lines (MOL) for solving the heat equation. The use of MOL yields a system of first–order differential equations with initial value. The solution of this system could be obtained in the form of exponential matrix function. Two approaches could be applied on this problem. The first approach is approximation of the exponential matrix by Taylor expansion, Padé and limit approximations. Using this approach leads to create various explicit and implicit finite difference methods with different stability region and order of accuracy up to six for space and superlinear convergence for time variables. Also, the second approach is a direct method which computes the exponential matrix by applying its eigenvalues and eigenvectors analytically. The direct approach has been applied on one, two and three‐dimensional heat equations with Dirichlet, Neumann, Robin and periodic boundary conditions.     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces

In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS.

     

非线性发展方程的H(o)lder连续惯性流形

引入非线性发展方程的H\"older连续惯性流形的概念,为原来惯性流形概念的推广和修正.惯性流形是有限维不变的Lipschiz流形,是研究发展方程解的长时间性态的合适工具,其缺点是需要谱间隙条件.提出H\"older连续惯性流形也是有限维不变的,但光滑性减弱为H\"older连续,不需要谱间隙条件.该流形与指数吸引子交集具有指数吸引性,无穷维动力系统可在H\"older连续惯性流形上约化为有限维常微分方程组.     

Mean-square exponential dichotomy of numerical solutions to stochastic differential equations

We present the ability of numerical simulations to reproduce the mean-square exponential dichotomy of stochastic differential equations. Under some conditions, we show that the mean-square exponential dichotomy of stochastic differential equations is equivalent to that of the numerical method for sufficient small step sizes     

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.     

G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性

研究了一类G-Brown运动驱动的中立型随机时滞微分方程的指数稳定性.在G-框架意义下,运用合适的Lyapunov-Krasovskii泛函,中立型时滞微分方程理论以及随机分析技巧,证明了所研究方程平凡解的p-阶矩指数稳定性,得到了所研究方程平凡解是p-阶矩指数稳定的充分条件.最后通过例子说明所得的结果.     

Markov调制奇异随机微分方程的指数稳定性

建立了Markov调制奇异随机微分方程的p阶指数稳定性和几乎必然指数稳定性的充要条件．     

Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

In this paper, we consider large‐scale nonsymmetric differential matrix Riccati equations with low‐rank right‐hand sides. These matrix equations appear in many applications such as control theory, transport theory, applied probability, and others. We show how to apply Krylov‐type methods such as the extended block Arnoldi algorithm to get low‐rank approximate solutions. The initial problem is projected onto small subspaces to get low dimensional nonsymmetric differential equations that are solved using the exponential approximation or via other integration schemes such as backward differentiation formula (BDF) or Rosenbrock method. We also show how these techniques can be easily used to solve some problems from the well‐known transport equation. Some numerical examples are given to illustrate the application of the proposed methods to large‐scale problems.     

Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps

In this paper, we consider a class of neutral stochastic partial differential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square as well as almost surely exponential stability of mild solutions are derived by means of the Banach fixed point principle. An example is provided to illustrate the effectiveness of the proposed result.     

Existence and exponential stability of periodic solution for impulsive delay differential equations and applications

In this paper, we consider a class of nonlinear impulsive delay differential equations. By establishing an exponential estimate for delay differential inequality with impulsive initial condition and employing Banach fixed point theorem, we obtain several sufficient conditions ensuring the existence, uniqueness and global exponential stability of a periodic solution for nonlinear impulsive delay differential equations. Furthermore, the criteria are applied to analyze dynamical behavior of impulsive delay Hopfield neural networks and the results show different behavior of impulsive system originating from one continuous system.     

On differential–algebraic equations in infinite dimensions

We consider a class of differential–algebraic equations (DAEs) with index zero in an infinite dimensional Hilbert space. We define a space of consistent initial values, which lead to classical continuously differential solutions for the associated DAE. Moreover, we show that for arbitrary initial values we obtain mild solutions for the associated problem. We discuss the asymptotic behaviour of solutions for both problems. In particular, we provide a characterisation for exponential stability and exponential dichotomies in terms of the spectrum of the associated operator pencil.     

Exponential stability of second-order stochastic evolution equations with Poisson jumps

This paper is concerned with the exponential stability problem of second-order nonlinear stochastic evolution equations with Poisson jumps. By using the stochastic analysis theory, a set of novel sufficient conditions are derived for the exponential stability of mild solutions to the second-order nonlinear stochastic differential equations with infinite delay driven by Poisson jumps. An example is provided to demonstrate the effectiveness of the proposed result.     

Moment exponential stability and integrability of stochastic functional differential equations

The main objective of the paper is to present sufficient conditions ensuring the pth mean and almost sure exponential stability, as well as the pth mean integrability of solutions to non-linear stochastic functional differential equations and, especially, to stochastic differential equations with time-varying lags. Some examples are given to illustrate the theoretical considerations.     

Well-posedness of the analytic Cauchy problem in classes of entire functions of finite order

We study the Cauchy problem for a system of complex linear differential equations in scales of spaces of functions of exponential type with an integral metric. Conditions under which this problem is well posed are obtained. These sufficient conditions are shown to be also necessary for the well-posedness of the Cauchy problem in the case of systems of ordinary differential equations with a parameter.     

Exponential stability of energy solutions to stochastic partial differential equations with variable delays and jumps

In this work, we investigate stochastic partial differential equations with variable delays and jumps. We derive by estimating the coefficients functions in the stochastic energy equality some sufficient conditions for exponential stability and almost sure exponential stability of energy solutions, and generalize the results obtained by Taniguchi [T. Taniguchi, The exponential stability for stochastic delay partial differential equations, J. Math. Anal. Appl. 331 (2007) 191-205] and Wan and Duan [L. Wan, J. Duan, Exponential stability of non-autonomous stochastic partial differential equations with finite memory, Statist. Probab. Lett. 78 (5) (2008) 490-498] to cover a class of more general stochastic partial differential equations with jumps. Finally, an illustrative example is established to demonstrate our established theory.     

Razumikhin-type Theorems on Exponential Stability of Stochastic Functional Differential Equations with Infinite Delay

The main aim of this paper is to study the stability of the stochastic functional differential equations with infinite delay. We establish several Razumikhin-type theorems on the exponential stability for stochastic functional differential equations with infinite delay. By applying these results to stochastic differential equations with distributed delay, we obtain some sufficient conditions for both pth moment and almost surely exponentially stable. Finally, some examples are presented to illustrate our theory.