A Numerical Method for Establishing Liapunov Stability of Linear Second Order Non-autonomous Dynamical Systems

A numerical method which basically utilizes the Liapunov DirectMethod is presented which establishes sufficient conditionsfor the stability of the null solution of the nonautonomousdifferential equation x+b(t)x+a(t)x = 0. This is accomplishedwithout the unstructured task of searching for given Liapunovfunctions for the particular problem being analysed. The methodis simple and straightforward to apply, and can be used to obtainstability results as long as b is not identically zero. Thenumerical approach is demonstrated by applying it to the dampedMathieu equation, the Euler equation, and an equation with quasiperiodiccoefficients.     

Weighted Positivity of Second Order Elliptic Systems

Integral inequalities that concern the weighted positivity of a differential operator have important applications in qualitative theory of elliptic boundary value problems. Despite the power of these inequalities, however, it is far from clear which operators have this property. In this paper, we study weighted integral inequalities for general second order elliptic systems in ℝ ⁿ (n ≥ 3) and prove that, with a weight, smooth and positive homogeneous of order 2–n, the system is weighted positive only if the weight is the fundamental matrix of the system, possibly multiplied by a semi-positive definite constant matrix.      

Quadratic Eigenvalue Problems for Second Order Systems

We consider the spectral structure of a quadratic second order system boundary-value problem. In particular we show that all but a finite number of the eigenvalues are real and semi-simple. We develop the eigencurve theory for these problems and show that the order of contact between an eigencurve and the parabola gives the Jordan chain associated with the eigenvector corresponding to that eigencurve. Following this we use our knowledge of the eigencurves to obtain eigenvalue asymptotics. Finally the completeness of the eigenfunctions is studied using operator matrix techniques. It should be noted here that the usual left definiteness assumptions have been overcome in this study.     

Attractors for Second Order Lattice Dynamical Systems

We consider the existence and the approximation of the global attractor for second order damped lattice dynamical systems.     

Numerical-Analytic Methods for Differential-Algebraic Systems

The numerical-analytic method is useful to show that a differential equation with a boundary condition has a solution. In this paper we use this method to differential-algebraic systems with boundary conditions. Some existence results are formulated under assumptions that the corresponding functions satisfy Lipschitz conditions in matrix notation.     

Periodic Solutions of Nonautonomous Second Order Hamiltonian Systems

In this paper, we develop the local linking theorem given by Li and Willein by replacing the Palais-Smale condition with a Cerami one, and apply it to the study of the existence of periodic solutions of the nonautonomous second order Hamiltonian systems （H） ü＋A（t）u＋∨V（t, u）=0, u∈R^N, t∈R. We handle the case of superquadratic nonlinearities which differ from those used previously. Our results extend the theorems given by Li and Willem.     

半线性二阶系统的奇异摄动现象*

本文研究半线性二阶系统的奇异摄动现象,在适当的假设下,借助于构造特殊的不变域,证得向量边值问题解的存在及其当ε→0+时的渐近性质,在这种不变域中解呈现所谓的边界层现象和角层现象.     

Error-Controlled Model Order Reduction of Second Order Systems with Adaptive Choice of Shifts and Order

This contribution summarizes a novel approach to error-controlled model order reduction of second order systems, i. e. a new way to find a reduced order model of specified approximation quality. To this end, starting from a particular state space realization of the model, one performs multiple reduction steps and increments the reduced model until (computationally affordable) rigorous global error bounds fall below the prescribed limit. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

某些二阶系统正解的存在性

考察如下边值问题正解的存在性x″(t) λa(t) f (x(t) ,y(t) ) =0y″(t) λb(t) g(x(t) ,y(t) ) =0x(0 ) =x(1 ) =y(0 ) =y(1 ) =0其中 f ,g:R × R R ;a,b:[0 ,1 ] R .所有的函数都被假定是连续的 ,此外 f ,g满足某些增长性条件 .本文得到了一些正解的存在性结果 .     

The Dissipation Inequality for Differential-Algebraic Systems

We discuss the infinite time-horizon linear-quadratic optimal control problem for differential-algebraic equations. In contrast to previous approaches we do not impose any assumptions on the system except for impulse controllability. In particular, we show that the optimal control problem is feasible if and only if a dissipation inequality has a solution. Moreover, we discuss conditions under which the problem has a minimizing (instead of only an infimizing) solution and furthermore, when this minimizing solution is unique. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

一类二阶非自治Hamilton系统的周期解

在线性增长和次线性增长条件下,利用临界点理论中的极小作用原理和鞍点定理,研究了二阶非自治Hamilton系统周期解的存在性问题,获得了一些新的可解性条件．     

Periodic Solutions of Second Order Superlinear Singular Dynamical Systems

We study the existence of periodic solutions of second order superlinear dynamical systems with a singularity of repulsive type. The proof is based on a well-known fixed point theorem for completely continuous operators. We do not need to consider so-called strong force conditions. Recent results in the literature are generalized and significantly improved.     

Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems

The network approach to the modelling of complex technical systems results frequently in a set of differential-algebraic systems that are connected by coupling conditions. A common approach to the numerical solution of such coupled problems is based on the coupling of standard time integration methods for the subsystems. As a unified framework for the convergence analysis of such multi-rate, multi-method or dynamic iteration approaches we study in the present paper the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window. Preconditioning is used to guarantee stability of the coupled numerical methods. The theoretical results are applied to quasilinear problems from electrical circuit simulation and to index-3 systems arising in multibody dynamics.     

对称超二次二阶哈密尔顿系统的周期解

我们利用Ambrosetti-Rabinowitz对称形式的山路引理证明了给定周期T的对称超二次二阶哈密尔顿系统具有无穷多个反T/2-周期且奇的周期解.     

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.     

Multiplicity Results for Second Order Non-Autonomous Singular Dirichlet Systems

In this paper, we establish multiplicity results for second order non-autonomous singular Dirichlet systems. The proof is based on a well-known fixed point theorem in cones, and an existence principle proved in Agarwal and O’Regan (J. Differ. Equ. 175:393–414, 2001), which was established using a nonlinear alternative of Leray-Schauder type. Truncation techniques play an important role in the analysis. Some recent results in the literature are generalized and improved.