Stability of Runge-Kutta methods for delay differential systems with multiple delays

The stability of Runge-Kutta methods for systems of delay differentialequations (DDEs) with multiple delays is considered. The stabilityregions of explicit and implicit Runge-Kutta methods are discussedwhen they are applied to asymptotically stable linear DDEs withmultiple delays. A simple estimate on the stability regionsof explicit Runge-Kutta methods is presented. It is shown thatthe stable step-size for numerical integration of DDEs withmultiple delays can be easily selected by means of the estimate.     

Estimation of numerically stable step-size for neutral delay-differential equations via spectral radius

We derive the estimates of numerically stable step-size for systems of neutral delay-differential equations (NDDEs), which only need to be calculated the spectral radius of the corresponding matrices. The stable step-size for numerical integration of NDDEs can be easily selected by means of the estimates. The stability regions of both linear multistep methods and explicit Runge-Kutta methods are presented.     

New higher-order numerical one-step methods based on the Adomian and the modified decomposition methods

We develop new, higher-order numerical one-step methods and apply them to several examples to investigate approximate discrete solutions of nonlinear differential equations. These new algorithms are derived from the Adomian decomposition method (ADM) and the Rach-Adomian-Meyers modified decomposition method (MDM) to present an alternative to such classic schemes as the explicit Runge-Kutta methods for engineering models, which require high accuracy with low computational costs for repetitive simulations in contrast to a one-size-fits-all philosophy. This new approach incorporates the notion of analytic continuation, which extends the region of convergence without resort to other techniques that are also used to accelerate the rate of convergence such as the diagonal Padé approximants or the iterated Shanks transforms. Hence global solutions instead of only local solutions are directly realized albeit in a discretized representation. We observe that one of the difficulties in applying explicit Runge-Kutta one-step methods is that there is no general procedure to generate higher-order numeric methods. It becomes a time-consuming, tedious endeavor to generate higher-order explicit Runge-Kutta formulas, because it is constrained by the traditional Picard formalism as used to represent nonlinear differential equations. The ADM and the MDM rely instead upon Adomian’s representation and the Adomian polynomials to permit a straightforward universal procedure to generate higher-order numeric methods at will such as a 12th-order or 24th-order one-step method, if need be. Another key advantage is that we can easily estimate the maximum step-size prior to computing data sets representing the discretized solution, because we can approximate the radius of convergence from the solution approximants unlike the Runge-Kutta approach with its intrinsic linearization between computed data points. We propose new variable step-size, variable order and variable step-size, variable order algorithms for automatic step-size control to increase the computational efficiency and reduce the computational costs even further for critical engineering models.     

Variable stepsize implementation of multistep methods for y″=f(x,y,y′)

The Falkner method is a multistep scheme intended for the numerical solution of second-order initial value problems where the first derivative does appear explicitly. In this paper, we develop a procedure to obtain k-step Falkner methods (explicit and implicit) in their variable step-size versions, providing recurrence formulas to compute the coefficients efficiently. Considering a pair of explicit and implicit formulae, these may be implemented in predictor–corrector mode.     

Variable step-size implementation of sixth-order Numerov-type methods

The explicit sixth-order Numerov-type family of methods is considered. A new representative from this family is produced and equipped with a cheap step-size changing algorithm. Actually, after the completion of a step, this remains the same, halved, or doubled. The off-step points required for such technique are evaluated through local interpolant. Numerical tests over various problems illustrate the efficiency gained by this approach.     

Runge-Kutta方法关于时滞奇异摄动问题的误差分析

１．引言 用（,）表示Ｅｕｃｌｉｄｅａｎ空间的内积,｜｜·｜｜为相应范数,考虑时滞奇异摄动问题（ＳＰＰＤｓ）这里。∈,ｒ（ｒ＞０）是常数,　和　是给定的函数,ｆ：　　　　　　　　　　　和　　　　　　　　　　　　　　是给定的充分光滑的映射,它们满足下面的条件这里ｗ１和－ｗ２是具有适度大小的常数且　　　　　　　　　分别关于其它变量满足 Ｌｉｐｓｃｈｉｔｚ 条件．不失一般性,假设ｗ２＝１（参见［１］） 与经典 Ｌｉｐｓｃｈｉｔｚ条件相比,条件（１．２ａ）更弱．事实上,当（１．３）中的 Ｌ具有适度大小时,就能…     

Global Exponential Stability in Discrete-time Analogues of Delayed Cellular Neural Networks

A novel method called semi-discretization is employed in the formulation of discrete-time analogues of nonlinear delayed differential equations modelling cellular neural networks. The dynamical characteristics of the discrete-time analogues are studied. When the network parameters satisfy certain sufficient conditions which are independent of the delays, the discrete-time analogues for any choice on the discretization step-size are shown to be globally exponentially stable. The sufficient conditions are obtained by employing an appropriate form of Lyapunov sequences and these conditions correspond to those which have been obtained in the literature for the global exponential stability of continuous-time delayed cellular neural networks. Several examples and computer simulations are given to support our results and to demonstrate some of the advantages of the discrete-time analogues in numerically simulating their continuous-time counterparts.     

Stability of solutions to differential equations with several variable delays. II

We consider a class of scalar linear differential equations with several variable delays and constant coefficients. We treat coefficients and maximum admissible values of delays as parameters that define a family of equations from the class under consideration. We study domains in the parameter space, where fundamental solutions of all equations of the family are uniformly or exponentially stable and have a fixed sign. We establish explicit necessary and sufficient conditions for the stability and sign-definiteness of the equations family.     

Gradient Methods with Adaptive Step-Sizes

Motivated by the superlinear behavior of the Barzilai-Borwein (BB) method for two-dimensional quadratics, we propose two gradient methods which adaptively choose a small step-size or a large step-size at each iteration. The small step-size is primarily used to induce a favorable descent direction for the next iteration, while the large step-size is primarily used to produce a sufficient reduction. Although the new algorithms are still linearly convergent in the quadratic case, numerical experiments on some typical test problems indicate that they compare favorably with the BB method and some other efficient gradient methods.     

Asymptotic constancy for a linear differential system with multiple delays

In this letter we consider a linear differential system with multiple delays which has nonisolated equilibria. In order to study the asymptotic behavior of linear delay differential equations, characteristic equations are generally used. But it is hard to establish the properties of zeros of the characteristic equations, especially if there are multiple time delays. So we use the invariance principle combined with two functionals to show whether any solutions converge. One of the functionals plays the role of a Lyapunov functional, and the other is a conserved quantity. Furthermore we give explicit expressions for the limits of the solutions by using the conserved quantity.     

Conditional contractivity of Runge–Kutta methods for nonlinear differential equations with many variable delays

This paper deals with conditional contractivity properties of Runge–Kutta (RK) methods with variable step-size applied to nonlinear differential equations with many variable delays (MDDEs). The concepts of CRN_m(ω, H)- and BN_f(μ, ?)-stability are introduced. It is shown that the numerical solution produced by a BN_f(μ, ?)-stable Runge–Kutta method with an appropriate interpolation is contractive. In particular, these results are also novel for nonlinear differential equations with many constant delays or single variable delay. To obtain BN_f(μ, ?)-stable methods, (k, l)-algebraically stable Runge–Kutta methods are also investigated.     

Sufficient conditions for robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections

The problem of the robust stability of large-scale dynamical systems including delayed states and parameter perturbations in interconnections is considered. By using algebraic Riccati equations and some analytical methods, some sufficient conditions on linear decentralized state feedback controllers are derived so that the systems remain stable in the presence of delayed states and parameter perturbations. Such conditions give some bounds on the perturbations of interconnections with delayed states and uncertain parameters, and result in a quantitative measures of robustness for large-scale dynamical systems including delayed states and uncertain parameters in interconnections. The results obtained in this paper are applicable not only to large-scale systems with multiple time-varying delays, but also to large-scale systems without exact knowledge of the delays, i.e., large-scale systems with uncertain delays.     

Stability analysis for networked control systems with sampling,transmission protocols and input delays

In this paper, the stability problem is investigated for networked control systems. Input delays and multiple communication imperfections containing time-varying transmission intervals and transmission protocols are considered. A unified framework based on the hybrid systems with memory is proposed to model the whole networked control system. Hybrid systems with memory are used to model hybrid systems affected by delays and permit multiple jumps at a jumping instant. The stability analysis depends on the Lyapunov–Krasovskii functional approaches for hybrid systems with memory and the proposed stability theorem does not need strict decrease of the Lyapunov–Krasovskii functional during jumps. Based on the developed stability theorems, stability conditions for networked control systems are established. An explicit formula is given to compute the maximal allowable transmission interval. In the special case that the networked control system contains linear dynamics, an explicit Lyapunov functional is constructed and stability conditions in terms of linear matrix inequalities (LMI) are proposed. Finally, an example of a chemical batch reactor is given to illustrate the effectiveness of the proposed results.     

An asymptotic expansion for product integration applied to Cauchy principal value integrals

Product integration methods for Cauchy principal value integrals based on piecewise Lagrangian interpolation are studied. It is shown that for this class of quadrature methods the truncation error has an asymptotic expansion in integer powers of the step-size, and that a method with an asymptotic expansion in even powers of the step-size does not exist. The relative merits of a quadrature method which employs values of both the integrand and its first derivative and for which the truncation error has an asymptotic expansion in even powers of the step-size are discussed.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

具时变时滞和 Markovian转换的不确定奇异随机系统的鲁棒 H_∞ 滤波(英文)

本文处理了一类具与模式有关的时变时滞和 Markovian转换的不确定奇异随机系统的鲁棒H∞滤波问题.所考虑的系统包含参数不确定性,Markovian参数,随机扰动和与模式有关的时变时滞.本文的目的是设计一个滤波器以保证滤波错误系统是正则的、无脉冲的、鲁棒指数均方稳定的和可达到一个给定的 H∞扰动衰减水平.文章首先得到所求鲁棒指数H∞滤波器存在的充分条件,然后给出所求滤波器参数的显示表示.     

Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics

We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An embedded 3(2) pair of nonlinear methods for solving first order initial-value ordinary differential systems

In this paper, we present two families of second-order and third-order explicit methods for numerical integration of initial-value problems of ordinary differential equations. Firstly, a family of second-order methods with two free parameters is derived by considering a suitable rational approximation to the theoretical solution of the problem at some grid points. Imposing that the principal term of the local truncation error of this family vanishes, we obtain an expression for one of the parameters in terms of the other. With this approach, a new one-parameter family of third-order methods is obtained. By selecting any 3(2) pair of second and third order methods, they can be implemented as an embedded type method, thus leading to a variable step-size formulation. We have considered one 3(2) pair of second and third order methods and made a comparison of numerical results with several ode solvers which are currently used in practice. The comparison of numerical results shows that the embedded 3(2) pair outperforms the methods considered for comparison.     

On adaptive step-size selections in gradient methods

In recent years several proposals for the step-size selection have largely improved the gradient methods, in the case of both constrained and unconstrained nonlinear optimization. We introduce a new step-size rule with some crucial properties. We design step-size selection strategies where the new rule and a standard Barzilai-Borwein (BB) rule can be adaptively alternated to get meaningful convergence rate improvements in comparison with other BB-like gradient schemes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.