Minimizing a nonlinear function under a fuzzy max-t-norm relational equation constraint

This work studies a nonlinear optimization problem subject to fuzzy relational equations with max-t-norm composition. Since the feasible domain of fuzzy relational equations with more than one minimal solution is non-convex, traditional nonlinear programming methods usually cannot solve them efficiently. This work proposes a genetic algorithm to solve this problem. This algorithm first locates the feasible domain through the maximum solution and the minimal solutions of the fuzzy relational equations, to significantly reduce the search space. The algorithm then executes all genetic operations inside this feasible domain, and thus avoids the need to check the feasibility of each solution generated. Moreover, it uses a local search operation to fine-tune each mutated solution. Experimental results indicate that the proposed algorithm can accelerate the searching speed and find the optimal solution.     

Fuzzy relational equations in general framework

The aim of this paper is to provide a survey of issues regarding the problem of solving generalized fuzzy relational equations that are defined within a recently introduced framework of sup-preserving aggregation structures. Generalized fuzzy relational equations subsume the previously studied types of fuzzy relational equations, that is those based on either sup-t-norm or inf-residuum classes of compositions.     

Monotone methods for a discrete boundary problem

This paper is motivated by recent interests in space discrete Nagumo equations and is concerned with the existence of solutions of a nonlinear discrete boundary value problem. Monotone methods are used to derive the existence theorems. These methods, as is well known, provide constructive schemes for calculating the solutions.     

A Level Set Framework for Capturing Multi-Valued Solutions of Nonlinear First-Order Equations

We introduce a level set method for the computation of multi-valued solutions of a general class of nonlinear first-order equations in arbitrary space dimensions. The idea is to realize the solution as well as its gradient as the common zero level set of several level set functions in the jet space. A very generic level set equation for the underlying PDEs is thus derived. Specific forms of the level set equation for both first-order transport equations and first-order Hamilton-Jacobi equations are presented. Using a local level set approach, the multi-valued solutions can be realized numerically as the projection of single-valued solutions of a linear equation in the augmented phase space. The level set approach we use automatically handles these solutions as they appear     

A quasi-Newton modified LP-Newton method

We consider a method to solve constrained system of nonlinear equations based on a modification of the Linear-Programming-Newton method and replacing the first-order information with a quasi-Newton secant update, providing a computationally simple method. The proposed strategy combines good properties of two methods: the least change secant update for unconstrained system of nonlinear equations with isolated solutions and the Linear-Programming-Newton for constrained nonlinear system of equations with possible nonisolated solutions. We analyse the local convergence of the proposed method under a standard error bound condition proving its linear convergence for nonisolated solutions. Numerical experiments were done in order to show the claimed convergence rate.     

Infinite fuzzy relation equations with continuous t-norms

This study develops a concept of infinite fuzzy relation equations with a continuous t-norm. It extends the work by Xiong and Wang [Q.Q. Xiong, X.P. Wang, Some properties of sup-min fuzzy relational equations on infinite domains, Fuzzy Sets and Systems 151 (2005) 393-402]. We describe attainable (respectively, unattainable) solutions, which are closely related to minimal solutions to the equations. It is shown that a solution set comprises both attainable and unattainable solutions. The study offers a characterization of these solutions. Under some assumptions, the solution set is presented and discussed. Two applications are also given.     

Subanalytic solutions of linear difference equations and multidimensional hypergeometric sequences

We consider linear difference equations with polynomial coefficients over C and their solutions in the form of sequences indexed by the integers (sequential solutions). We investigate the C-linear space of subanalytic solutions, i.e., those sequential solutions that are the restrictions to Z of some analytic solutions of the original equation. It is shown that this space coincides with the space of the restrictions to Z of entire solutions and that the dimension of this space is equal to the order of the original equation.We also consider d-dimensional (d≥1) hypergeometric sequences, i.e., sequential and subanalytic solutions of consistent systems of first-order difference equations for a single unknown function. We show that the dimension of the space of subanalytic solutions is always at most 1, and that this dimension may be equal to 0 for some systems (although the dimension of the space of all sequential solutions is always positive).Subanalytic solutions have applications in computer algebra. We show that some implementations of certain well-known summation algorithms in existing computer algebra systems work correctly when the input sequence is a subanalytic solution of an equation or a system, but can give incorrect results for some sequential solutions.     

Concatenated solutions of delay-differential equations and their representation with time-delay feedback systems

Stimulated by the monodromy operator approach to time-delay systems (TDSs) developed recently, this article studies the conversion problems of delay-differential equations (DDEs) into the representation as time-delay feedback systems. We give two conversion methods including the conversion of initial conditions, where we show that each of the two methods corresponds, in general, to one of the two different definitions for the solutions of DDEs, called pseudo concatenated solutions and continuous concatenated solution. The study is actually closely related to the subtle behaviours of the solutions of DDEs under discontinuous initial conditions, and simple examples illustrating such subtleties as well as the validity of the conversion methods are also provided. The results of this article suggest that the ability of representing TDSs is higher in the representation as time-delay feedback systems than in the representation as DDEs.     

Variable-coefficient auxiliary equation method for exact solutions of non-linear evolution equations

In this paper, a variable-coefficient auxiliary equation method is proposed to seek more general exact solutions of non-linear evolution equations. Being concise and straightforward, this method is applied to the Kawahara equation, Sawada–Kotera equation and (2+1)-dimensional Korteweg–de Vries equations. As a result, many new and more general exact solutions are obtained including Jacobi elliptic, hyperbolic and trigonometric function solutions. It is shown that the proposed method provides a straightforward and effective method for non-linear evolution equations in mathematical physics.     

模糊关系不等式 A o X o B ≤ C 的解

对于普通的矩阵乘积, 当一个方程或者不等式有解时, 有可能存在无数多个解, 而直接求解它们又是很困 难的. 同样, 对于有限论域上采用最大–最小合成算子的模糊关系方程或者不等式也存在着类似的问题. 不幸的是, 研究此类问题的文献相对较少. 本文致力于研究模糊关系不等式A o X o B ≤ C的一种新求解方法. 首先, 利用两 个重要的公式, 将所考虑的模糊关系不等式转化成较简单的形式. 对于模糊关系不等式的可解性给出一个充分必 要条件. 它表明模糊关系不等式A o X o B ≤ C的解可以由有限个节点解来刻画. 然后, 利用矩阵的半张量积, 给出 具体的求解算法. 最后, 介绍了具有模糊关系不等式限制的格化线性规划, 来说明本文所提出方法的有效性.     

PARAMETRIC SOLUTIONS TO THE GENERALIZED SYLVESTER MATRIX EQUATION AX ‐ XF = BY AND THE REGULATOR EQUATION AX ‐ XF = BY + R

In this paper, explicit parametric solutions to the generalized Sylvester matrix equation AX ‐ XF = BY and the regulator matrix equation AX ‐ XF = BY + R are proposed without any transformation and factorization. The proposed solutions are presented in terms of the Krylov matrix of matrix pair (A, B), a symmetric operator and the generalized observability matrix of matrix pair (Z, F) where Z is an arbitrary matrix and is used to denote the degree of freedom in the solution. Due to its elegant form and convenient computation, these proposed solutions will play an important role in solving and analyzing these types of equations in control systems theory.     

Asymptotic properties of solutions to discrete Coulomb equations

We construct two finite-difference models for the Coulomb differential equation which arises in the quantum mechanics analysis of the scattering of two charged point particles. These difference equations correspond to the standard and Mickens-Ramadhani schemes for the Coulomb equation. Our major goal is to determine the first two terms in the asymptotic solutions and compare them to the corresponding solutions of the Coulomb differential equation. In particular, the form of the anomalous phase term is examined.     

Necessary conditions for constant solutions of coupled Riccati equations in Nash games

The asymptotic behaviour of solutions as t → ∞ for coupled matrix Riccati equations occurring in open-loop linear-quadratic Nash games is studied in this paper. A general formula representing all possible solutions is given. Necessary conditions for constant real solutions are derived and an estimate for the rate of convergence is obtained. Two examples illustrate these results.     

Optimization of fuzzy relational equations with max-av composition

Max-min and max-product compositions are commonly utilized to optimize a linear objective function subject to fuzzy relational equations. Both are members in the class of max-t-norm composition. In this study, the max-av composition is considered for the same optimization model, which does not belong to the max-t-norm composition. However, max-av composition generates some properties of the solution set that are similar to the max-product composition. Thanks to these properties, a simple value matrix with rules can be applied to reduce problem size. Thus, this study proposes an efficient procedure for obtaining optimal solutions without decomposing the problem into two sub-problems or finding all the potential minimal solutions.     

An explicit solution to right factorization with application in eigenstructure assignment

Based on the well-known Leverrier algorithm, a simple explicit solution to right factorization of a linear system is established.This solution is expressed by the controllability matrix of the given system and a symmetric operator matrix.Applications of this solution to a type of generalized Sylvester matrix equations and the problem of parametric eigenstructure assignment by state feedback are investigated,and general complete parametric solutions to these two problems are deduced. These new solutions are simple,and possess desirable structural properties which render the solutions readily implementable.An example demonstrates the effect of the proposed results.     

Almost Sure Practical Exponential Stability of Nonlinear Disturbed Stochastic Systems with Guaranteed Decay Rate

In this paper, we consider nonlinear stochastic differential equations driven by multiplicative noises and affected by exogenous disturbances. Sufficient conditions are investigated for almost sure practical exponential stability of the non trivial solutions of these equations. A lower bound of the decay rate of these solutions is guaranteed.     

Some new solitonary solutions of the modified Benjamin-Bona-Mahony equation

In this paper, we use the exp-function method to construct some new soliton solutions of the Benjamin-Bona-Mahony and modified Benjamin-Bona-Mahony equations. These equations have important and fundamental applications in mathematical physics and engineering sciences. The exp-function method is used to find the soliton solution of a wide class of nonlinear evolution equations with symbolic computation. This method provides the concise and straightforward solution in a very easier way. The results obtained in this paper can be viewed as a refinement and improvement of the previously known results.     

几个非线性偏微分方程的非古典对称及相似解

研究了一些非线性偏微分方程的非古典势对称和非古典对称,得到了某些方程的新的势对称和新的对称,同时也得到了其伴随系统的新的对称,并求出了一些相似解．这些解对进一步研究这些非线性偏微分方程所描述的物理现象具有广泛的应用价值．     

The oscillation of partial difference equations with continuous arguments

For the partial difference equations

and

we shall obtain sufficient conditions for the oscillation of all solutions of these equations.     

Relational heuristics for the design of deterministic programs

Summary It has been shown that the design of deterministic programs can be formulated as the resolution of relational equations. Because relational calculi are not sufficiently structured, there are no algorithmic solutions to relational equations. In this paper, we formulate some heuristic solutions to these equations.Part of this work was carried out while the first and second author were at Laval University in Québec. Canada: it was supported by the National Research Council of Canada through a research grant to the first author and a scholarship to the second author. Presently, this research is supported by a grant to the first and third authors from the Tunisian Council on Scientific and Technical Research