Stability analysis of a new kind n-unit series repairable system

In this paper, we deal with a new model of an n-unit series repairable system, in which a concept of a repairman with multiple-delayed vacation is introduced and the impact on the system reliability due to a replaceable facility is also considered. This paper is devoted to studying the unique existence and stability of the system solution. C₀-semigroup theory is used to prove the existence of a unique nonnegative time-dependent solution of the system. Then by analyzing the spectra distribution of the system operator, we prove that the dynamic solution of the system asymptotically converges to the nonnegative steady-state solution which is the eigenfunction corresponding to eigenvalue 0 of the system operator. Furthermore, we discuss the exponential stability of the system in a special case. Some reliability indices of the system are also studied and the optimal vacation time is analyzed at the end of the paper.     

具有分布反馈和边界反馈的非均质Timoshenko梁的指数镇定性

讨论具有分布反馈控制和边界反馈控制的非均质Timoshenko梁的指数镇定问题.首先利用已有的关于线性分布参数系统的渐进稳定性判据,证明所论梁系统的能量可仅由一个分布反馈控制指数镇定.进而利用频域分片乘子方法,在所论梁系统同时具有分布反馈控制和边界反馈控制的条件下,证明其相应的闭环系统能量指数稳定.     

Asymptotic Stability of a Stationary Flowing Regime of an Ideal Incompressible Fluid

We give sufficient conditions for asymptotic stability of a stationary solution to a flowing problem of a homogeneous incompressible fluid through a given planar domain. We consider a planar problem for the Euler equation and boundary conditions for the curl and the normal component of the velocity; moreover, the latter is given on the whole boundary of the flow domain and the curl is given only on the inlet part of the boundary. We establish asymptotic stability of a stationary flow (in linear approximation), assuming it to have no rest points and to satisfy some smallness condition which means that the perturbations leave the flow domain before they become to affect the main flow. In particular, we prove asymptotic stability for an arbitrary stationary flow in a rectangular canal close to the Couette flow without rest points. Moreover, we show that stability of the main flow in the L ₂-norm under curl perturbations implies its stability in higher-order norms depending, for example, on the derivatives of the curl.     

Existence of positive solutions to n-point nonhomogeneous boundary value problem

In this paper, we are concerned with the existence of positive solutions to a n-point nonhomogeneous boundary value problem. By using the Krasnoselskii's fixed point theorem in Banach spaces, some sufficient conditions guaranteeing the existence of positive solution is established for the n-point nonhomogeneous boundary value problem.     

Exponential asymptotic property of a parallel repairable system with warm standby under common-cause failure

We discuss exponential asymptotic property of the solution of a parallel repairable system with warm standby under common-cause failure. This system can be described by a group of partial differential equations with integral boundary. First we show that the positive contraction C₀-semigroup T(t) [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] which is generated by the operator corresponding to these equations is a quasi-compact operator. Then by using [Weiwei Hu, Asymptotic stability analysis of a parallel repairable system with warm standby under common-cause failure, Acta Anal. Funct. Appl. 8 (1) (2006) 5-20] that 0 is an eigenvalue of the operator with algebraic index one and the C₀-semigroup T(t) is contraction, we conclude that the spectral bound of the operator is zero. By using the above results the exponential asymptotical stability of the time-dependent solution of the system follows easily.     

Exponential stability for a nonlinear one-dimensional heat-conductive viscous real gas

In this paper, we prove the existence and the exponential stability in H₊ⁱ (i=1,2), an incomplete metric subspace of Hⁱ×Hⁱ×Hⁱ (i=1,2), of a nonlinear C₀-semigroup S(t) for a nonlinear one-dimensional heat-conductive viscous real gas in bounded domain Ω=(0,1). The results in this paper improve those previously related results.     

Nonlinear problems arising in electrostatic actuation in MEMS

In this paper we study the nonlinear problem arising in electrostatic actuation of MEMS. We show that the existence and non-existence of the solution of this problem depend on the value of the physical parameters of the equation. In addition we consider the corresponding initial value problem and we derived the existence of periodic solution, stability of steady states and the ω-limit set.     

Convergence rate of the solution toward boundary layer solution for initial-boundary value problem of the 2-D viscous conservation laws

In this paper, we study the large time behavior of the solution to the initial boundary value problem for 2-D viscous conservation laws in the space x ? bt. The global existence and the asymptotic stability of a stationary solution are proved by Kawashima et al. [1]. Here, we investigate the convergence rate of solution toward the boundary layer solution with the non-degenerate case where f′(u₊) − b < 0. Based on the estimate in the H² Sobolev space and via the weighted energy method, we draw the conclusion that the solution converges to the corresponding boundary layer solution with algebraic or exponential rate in time, under the assumption that the initial perturbation decays with algebraic or exponential in the spatial direction.     

Explicit solution of the finite time L2-norm polynomial approximation problem

The aim of this paper is to present a new approach to the finite time L₂-norm polynomial approximation problem. A new formulation of this problem leads to an equivalent linear system whose solution can be investigated analytically. Such a solution is then specialized for a polynomial expressed in terms of Laguerre and Bernstein basis.     

Robust H∞ control for linear Markovian jump systems with unknown nonlinearities

This paper studies the problem of stochastic stability and disturbance attenuation for a class of linear continuous-time uncertain systems with Markovian jumping parameters. The uncertainties are assumed to be nonlinear and state, control and external disturbance dependent. A sufficient condition is provided to solve the above problem. An H_∞ controller is designed such that the resulting closed-loop system is stochastically stable and has a disturbance attenuation γ for all admissible uncertainties. It is shown that the control law is in terms of the solutions of a set of coupled Riccati inequalities. A numerical example is included to demonstrate the potential of the proposed technique.     

Stabilization of nonuniform Euler-Bernoulli beam with locally distributed feedbacks

In this article, we study the stabilization problem of a nonuniform Euler-Bernoulli beam with locally distributed feedbacks. Firstly, using the semi-group theory, we establish the well-posedness of the associated closed loop system. Then by proving the uniqueness of the solution of a related ordinary differential equations, we derive the asymptotic stability of the closed loop system. Finally, by means of the piecewise frequency domain multiplier method, we prove that the corresponding closed loop system can be exponentially stabilized by only one of the two distributed feedback controls proposed in this paper.     

The well-posedness and stability of a repairable standby human–machine system

In this paper, the solution of a standby human–machine system is investigated. By using the method of functional analysis, especially, the linear operator theory and the C₀ semigroup theory on Banach space, we prove the well-posedness and the existence of a positive solution of the system. And under some appropriate hypotheses, we study the asymptotic stability of solution of the system.     

Positive solution for m-point boundary value problems

In this paper, we investigate an m-point boundary value problem with sign changing nonlinearity. The existence of an interval of parameters which ensures the problem has at least one positive solution is determined by constructing available operator and combining the method of lower solution with the method of topology degree. Moreover, the associated Green's function for the above problem is also given.     

Multiple positive solutions of nonhomogeneous elliptic systems with strongly indefinite structure and critical Sobolev exponents

In this paper, we study the existence of multiple positive solutions to some Hamiltonian elliptic systems −Δv=λu+u^p+εf(x), −Δu=μv+v^q+δg(x) in Ω;u,v>0 in Ω; u=v=0 on ∂Ω, where Ω is a bounded domain in R^N (N?3); 0?f, g∈L∞(Ω); 1/(p+1)+1/(q+1)=(N−2)/N, p,q>1; λ,μ>0. Using sub- and supersolution method and based on an adaptation of the dual variational approach, we prove the existence of at least two nontrivial positive solutions for all λ,μ∈(0,λ₁) and ε,δ∈(0,δ₀), where λ₁ is the first eigenvalue of the Laplace operator −Δ with zero Dirichlet boundary conditions and δ₀ is a positive number.     

A semigroup approach to the Gnedenko system with single vacation of a repairman

We investigate the Gnedenko system with one repairman who can take vacations. Our main focus is on the time asymptotic behaviour of the system. Using C ₀-semigroup theory for linear operators we first prove the well-posedness of the system and the existence of a unique positive dynamic solution given an initial value. Then by analysing the spectral distribution of the system operator and taking into account the irreducibility of the semigroup generated by the system operator we show that the dynamic solution converges strongly to the steady state solution. Thus we obtain asymptotic stability of the dynamic solution.     

Generalized vector quasi-equilibrium problems with applications

In this paper, we consider the generalized vector quasi-equilibrium problem with or without involving Φ-condensing maps and prove the existence of its solution by using known fixed point and maximal element theorems. As applications of our results, we derive some existence results for a solution to the vector quasi-optimization problem for nondifferentiable functions and vector quasi-saddle point problem.     

Asymptotic stability of the rarefaction wave for the generalized KdV–Burgers–Kuramoto equation

In this paper, we will study the Cauchy problem for the generalized KdV–Burgers–Kuramoto equation, which represents a dissipative, stroboscopic and unstable system in physics. When the initial data is a small disturbance of a rarefaction wave of the inviscid Burgers equation, we prove the global existence of the solution to the corresponding Cauchy problem and asymptotic stability of the rarefaction wave. The analysis is based on a priori estimates and the L²$L^2$-energy method.     

Collective fixed points and maximal elements with applications to abstract economies

In this paper, we first establish collective fixed points theorems for a family of multivalued maps with or without assuming that the product of these multivalued maps is Φ-condensing. As an application of our collective fixed points theorem, we derive the coincidence theorem for two families of multivalued maps defined on product spaces. Then we give some existence results for maximal elements for a family of L_S-majorized multivalued maps whose product is Φ-condensing. We also prove some existence results for maximal elements for a family of multivalued maps which are not L_S-majorized but their product is Φ-condensing. As applications of our results, some existence results for equilibria of abstract economies are also derived. The results of this paper are more general than those given in the literature.     

An obstacle problem for nonlinear hemivariational inequalities at resonance

In this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p-Laplacian. Using a variational approach based on the nonsmooth critical point theory for locally Lipschitz functionals defined on a closed, convex set, we prove two existence theorems. In the second theorem we have a pointwise interpretation of the obstacle problem, assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion.     

Nonlinear eigenvalue Neumann problems with discontinuities

In this paper we study nonlinear eigenvalue problems with Neumann boundary conditions and discontinuous terms. First we consider a nonlinear problem involving the p-Laplacian and we prove the existence of a solution for the multivalued approximation of it, then we pass to semilinear problems and we prove the existence of multiple solutions. The approach is based on the critical point theory for nonsmooth locally Lipschitz functionals.