On stability of a class of positive linear functional difference equations

We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron–Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae. Pham Huu Anh Ngoc and Toshiki Naito are supported by the Japan Society for Promotion of Science (JSPS) ID No. P 05049.     

On a class of positive linear differential equations with infinite delay

We give an explicit criterion for positivity of the solution semigroup of linear differential equations with infinite delay and a Perron-Frobenius type theorem for positive equations. Furthermore, a novel criterion for the exponential asymptotic stability of positive equations is presented. Finally, we provide a sufficient condition for the exponential asymptotic stability of positive equations subjected to structured perturbations. A simple example is given to illustrate the obtained results.     

Stability and robust stability of positive Volterra systems

We study positive linear Volterra integro‐differential systems with infinitely many delays. Positivity is characterized in terms of the system entries. A generalized version of the Perron–Frobenius theorem is shown; this may be interesting in its own right but is exploited here for stability results: explicit spectral criteria for L¹‐stability and exponential asymptotic stability. Also, the concept of stability radii, determining the maximal robustness with respect to additive perturbations to L¹‐stable system, is introduced and it is shown that the complex, real and positive stability radii coincide and can be computed by an explicit formula. Copyright © 2011 John Wiley & Sons, Ltd.     

Exponential stability of slowly varying discrete systems with multiple state delays

We give sufficient conditions for the exponential stability of a class of perturbed time‐varying difference equations with multiple delays and slowly varying coefficients. Under appropriate growth conditions on the perturbations, combined with the ‘freezing’ technique, we establish explicit conditions for global exponential stability. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability of difference Volterra equations: Direct Liapunov method and numerical procedure

A procedure to construct Liapunov functionals for discrete Volterra equations is proposed. Using this procedure stability conditions are derived for general Volterra difference equations. Some applications of the proposed procedure for obtaining stability conditions for linear multistep methods for Volterra integro-differential equations are presented.     

New criteria for exponential stability of nonlinear time‐varying differential systems

General nonlinear time‐varying differential systems are considered. An explicit criterion for exponential stability is presented. Furthermore, an explicit robust stability bound for systems subjected to nonlinear time‐varying perturbations is given. In particular, it is shown that the generalized Aizerman conjecture holds for positive linear systems. Some examples are given to illustrate obtained results.Copyright © 2012 John Wiley & Sons, Ltd.     

Robust stability of delay difference systems under fractional perturbations in infinite-dimensional spaces

In this article we study the robust stability of difference systems with delays under fractional perturbations in infinite-dimensional spaces. First, the estimates of the complex stability radius are addressed. Second, it is shown that for positive linear systems, the complex, real and positive stability radii coincide and can be computed by a simple formula. Finally, a simple example is given to illustrate the obtained results.     

Stability radii of higher order positive difference systems

In this paper we study stability radii of positive polynomial matrices under affine perturbations of the coefficient matrices. It is shown that the real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by some examples.     

Strong stability radii of positive linear time‐delay systems

In this paper, we study robustness of the strong delay‐independent stability of linear time‐delay systems under multi‐perturbation and affine perturbation of coefficient matrices via the concept of strong delay‐independent stability radius (shortly, strong stability radius). We prove that for class of positive time‐delay systems, complex and real strong stability radii of positive linear time‐delay systems under multi‐perturbations (or affine perturbations) coincide and they are computed via simple formulae. Apart from that, we derive solution of a global optimization problem associated with the problem of computing of the strong stability radii of a positive linear time‐delay system. An example is given to illustrate the obtained results. Copyright © 2005 John Wiley & Sons, Ltd.     

Finite-time stability and stabilization of time-delay systems

Finite-time stability and stabilization of retarded-type functional differential equations are developed. First, a theoretical result on finite-time stability inspired by the theory of differential equations, using Lyapunov functionals, is given. As it may appear not easily usable in practice, we show how to obtain finite-time stabilization of linear systems with delays in the input by using an extension of Artstein’s model reduction to nonlinear feedback. With this approach, we give an explicit finite-time controller for scalar linear systems and for the chain of integrators with delays in the input.     

Optimal control of Volterra type stochastic difference equations

Many processes in automatic regulation, physics, etc. can be modelled by stochastic difference equations. One of the main problems of the theory of difference equations and their applications is connected with stability and optimal control [1]. In this paper we discuss the optimal control of second-kind Volterra type stochastic difference equations. In [2–9] for Volterra type stochastic integral equations, analogous results were obtained.     

Stability radii of positive linear systems under fractional perturbations

In this paper we study stability radii of positive linear discrete‐time systems under fractional perturbations. It is shown that real and complex stability radii coincide and can be computed by a simple formula. From the obtained results, we apply to derive estimates and computable formulae for the stability radii of positive linear delay systems. Finally, a simple example is given to illustrate the obtained results. Copyright © 2008 John Wiley & Sons, Ltd.     

On exponential stability of linear non-autonomous functional differential equations of neutral type

General linear non-autonomous functional differential equations of neutral type are considered. A novel approach to exponential stability of neutral functional differential equations is presented. Consequently, explicit criteria are derived for exponential stability of linear non-autonomous functional differential equations of neutral type. A brief discussion to the obtained results and illustrative examples are given.     

Markov跳变系统的随机稳定性和鲁棒性

对一类由Markov过程驱动的跳变性系统,给出均方差渐近／指数／随机稳定的线性矩阵不等式（LinearMatrixInequality,LMI）刻画。与已有的稳定性判据相比,论文提出的判据更方便验证。在此基础上,探讨在子系统和转移概率矩阵受扰时的鲁棒稳定问题,给出鲁棒裕度的显示估计。     

Euler-like discrete models of the logistic differential equation

To understand the behavior of difference schemes on nonlinear differential equations, it seems desirable to extend the standard linear stability theory into a nonlinear theory. As a step in that direction, we investigate the stability properties of Euler-related integration algorithms by checking how they preserve and violate the dynamical structure of the logistic differential equation.Among the schemes considered are two linearly implicit nonstandard schemes which are adjoint to each other. We find that these schemes are superior to explicit schemes when they are stable and the blow-up time has not passed: for these λh-values they are dynamically faithful. When these schemes ‘turn unstable’, however, they have much less desirable properties than explicit or fully implicit schemes: they become simultaneously superstable and unstable. This is explained by the fact that these schemes are not self-adjoint: the linearly implicit self-adjoint scheme is dynamically faithful in an Euler-typical range of step sizes and gives correct stability for all step sizes.     

Convergence to equilibria in recurrence equations

In this paper, we deal with linear and nonlinear perturbations of first-order recurrence systems with constant coefficients having infinitely many equilibria. We give sufficient conditions for the asymptotic constancy of the solutions of the perturbed equation. As a consequence of our main theorem, we obtain sufficient conditions for systems of higher-order difference equations to have asymptotic equilibrium.     

Fractional convolution quadrature based on generalized Adams methods

In this paper we present a product quadrature rule for Volterra integral equations with weakly singular kernels based on the generalized Adams methods. The formulas represent numerical solvers for fractional differential equations, which inherit the linear stability properties already known for the integer order case. The numerical experiments confirm the valuable properties of this approach.     

Exponential stability criteria for discrete time‐delay systems

In this paper, we study the exponential stability of linear discrete time‐delay systems with slowly varying coefficients and nonlinear perturbations. We establish the robustness of the exponential stability in Hilbert spaces, in the sense that the exponential stability for a given linear equation persists under sufficiently small perturbations. As an application of the main results, we discuss the exponential stability of a general nonlinear system. The main novelty of this work is that we always consider the exponential behavior of solutions with respect to an specific ball. Copyright © 2013 John Wiley & Sons, Ltd.     

Finite difference method for time–space linear and nonlinear fractional diffusion equations

ABSTRACT

In this paper a finite difference method is presented to solve time–space linear and nonlinear fractional diffusion equations. Specifically, the centred difference scheme is used to approximate the Riesz fractional derivative in space. A trapezoidal formula is used to solve a system of Volterra integral equations transformed from spatial discretization. Stability and convergence of the proposed scheme is discussed which shows second-order accuracy both in temporal and spatial directions. Finally, examples are presented to show the accuracy and effectiveness of the schemes.     

线性时变系统的区间稳定性和鲁棒稳定性

用矩阵测度研究了线性时变系统的区间稳 定性和具有非线性时变摄动的线性时变系统的鲁棒稳定性,得到了它们稳定的判别准则,所 得结果与文\[1~5\]的结果互不包含．