Relative ageing for two parallel systems and related problems

Kalashnikov and Rachev [1] have proposed a partial ordering of two life distributions which is equivalent to an increasing hazard (failure rate) ratio, when the ratio exists. The phenomenon of crossing hazards has received considerable attention in recent years. Recently, Sengupta and Deshpande [2) have studied this and two other models of relative ageing. In this paper, we consider the relative ageing properties of two parallel systems with identical but different number of components. We also compare the variances of the two life distributions having the same mean but with increasing hazard ratio. Several examples are given to illustrate the results.     

Analysis of reliability systems via Gini-type index

Different strategies of reliability theory for the analysis of coherent systems have been studied by various researchers. Here, the Gini-type index is utilized as an applicable tool for the study and comparison of the ageing properties of complex systems. A new stochastic order in terms of Gini-type index is introduced to compare the speed of ageing of components and systems. The parallel-series and series-parallel systems with shared components are studied by their corresponding Gini-type indexes. Also, the generalization of Gini-type index for the multidimensional case is discussed, and is used to compare components lifetimes properties in the presence of other dependent components. It is shown that the ageing properties of a component lifetime can differ when the other components are working or have already failed. Numerous illustrative examples are given for better intuition of Gini-type and generalized Gini-type indexes throughout the paper.     

The solution of contact problems of creep theory for combined ageing foundations

Solutions are presented of certain plane and axisymmetric contact problems on the frictionless impression of a rigid stamp into a two-layered ageing viscoelastic foundation. It is assumed that the upper layer is thin relative to the contact domain, and inhomogeneously ageing. The rheological properties of the lower layer are described by the equations of linear creep theory for ageing materials. The layers are mutually rigidly adherent. The contact domain does not change with time. Depending on the relationships between the moduli of the instantaneous elastic strains of the layers, the mixed problems reduce to integral equations of the first or second kinds containing Fredholm and Volterra operators. An analytic method is proposed for solving such equations which enables an expansion to be obtained for the fundamental characteristics of the contact interaction for a force varying with time in an arbitrary manner and acting on the stamp. Cases are investigaged for the artificial and natural ageing of a two-layer foundation.     

Some results on the proportional reversed hazards model

The proportional reversed hazards model consists in describing random failure times by a family {[F(x)]^θ, θ>0} of distribution functions, where F(x) is a baseline distribution function. We show various results on this model related to some topics in reliability theory, including ageing notions of random lifetimes, comparisons based on stochastic orders, and relative ageing of distributions.     

Dynamically similar control systems and a globally minimum gain control technique: IMSC

The concept of dynamically similar control systems is introduced. The necessary and sufficient conditions to minimize a quadratic modal gain measure are given for dynamically similar closed-loop control systems. The globally minimum modal gain is obtained when the independent modal space control (IMSC) is used. Corollaries of the results for the control of infinite-dimensional structural distributed parameter systems (DPS) are given. Based on the results, a modal interaction parameter (MIP) is defined for all control systems. The minimum value of MIP is zero and uniquely corresponds to the IMSC. A nonzero value of MIP corresponds to all other coupled control (CC) designs and implies suboptimality relative to the IMSC design. The relative optimality of the real-space gain matrices of the IMSC and the CC designs depends on the actuator locations for the IMSC. Based on this, a real-space interaction parameter (RIP) is defined. A positive value of RIP renders IMSC optimal in its real-space gain matrix. The MIP and RIP are indications of suboptimality of a particular control technique and can be used to tune-up the control design via actuator locations. Actuator distribution criteria are suggested for both CC and IMSC designs, based on the values of MIP and RIP, respectively.This work was supported by the National Science Foundation, Grant No. MEA-82-04920.     

Relative priority policies for minimizing the cost of queueing systems with service discrimination

This paper considers several single-server two-class queueing systems with different cost functions. Customers in the two classes are discriminated by service rates and relative priorities. Most attention is focused on the ones with general quadratic bivariable and exponential cost functions that are usually applied in the relatively complicated systems. To the best of the authors’ knowledge, there is no literature analyzing these two kinds of cost functions on the subject of relative priority. We explicitly present the conditions under which relative priority outperforms absolute priority for reducing system cost and further provide the method to find the optimal DPS policy. Moreover, we also discuss variations where service rates of the two classes are decision variables under service equalization and service discrimination disciplines, respectively.     

Adaptive robust control of artificial swarm systems

We consider an artificial swarm system consisting of multi-agents. The agents may interact with each other based on their relative positions. Each agent exhibits a repulsion/attraction behavior toward another agent, which mimics some biological swarm systems. The performance of each individual agent is the accumulation of these respective considerations toward other agents. The overall performance of the artificial swarm system mimics the aggregation and formation in biological systems. We propose an adaptive robust control for each agent toward achieving the performance. The control can withstand uncertainty, which is time-varying, nonlinear, and without known bound. The controlled system converges to the desirable swarm system performance regardless of the uncertainty.     

On the stability of inhomogeneously viscoelastic reinforced bars

There is investigated the stability of inhomogeneously ageing reinforced viscoelastic bars. It is assumed that the strains and stresses in the reinforcement are related by Hooke's law. The properties of the matrix material are described by equations of the theory of viscoelasticity of inhomogeneously ageing solids /1,2/. Under different boundary conditions for the ends of the bar and loading methods an expression is set up for the critical force in stability problems in an infinite time interval. The stability definition taken corresponds to the Liapunov stability definition for the motion of dynamical systems. Estimates of the critical time when the magnitude of the deflection of a viscoelastic bar reaches a given value are obtained for stability problems in a finite time interval. The formulation for the stability problem in a finite time interval starts from the definition of stability of motion of dynamical systems by taking its beginning from the Chetaev work. The dependence of the critical time on the inhomogeneity and the reinforcing parameter is investigated numerically. The stability of viscoelastic unreinforced bars was studied in /3,4/, A survey and bibliography of research associated with the stability problem for viscoelastic bars are available in /5–8/.     

On the stability of inhomogeneously ageing viscoelastic plates

A method of investigation is proposed and conditions are set up for the stability of viscoelastic inhomogeneously ageing plates of arbitrary shape with a common creep kernel. The form of the stability conditions is found as a function of the surface forces. The stability problem is examined numerically in a finite time interval. The paper touches on the investigations in /1–3/. (See the bibliography of research on the stability of homogeneous viscoelastic systems in /1–5/, for example.)     

小参数时变非线性系统的技术稳定性

分析一类含小参数的时变非线性系统关于给定状态约束集合的技术稳定性.根据向量微分比较原理和基本的单调性准则,利用向量V函数方法给出由系统系数表达的技术稳定性判据.并讨论了基于派生系统和线性化方法研究非线性系统技术稳定性的条件.另外,对于派生时变线性系统的指数稳定性给出了简单的代数判据.最后给出示例说明文中方法.     

Analysis of stochastic min-max-plus systems: Results and conjectures

Systems in which the operations min, max and addition appear simultaneously are called min-max-plus systems. Such systems, which are extensions of timed discrete event systems (which on their turn are based on the max-plus algebra, i.e., on the operations max and addition only), have been studied for some years now [1–3]. In these references only deterministic systems were studied. In the current paper, some stochastic extensions will be considered. It will be shown that extensions of eigenvalues, Lyapunov coefficients, exist for these stochastic systems. Some conjectures will be given which are supported by characteristic examples.     

对一类非线性扰动方程的分析

本文研究了一类非线性扰动方程，确定自治扰动方程双曲不动点的稳定和不稳定流形的相对位置，给出了存在唯一极限环的参数范围，并应用于电机工程和力学中一类非线性系统。     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

On the properties of small sample of GM(1,1) model

In grey prediction modeling, the more samples selected the more errors. This paper puts forward new explanations of “incomplete information and small sample” of grey systems and expands the suitable range of grey system theory. Based on the geometric sequence, it probes into the influence on the relative errors by selecting the different sample sizes. The research results indicate that to the non-negative increasing monotonous exponential sequence, the more samples selected, the more average relative errors. To the non-negative decreasing monotonous exponential sequence, a proper sample number exists that has the least average relative error. When the initial value of the sequence of raw data of new information GM(1,1) model changes, the development coefficient remains unchanged. The segmental correction new information GM(1,1) model (SNGM) can obviously improve the simulation accuracy. It puts forward the mathematic proofs that the small sample usually has more accuracy than the large sample when establishing GM(1,1) model in theory.     

Relative approximate controllability of abstract functional systems with infinite delay and delayed control

This paper is concerned with the relative approximate controllability of functional systems with infinite delay and delayed control in Hilbert spaces. In particular, we begin with studying some criteria of the controllability of linear systems. Based on those results, sufficient conditions are derived for the relative approximate controllability of nonlinear functional systems. Finally, an example is included to illustrate the effectiveness of the proposed methods. Copyright © 2016 John Wiley & Sons, Ltd.     

Stability analysis for a class of functional differential equations and applications

The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov-Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems.     

Ageing concepts: An approach based on quantile function

The notion of ageing plays an important role in reliability analysis and in identifying life distributions. Most of the ageing concepts existing in the literature are described on the basis of measures defined in terms of the distribution function. Recently, the role of quantile functions has also been identified as lifetime models, and reliability functions based on distribution functions were redefined in terms of quantile functions. In the present paper, we redefine some important popular ageing concepts using quantile functions. The uses of new definitions are illustrated by discussing ageing properties of some quantile function models.     

Existence of infinitely many homoclinic orbits to Aubry sets for positive definite Lagrangian systems

In this paper, we study the problem of homoclinic orbits to Aubry sets for time-periodic positive definite Lagrangian systems. We show that there are infinitely many homoclinic orbits to some Aubry set under the conditions that the associated Mather set is uniquely ergodic and the first relative homology group of the projection of this Aubry set is nonzero.