A method for discrete stochastic MADM problems based on the ideal and nadir solutions

Many real life decision making problems can be modeled as discrete stochastic multi-attribute decision making (MADM) problems. A novel method for discrete stochastic MADM problems is developed based on the ideal and nadir solutions as in the classical TOPSIS method. In a stochastic MADM problem, the evaluations of the alternatives with respect to the different attributes are represented by discrete stochastic variables. According to stochastic dominance rules, the probability distributions of the ideal and nadir variates, both are discrete stochastic variables, are defined and determined for a set of discrete stochastic variables. A metric is proposed to measure the distance between two discrete stochastic variables. The ideal solution is a vector of ideal variates and the nadir solution is a vector of nadir variates for the multiple attributes. As in the classical TOPSIS method, the relative closeness of an alternative is determined by its distances from the ideal and nadir solutions. The rankings of the alternatives are determined using the relative closeness. Examples are presented to illustrate the effectiveness of the proposed method. Through the examples, several significant advantages of the proposed method over some existing methods are discussed.     

Monotone and numerical-analytic methods for differential equations

The method of lower and upper solutions combined with monotone iterative techniques is used for ordinary differential equations with integral boundary conditions showing the existence of extremal solutions. Some existence results are also formulated using the numerical-analytic method. Sufficient conditions for existence are given.     

Four positive periodic solutions of a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting

In this paper, by employing the continuation theorem of coincidence degree theory, we establish an easily verifiable criteria for the existence of at least four positive periodic solutions for a discrete time delayed predator–prey system with nonmonotonic functional response and harvesting.     

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem.Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense.An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented.     

A higher-order compact ADI method with monotone iterative procedure for systems of reaction-diffusion equations

This paper is concerned with an existing compact finite difference ADI method, published in the paper by Liao et al. (2002) [3], for solving systems of two-dimensional reaction-diffusion equations with nonlinear reaction terms. This method has an accuracy of fourth-order in space and second-order in time. The existence and uniqueness of its solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear reaction terms. The convergence of the finite difference solution to the continuous solution is proved. An efficient monotone iterative algorithm is presented for solving the resulting discrete system, and some techniques for the construction of upper and lower solutions are discussed. An application using a model problem gives numerical results that demonstrate the high efficiency and advantages of the method.     

离散捕食系统的持久性和全局稳定性

研究一类带有Beddington．DeAngelis功能性反应和时滞的多维离散食饵．捕食系统的持久性和全局稳定性。利用差分方程的比较原理,给出了系统持久性的充分条件;应用估值方法推导得到系统全局稳定性的充分条件;使用不动点理论,证明了系统周期解的存在性和全局稳定性。     

基于Radau伪谱法的非线性最优控制问题的收敛性

在过去的10年里,伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线性最优控制问题的一种高效、灵活的数值解法.本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究.证明了原最优控制问题的离散解存在、存在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解.在此基础上,证明了Radau伪谱法的收敛性.本文结论与现有文献相比,去掉了一些必要条件,更适合一般的非线性时不变系统.     

On the existence,uniqueness and nature of Carathéodory and Filippov solutions for bimodal piecewise affine dynamical systems

In this paper, we deal with the well-posedness (in the sense of existence and uniqueness of solutions) and nature of solutions for discontinuous bimodal piecewise affine systems in a differential inclusion setting. First, we show that the conditions guaranteeing uniqueness of Filippov solutions in the context of general differential inclusions are quite restrictive when applied to bimodal piecewise affine systems. Later, we present a set of necessary and sufficient conditions for uniqueness of Filippov solutions for bimodal piecewise affine systems. We also study the so-called Zeno behavior (possibility of infinitely many switchings within a finite time interval) for Filippov solutions.     

A polyhedral approach to solving multicriterion combinatorial optimization problems over sets of polyarrangements

Multicriterion discrete optimization problems over feasible combinatorial sets of polyarrangements are considered. Structural properties of feasible domains and different types of efficient solutions are investigated. Based on the ideas of Euclidean combinatorial optimization and the major criterion method, a polyhedral approach to the solution of the problems is developed and substantiated. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 118-126, May-June 2009.     

On periodic solutions of adaptive systems in the presence of periodic forcing terms

We consider a discrete-time system consisting of a linear plant and a periodically forced feedback controller whose parameters are slowly adapted. Using degree theory, we give sufficient conditions for the existence of periodic solutions. Using linearization methods, we give sufficient conditions for their (in)stability provided the adaptation is slow enough. We then study when the degree theoretic conditions for the existence are satisfied byd-steps-ahead adaptive controllers in the presence of unmodeled dynamics and a persistently exciting periodic reference output.     

线性离散周期奇异系统的可解性与渐近稳定性

讨论了线性离散周期奇异系统初值问题的可解性和渐近稳定性问题.首先分析总结了线性离散变系数奇异系统可解性及其广义状态解的一般概念.在此基础上,定义了线性离散变系数奇异系统的一致渐近稳定性,并通过增加系统维数把线性离散周期奇异系统转化为线性定常奇异系统,从而得到了线性离散周期奇异系统可解和一致渐近稳定的充要条件.     

Weighted convergent and bounded solutions of difference systems

Using a “summable” dichotomy and fixed-point theorems, we prove the existence of convergent and bounded solutions of nonlinear difference systems.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.     

Global existence and uniqueness of positive solutions and optimal control for a novel model of pest control

In the present paper, a novel model of pest control governed by coupled sets of partial and ordinary differential equations is investigated in detail. Global existence and uniqueness of positive solutions to the above model are obtained. Also, the existence of optimal control is proved, and the first-order and second-order optimality conditions satisfied by optimal control are verified. In particular, the optimal control is demonstrated to be of Bang-Bang form.     

Asymptotic behaviour of solutions of systems of discrete equations via Liapunov type technique

The second Liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems. In the presented paper, a Liapunov-type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations. Conditions for the existence of continuum of solutions, the graphs of which remain within a given prescribed set, are formulated for the general systems of discrete equations Δu(k) = F(k, u(k)). An additional advantage of the presented approach consists of the fact that no assumption concerning the existence of the trivial solution (or the existence of an equilibrium point) of systems considered is assumed. Moreover, the asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae. Several illustrative examples are considered, too.     

广义非线性系统的极限值问题

应用单调迭代法和上下解的方法讨论了广义非线性系统的极限值问题 ,给出了解存在性的构造性证明 ,所构造的逼近序列是线性系统的解 ,因此较易实现数值计算 .并且所得结果推广了非线性微分系统的结果 .     

Discrete noiseless duel with a skewsymmetric payoff function on the unit square for models of socioeconomic competitive processes with a finite number of pure strategies

A discrete noiseless duel is defined on the unit square; each player in the duel has a finite number of pure strategies uniformly distributed on the unit segment. The theorem on the existence of individual solutions of the discrete noiseless duel in pure strategies is proved. The construction of a program procedure for solving the discrete noiseless duel is presented.     

Rate of convergence for the Legendre pseudospectral optimal control of feedback linearizable systems

Pseudospectral (PS) computational methods for nonlinear constrained optimal control have been applied to many industrial-strength problems, notably, the recent zero-propellant-maneuvering of the international space station performed by NASA. In this paper, we prove a theorem on the rate of convergence for the optimal cost computed using a Legendre PS method. In addition to the high-order convergence rate, two theorems are proved for the existence and convergence of the approximate solutions. Relative to existing work on PS optimal control as well as some other direct computational methods, the proofs do not use necessary conditions of optimal control. Furthermore, we do not make coercivity type of assumptions. As a result, the theory does not require the local uniqueness of optimal solutions. In addition, a restrictive assumption on the cluster points of discrete solutions made in existing convergence theorems is removed.     

Numerical solutions of a nonlinear reaction-diffusion system

This paper is concerned with finite difference solutions of a coupled system of nonlinear reaction-diffusion equations. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. The existence and uniqueness of a non-negative finite difference solution and three monotone iterative algorithms for the computation of the solutions are given. It is shown that the time-dependent problem has a unique non-negative solution, whereas the steady-state problem may have multiple non-negative solutions depending on the parameters in the problem. The different non-negative steady-state solutions can be computed from the monotone iterative algorithms by choosing different initial iterations. Also discussed is the asymptotic behaviour of the time-dependent solution in relation to the steady-state solutions. The asymptotic behaviour result gives some conditions ensuring the convergence of the time-dependent solution to a positive or semitrivial non-negative steady-state solution. Numerical results are given to demonstrate the theoretical analysis results.     

On the existence of a global minimum in inverse parameters identification by Self-Optimizing inverse analysis method

In this paper, a mathematical proof of the existence of a global minimum of Self-Optim (Self-Optimizing Inverse Analysis Method) cost functional is presented based upon weak-solution theory of partial differential equations. The Self-Optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem. Furthermore, discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified. This paper provides a rigorous mathematical foundation for applications of the Self-Optim method to various inverse problems in mechanics.