分裂映射的两侧逼近区间割线法

对于求解非线性方程组的区间迭代法,若利用Moore检验,可判断解的存在唯一性.[2]中在偏序下给出的区间Newton型方法,也有同样特性,本文利用f:D?R~n→R~n的斜度构造的区间割线算子,也可用于检验方程组解的存在唯一性,但它不用计算f的导数,针对f的不同分裂,还可以构造不同的两侧逼近割线法.分裂得当,便于求逆,使计算     

An adaptive numerical method to handle blow-up in a parabolic system

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

An interval algorithm for sensitivity analysis of coupled vibro-acoustic systems

Sensitivity analysis is a vital part in the optimization design of coupled vibro-acoustic systems. A new interval sensitivity-analysis method for vibro-acoustic systems is proposed in this paper. This method relies on only interval perturbation analysis instead of partial derivatives and difference operations. For strongly nonlinear systems, in particular, this methodology requires parameter variation over narrower ranges in comparison with other methods. To implement sensitivity analysis based on this method, the interval ranges of the responses of the vibro-acoustic system with interval parameters should first be obtained. Therefore, an interval perturbation-analysis method is presented for obtaining the interval bounds of the sound-pressure responses of a coupled vibro-acoustic system with interval parameters. The interval perturbation method is then compared with the Monte Carlo method, which can be taken as the benchmark for comparative accuracy. Two numerical examples involving sensitivity analysis of vibro-acoustic systems illustrate the feasibility and effectiveness of the proposed interval-based method.     

Optimal synthesis in a control problem with Lipschitz input data

We propose a numerical algorithm for constructing an optimal synthesis in the control problem for a nonlinear system on a fixed time interval. We estimate the difference between the values of the cost functional on optimal trajectories and on the trajectories constructed according to this algorithm. The operation of the algorithm is illustrated by solving model examples on the plane.     

Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint-propagation techniques

We investigate solution techniques for numerical constraint-satisfaction problems and validated numerical set integration methods for computing reachable sets of nonlinear hybrid dynamical systems in the presence of uncertainty. To use interval simulation tools with higher-dimensional hybrid systems, while assuming large domains for either initial continuous state or model parameter vectors, we need to solve the problem of flow/sets intersection in an effective and reliable way. The main idea developed in this paper is first to derive an analytical expression for the boundaries of continuous flows, using interval Taylor methods and techniques for controlling the wrapping effect. Then, the event detection and localization problems underlying flow/sets intersection are expressed as numerical constraint-satisfaction problems, which are solved using global search methods based on branch-and-prune algorithms, interval analysis and consistency techniques. The method is illustrated with hybrid systems with uncertain nonlinear continuous dynamics and nonlinear invariants and guards.     

Sampled-data control of uncertain switched neutral nonlinear systems under asynchronous switching

The robust exponential stabilization for a class of the uncertain switched neutral nonlinear systems with time-varying delays based on the sampled-data control is investigated in this paper. The closed-loop system with sampled-data control is modeled as a continuous time system with a time-varying piecewise continuous control input delay. Considering the relationship between the sampling period and the dwell time of two switching instants, sampling interval with no switching and sampling interval with one switching are discussed, respectively. By Wirtinger-based inequality, Wirtinger-based double integral inequality, and free-weighting matrix technique, some delay-dependent sufficient conditions are given to guarantee the exponential stability of uncertain switched neutral nonlinear systems under asynchronous switching. In addition, sampled-data controllers can also be designed by special operations of matrices. Finally, two numerical examples are used to show the effectiveness of the approach proposed in this paper.     

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality

A numerical approach to design control invariant sets for constrained nonlinear discrete-time systems with guaranteed optimality is proposed in this paper. The addressed approach is based on the fact that zonotopes are more flexible for representing sets than boxes in interval analysis. Then the solver of set inversion via interval analysis is extended to set inversion via zonotope geometry by introducing the novel idea of bisecting zonotopes. The main feature of the extended solver of set inversion is the bisection and the evolution of a zonotope rather than a box. Thus the shape of admissible domains for set inversion can be broadened from boxes to zonotopes and the wrapping effect can be reduced as well by using the zonotope evolution instead of the interval evolution. Combined with global optimization via interval analysis, the extended solver of set inversion via zonotope geometry is further applied to design control invariant sets for constrained nonlinear discrete-time systems in a numerical way. Finally, the numerical design of a control invariant set and its application to the terminal control of the dual-mode model predictive control are fulfilled on a benchmark Continuous-Stirred Tank Reactor example.     

Delay Range-Dependent Stability Analysis for Markovian Jumping Stochastic Systems with Nonlinear Perturbations

This article addresses the problem of delay-dependent stability for Markovian jumping stochastic systems with interval time-varying delays and nonlinear perturbations. The delay is assumed to be time-varying and belongs to a given interval. By resorting to Lyapunov–Krasovskii functionals and stochastic stability theory, a new delay interval-dependent stability criterion for the system is obtained. It is shown that the addressed problem can be solved if a set of linear matrix inequalities (LMIs) are feasible. Finally, a numerical example is employed to illustrate the effectiveness and less conservativeness of the developed techniques.     

A semi-analytic approach for the nonlinear dynamic response of circular plates

This paper presents a new semi-analytic perturbation differential quadrature method for geometrically nonlinear vibration analysis of circular plates. The nonlinear governing equations are converted into a linear differential equation system by using Linstedt–Poincaré perturbation method. The solutions of nonlinear dynamic response and the nonlinear free vibration are then sought through the use of differential quadrature approximation in space domain and analytical series expansion in time domain. The present method is validated against analytical results using elliptic function in several examples for both clamped and simply supported circular plates, showing that it has excellent accuracy and convergence. Compared with numerical methods involving iterative time integration, the present method does not suffer from error accumulation and is able to give very accurate results over a long time interval.     

A time-decomposition algorithm for a stiff linear two-point boundary-value problem and its application to a nonlinear optimal control problem

An algorithm is proposed to solve a stiff linear two-point boundary-value problem (TPBVP). In a stiff problem, since some particular solutions of the system equation increase and others decrease rapidly as the independent variable changes, the integration of the system equation suffers from numerical errors. In the proposed algorithm, first, the overall interval of integration is divided into several subintervals; then, in each subinterval a sub-TPBVP with arbitrarily chosen boundary values is solved. Second, the exact boundary values which guarantee the continuity of the solution are determined algebraically. Owing to the division of the integration interval, the numerical error is effectively reduced in spite of the stiffness of the system equation. It is also shown that the algorithm is successfully imbedded into an interaction-coordination algorithm for solving a nonlinear optimal control problem.The authors would like to thank Mr. T. Sera and Mr. H. Miyake for their help with the calculations.     

Multipoint solution of two-point boundary-value problems

The purpose of this paper is to report on the application of multipoint methods to the solution of two-point boundary-value problems with special reference to the continuation technique of Roberts and Shipman. The power of the multipoint approach to solve sensitive two-point boundary-value problems with linear and nonlinear ordinary differential equations is exhibited. Practical numerical experience with the method is given.Since employment of the multipoint method requires some judgment on the part of the user, several important questions are raised and resolved. These include the questions of how many multipoints to select, where to specify the multipoints in the interval, and how to assign initial values to the multipoints.Three sensitive numerical examples, which cannot be solved by conventional shooting methods, are solved by the multipoint method and continuation. The examples include (1) a system of two linear, ordinary differential equations with a boundary condition at infinity, (2) a system of five nonlinear ordinary differential equations, and (3) a system of four linear ordinary equations, which isstiff.The principal results are that multipoint methods applied to two-point boundary-value problems (a) permit continuation to be used over a larger interval than the two-point boundary-value technique, (b) permit continuation to be made with larger interval extensions, (c) converge in fewer iterations than the two-point boundary-value methods, and (d) solve problems that two-point boundary-value methods cannot solve.     

An efficient algorithm for finding all solutions of separable systems of nonlinear equations

An efficient algorithm is proposed for finding all solutions of systems of nonlinear equations with separable mappings. This algorithm is based on interval analysis, the dual simplex method, the contraction method, and a special technique which makes the algorithm not require large memory space and not require copying tableaus. By numerical examples, it is shown that the proposed algorithm could find all solutions of a system of 2000 nonlinear equations in acceptable computation time. AMS subject classification (2000)  65H10, 65G10     

A cell exclusion algorithm for determining all the solutions of a nonlinear system of equations

A new group of methods named cell exclusion algorithms (CEAs) is developed for finding all the solutions of a nonlinear system of equations. These types of algorithms, different in principle from those of homotopy, interval and cell-mapping-dynamical-analysis approaches, are based on cellular discretization and the use of a certain simple necessity test of the solutions. The main advantages of the algorithms are their simplicity, reliability, and general applicability. Having all features of interval techniques (but without using interval arithmetic) and with complexity O(log(1/)), the algorithms improve significantly on both the interval algorithms and the cell mapping techniques. Theoretical analysis and numerical simulations both demonstrate that CEAs are very efficient.     

On the self-adjoint nonlinear eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities

Certain properties of the nonlinear self-adjoint eigenvalue problem for Hamiltonian systems of ordinary differential equations with singularities are examined. Under certain assumptions on the way in which the matrix of the system and the matrix specifying the boundary condition at a regular point depend on the spectral parameter, a numerical method is proposed for determining the number of eigenvalues lying on a prescribed interval of the spectral parameter.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 1: analysis

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations. The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Interval bounds on the solutions of semi-explicit index-one DAEs. Part 2: computation

This article presents two methods for computing interval bounds on the solutions of nonlinear, semi-explicit, index-one differential-algebraic equations (DAEs). Part 1 presents theoretical developments, while Part 2 discusses implementation and numerical examples. The primary theoretical contributions are (1) an interval inclusion test for existence and uniqueness of a solution, and (2) sufficient conditions, in terms of differential inequalities, for two functions to describe componentwise upper and lower bounds on this solution, point-wise in the independent variable. The first proposed method applies these results sequentially in a two-phase algorithm analogous to validated integration methods for ordinary differential equations (ODEs). The second method unifies these steps to characterize bounds as the solutions of an auxiliary system of DAEs. Efficient implementations of both are described using interval computations and demonstrated on numerical examples.     

Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new higher-efficiency interval method for the response bound estimation of nonlinear dynamic systems, whose uncertain parameters are bounded. This proposed method uses sparse regression and Chebyshev polynomials to help the interval analysis applied on the estimation. It is also a non-intrusive method which needs much fewer evaluations of original nonlinear dynamic systems than the other Chebyshev polynomials based interval methods. By using the proposed method, the response bound estimation of nonlinear dynamic systems can be performed more easily, even if the numerical simulation in nonlinear dynamic systems is costly or the number of uncertain parameters is higher than usual. In our approach, the sparse regression method “elastic net” is adopted to improve the sampling efficiency, but with sufficient accuracy. It alleviates the sample size required in coefficient calculation of the Chebyshev inclusion function in the sampling based methods. Moreover, some mature technologies are adopted to further reduce the sample size and to guarantee the accuracy of the estimation. So that the number of sampling, which solves the certain ordinary differential equations (ODEs), can be reduced significantly in the Chebyshev interval method. Three numerical examples are presented to illustrate the efficiency of proposed interval method. In particular, the last two examples are high dimension uncertain problems, which can further exhibit the ability to reduce the computational cost.     

偏序下的区间松弛法

§1. 引言 本文给出了求解非线性方程组 f(x)=0,f:D?R~n→R~m (1.1)在偏序下的区间松弛法,它是在[1]的基础上将区间迭代与Newton-SOR 迭代结合得到的一种便于计算且收敛较快的序区间N-SOR松弛法,也是单调N-SOR迭代法的推广.§2给出了偏序下的区间Krawczyk算子,它是区间 Newton算子的推广,同样具     

Error Estimation for Nonlinear Complementarity Problems via Linear Systems with Interval Data

For the nonlinear complementarity problem, we derive norm bounds for the error of an approximate solution, generalizing the known results for the linear case. Furthermore, we present a linear system with interval data, whose solution set contains the error of an approximate solution. We perform extensive numerical tests and compare the different approaches.