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Damghan University

University Blvd, Damghan, IR

Akbar Hashemi Borzabadi

Associate Professor of Mathematics

DOI: 10.1080/00207160.2017.1343472

An adaptive nonmonotone spectral gradient method for the solution of distributed optimal control problem (OCP) for the viscous Burgers equation is presented in a black-box framework. Regarding the implicit function theorem, the OCP is transformed into an unconstrained nonlinear optimization problem (UNOP). For solving UNOP, an adaptive nonmonotone Barzilai–Borwein gradient method is proposed in which to make a globalization strategy, first an adaptive nonmonotone strategy which properly controls the degree of nonmonotonicity is presented and then is incorporated into an inexact line search approach to construct a more relaxed line search procedure. Also an adjoint technique is used to effectively evaluate the gradient. The low memory requirement and the guaranteed convergence property make the proposed method quite useful for large-scale OCPs. The efficiency of the presented method is supported by numerical experiments. © 2017, © 2017 Informa UK Limited, trading as Taylor & Francis Group.

AUTHOR KEYWORDS: 65K10; 90C30; Distributed optimal control; finite elements method; implicit function theorem; nonmonotone Barzilai–Borwein method; viscous Burgers equation

INDEX KEYWORDS: Gradient methods; Nonlinear programming; Numerical methods; Optimal control systems; Partial differential equations, Distributed optimal control; Distributed optimal control problems; Guaranteed convergence; Implicit function theorem; Non-linear optimization problems; Nonmonotone; Spectral gradient method; Viscous Burgers equation, Finite element method

PUBLISHER: Taylor and Francis Ltd.

DOI: 10.1007/s40314-016-0399-4

In the present paper, we prove necessary optimality conditions of Pontryagin type for a class of fuzzy optimal control problems. The new results are illustrated by computing the extremals of two fuzzy optimal control systems, which improve recent results of Najariyan and Farahi. © 2016, SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional.

AUTHOR KEYWORDS: Fuzzy Hamiltonian function; Fuzzy Pontryagin maximum principle; Fuzzy variational problems

PUBLISHER: Springer Science and Business Media, LLC

DOI: 10.1016/j.fss.2018.04.007

Using the concepts of derivative and integral of fuzzy functions in the sense of fuzzification, this paper is devoted to studying a new version fuzzy fundamental theorem of calculus as well as a new variant of fuzzy Taylor formula with an integral remainder in the univariate and multivariate cases. Here, the fuzzification of derivative and integral means using Zadeh's extension principle on the corresponding classical operators. Indeed, by presenting appropriating symbols, it is shown in this work, contrary to what was supposed to be, Zadeh's extension principle is capable of making the ability to compute and introduce many quantities and concepts in univariate and multivariate calculus such as integral, derivative, Taylor expansion and etc. © 2018 Elsevier B.V.

AUTHOR KEYWORDS: Fuzzy derivative and integral; Fuzzy fundamental theorem of calculus; Fuzzy Taylor formula; Zadeh's extension

INDEX KEYWORDS: Fuzzy set theory, Fundamental theorem of calculus; Fuzzy derivatives; Fuzzy function; Integral operators; Taylor expansions; Taylor formula; Zadeh's extension; Zadeh's extension principles, Calculations

PUBLISHER: Elsevier B.V.

DOI: 10.1007/s13370-016-0451-y

The stable equilibrium configuration of structures is a main goal in structural optimization. This goal may be achieved through minimizing the potential energy function. In the real world, sometimes, the input data and parameters of structural engineering design problems may be considered as fuzzy numbers which lead us to develop structural optimization methods in a fuzzy environment. In this regard, the present paper is intended to propose a fuzzy optimization scheme according to the nonmonotone globalization technique, the Barzilai–Borwein (BB) gradient method and the generalized Hukuhara differentiability (gH-differentiability). In fact, using the best benefits of BB-like methods i.e., simplicity, efficiency and low memory requirements, a modified global Barzilai–Borwein (GBB) gradient method is proposed for obtaining a non-dominated solution of the unconstrained fuzzy optimization related to the two bar asymmetric shallow truss in a fuzzy environment. The global convergence to first-order stationary points is also proved and the R-linear convergence rate is established under suitable assumptions. Furthermore, some numerical examples are given to illustrate the main results. © 2016, African Mathematical Union and Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Barzilai–Borwein gradient method; Generalized Hukuhara differentiability; Nonmonotone line search; Truss; Unconstrained fuzzy optimization

PUBLISHER: Springer Verlag

DOI: 10.1093/imamci/dnv061

This paper contributes a suitable hybrid iterative scheme for solving optimal control problems governed by integro-differential equations. The technique is based upon homotopy analysis and parametrization methods. Comparison of the obtained results of the proposed method with other methods shows that the method is reliable and capable of providing analytic treatment for solving such equations. Convergence analysis is presented. Some illustrative examples are given to demonstrate the accuracy of the proposed method. © The authors 2015. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications.

AUTHOR KEYWORDS: homotopy analysis method; integro-differential equations; optimal control problems

INDEX KEYWORDS: Differential equations; Integrodifferential equations; Optimal control systems, Convergence analysis; Direct approach; Homotopy analysis; Homotopy analysis methods; Iterative schemes; Optimal control problem; Optimal controls; Parametrizations, Iterative methods

PUBLISHER: Oxford University Press

DOI: 10.1080/02331934.2017.1287702

The Barzilai–Borwein (BB) gradient method has received many studies due to its simplicity and numerical efficiency. By incorporating a nonmonotone line search, Raydan (SIAM J Optim. 1997;7:26–33) has successfully extended the BB gradient method for solving general unconstrained optimization problems so that it is competitive with conjugate gradient methods. However, the numerical results reported by Raydan are poor for very ill-conditioned problems because the effect of the degree of nonmonotonicity may be noticeable. In this paper, we focus more on the nonmonotone line search technique used in the global Barzilai–Borwein (GBB) gradient method. We improve the performance of the GBB gradient method by proposing an adaptive nonmonotone line search based on the morphology of the objective function. We also prove the global convergence and the R-linear convergence rate of the proposed method under reasonable assumptions. Finally, we give some numerical experiments made on a set of unconstrained optimization test problems of the CUTEr collection. The results show the efficiency of the proposed method in the sense of the performance profile introduced (Math Program. 2002;91:201–213) by Dolan and Moré. © 2017 Informa UK Limited, trading as Taylor & Francis Group.

AUTHOR KEYWORDS: Barzilai–Borwein gradient method; global convergence; nonmonotone line search; Unconstrained optimization

PUBLISHER: Taylor and Francis Ltd.

In this paper, Haar wavelet benefits are applied to the optimal control of linear time-delay systems. A discretized form of optimal control problem at collocation points based on some useful properties of Haar wavelets transforms original problem into a nonlinear programming (NLP). The given numerical examples show the accuracy of the presented scheme in comparison with some other methods. © School of Engineering, Taylor’s University.

AUTHOR KEYWORDS: Discretization; Haar wavelet; Linear timedelay system; Nonlinear programming; Optimal control problem

PUBLISHER: Taylor's University

DOI: 10.1007/s40815-016-0165-1

In this paper, based on the parametric representation of fuzzy-valued function, an unconstrained fuzzy-valued optimization problem is converted to a general unconstrained optimization problem. Two solutions for the unconstrained fuzzy-valued optimization problem are proposed which are parallel to the concept of efficient solution in the case of multi-objective programming problem. Also it is proven that the optimal solution of its corresponding general unconstrained optimization problem is the optimal solution of original problem. Finally, some numerical examples are given to illustrate the discussed suitability scheme. In the first example, the various solutions are discussed in details and a classic motivating example in the mathematical study of variational inequalities, namely the Elliptic Obstacle Problem, is expressed in the second one. Non-convex Fuzzy Rosenbrock Function have been solved in third example. © 2016 Taiwan Fuzzy Systems Association and Springer-Verlag Berlin Heidelberg.

AUTHOR KEYWORDS: Fuzzy matrix; Fuzzy optimization; Fuzzy-valued convex function

INDEX KEYWORDS: Functions; Optimal systems; Optimization; Variational techniques, Convex functions; Elliptic obstacle problem; Fuzzy matrix; Fuzzy optimization; Multi-objective programming problem; Parametric representations; Unconstrained optimization problems; Variational inequalities, Multiobjective optimization

PUBLISHER: Springer Berlin Heidelberg

DOI: 10.1109/CFIS.2015.7391653

This study proposes a novel technique for solving linear programming problems in a fully fuzzy environment. A modified version of the well-known dual simplex method is used for solving fuzzy linear programming problems. The use of a ranking function together with the Gaussian elimination process helps in solving linear programming problems in a fully uncertain environment. The proposed algorithm is flexible, easy and reasonable. © 2015 IEEE.

AUTHOR KEYWORDS: Fuzzy linear programming problems (FLP); Fuzzy simplex method; Ranking and selection

INDEX KEYWORDS: Fuzzy set theory; Intelligent systems; Linear programming, Fully fuzzy linear programming; Fuzzy environments; Fuzzy linear programming problems; Fuzzy simplex; Gaussian elimination; Linear programming problem; Ranking and selection; Uncertain environments, Problem solving

PUBLISHER: Institute of Electrical and Electronics Engineers Inc.

DOI: 10.1109/CFIS.2015.7391703

We prove necessary optimality conditions of Euler-Lagrange type for both fuzzy unconstrained and constrained fractional variational problems where the fuzzy fractional derivative is described in the Jumarie's Riemann-Liouville sense. The new results are illustrated by computing the extremals of two fuzzy variational problems. © 2015 IEEE.

INDEX KEYWORDS: Intelligent systems, Euler-Lagrange; Fractional derivatives; Necessary optimality condition; New results; Riemann-liouville h-differentiability; Riemann-Liouville sense; Variational problems, Variational techniques

PUBLISHER: Institute of Electrical and Electronics Engineers Inc.

This paper presents the necessary optimality conditions of Euler–Lagrange type for variational problems with natural boundary conditions and problems with holonomic constraints where the fuzzy fractional derivative is described in the combined Caputo sense. The new results are illustrated by computing the extremals of two fuzzy variational problems. © 2016 Department of Mathematics, University of Osijek.

AUTHOR KEYWORDS: Fuzzy fractional euler–Lagrange equations; Fuzzy fractional holonomic problem; Fuzzy fractional variational problem

PUBLISHER: Udruga Matematicara Osijek

DOI: 10.1093/imamci/dnt032

In this paper, a wavelet collocation approach for finding approximate optimal control of the non-linear time-delay systems is introduced. A discretized form of the optimal control problem at collocation points based on some useful properties of Haar wavelets transforms the original problem into one of non-linear programming. Some numerical examples are presented to show the proficiency of the proposed scheme in comparison with another method. © The authors 2013.

AUTHOR KEYWORDS: Approximation; Discretization; Haar wavelet; Non-linear time-delay system; Optimal control

INDEX KEYWORDS: Mathematical transformations; Nonlinear programming; Numerical methods; Optimal control systems; Time delay, Approximation; Discretizations; Haar wavelets; Non linear; Optimal controls, Delay control systems

PUBLISHER: Oxford University Press

In this paper, a reliable iterative approach, for solving a wide range of linear and nonlinear Volterra-Fredholm integral equations is established. First the approach considers a discretized form of the integral terms where considering some conditions on the kernel of the integral equation it is proved that solution of the discretized form converges to the exact solution of the problem. Then the solution of the discretized form is approximated by an iterative scheme. Comparison of the approximate solution with exact solution shows that the used approach is easy and practical for some classes of linear and nonlinear Volterra-Fredholm integral equations. © 2015 Academic Center for Education, Culture and Research TMU.

AUTHOR KEYWORDS: Approximation; Discretization; Volterra-Fredholm integral equation

PUBLISHER: Academic Center for Education, Culture and Research

DOI: 10.1504/IJCSM.2015.072967

In this paper, a reliable algorithm for solving Schrödinger equations is established. By second-order central difference scheme, the second-order spatial partial derivative of the Schrödinger equations are reduced to a system of first-order ordinary differential equations, that are solved by an efficient algorithm. The comparison of the numerical solution and the exact solution for some test cases shows that the given algorithm is easy and practical for extracting good approximate solutions of Schrödinger equations. Copyright © 2015 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Approximation; Finite difference method; Numerical algorithm; Schrödinger equation

INDEX KEYWORDS: Approximation algorithms; Differential equations; Finite difference method; Numerical methods; Ordinary differential equations, Algorithm for solving; Approximate solution; Approximation; Central difference scheme; Dinger equation; First order ordinary differential equations; Numerical algorithms; Partial derivatives, Algorithms

PUBLISHER: Inderscience Enterprises Ltd.

DOI: 10.1504/IJMMNO.2013.055198

In this paper, a numerical scheme for obtaining approximate solutions of optimal control problems governed by SchrÖdinger equations is presented. In this method, by considering a partition of the control space, discrete form of the problem is converted to a quasi assignment problem. Then using an evolutionary algorithm, approximate optimal control is obtained as a piecewise linear function. The comparison between the numerical solution and the exact solution for the test cases shows the good accuracy of the presented method. © 2013 Inderscience Enterprises Ltd.

AUTHOR KEYWORDS: Approximation.; Discretisation; Evolutionary algorithm; Optimal control problem; SchrÖdinger equation

DOI: 10.3233/IFS-2012-0646

This paper presents a novel approach for finding positive solution of fully fuzzy polynomial equations systems employing Gröbner bases benefits. First, the system of fully fuzzy polynomial equations is converted to an equivalent crisp polynomial equations system. Then, using Gröbner basis properties a criteria for determining existence and finding positive fuzzy solutions of original system is introduced. The big advantage of the proposed method lies in the fact that it attains all positive fuzzy solutions of problem at a time. Finally, some applied numerical examples are illustrated to demonstrate the proficiency of the given approach. © 2013 - IOS Press and the authors. All rights reserved.

AUTHOR KEYWORDS: affine varieties; fully fuzzy polynomial equations systems; Fuzzy numbers; Gröbner bases

INDEX KEYWORDS: affine varieties; Basis property; Fuzzy numbers; Numerical example; Original systems; Polynomial equation; Positive solution, Fuzzy sets, Polynomials

In this paper a computational approach to acquire approximate optimal control of Fisher's equation based on useful properties of Haar wavelets is presented. By a discretization form of state and control variables in collocation points the main problem is transformed into a nonlinear programming (NLP). An appropriate technic of optimization has been implemented to untangle the created nonlinear programming problem. The method competency has illustrated presenting numerical results. © 2013 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Approximation; Fisher's equation; Haar wavelet; Nonlinear programming; Optimal control

PUBLISHER: Institute of Advanced Scientific Research, Inc.

DOI: 10.1080/00207160.2012.705279

In this paper, a novel hybrid method based on two approaches, evolutionary algorithms and an iterative scheme, for obtaining the approximate solution of optimal control governed by nonlinear Fredholm integral equations is presented. By converting the problem to a discretized form, it is considered as a quasi-assignment problem and then an iterative method is applied to find an approximate solution for the discretized form of the integral equation. An analysis for convergence of the proposed iterative method and its implementation for numerical examples are also given. © 2012 Copyright Taylor and Francis Group, LLC.

AUTHOR KEYWORDS: Approximation; Discretization; Evolutionary algorithm; Fredholm integral equation; Iterative method; Optimal control

INDEX KEYWORDS: Approximate solution; Approximation; Discretizations; Fredholm integral equations; Iterative schemes; Nonlinear Fredholm integral equation; Numerical example; Optimal controls, Control; Evolutionary algorithms; Integral equations; Numerical methods, Iterative methods

This paper presents a hybrid scheme based on Dinkelbach approach and wavelet collocation method to extract approximate solutions of fractional optimal control problems (FOCP)'s. First Dinkelbach approach is considered to linearize the problem, then it is tried by combination of collocation wavelet approach and a numerical scheme of solving nonlinear equations, an iterative approach be proposed to obtain approximate optimal trajectory and control functions. Finally, numerical examples are listed to show the efficiency of the given approach. © 2011 Institute of Advanced Scientic Research.

AUTHOR KEYWORDS: Fractional programming; Haar wavelet; Nonlinear programming; Optimal control

DOI: 10.1007/s10852-011-9166-0

In this paper, optimal control problem (OCP) governed by the heat equation with thermal sources is considered. The aim is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective function. To obtain an approximate solution of this problem, a partition of the time-control space is considered and the discrete form of the problem is converted to a quasi assignment problem. Then by using an evolutionary algorithm, an approximate optimal control function is obtained as a piecewise linear function. Numerical examples are given to show the proficiency of the presented algorithm. © 2011 Springer Science+Business Media B.V.

AUTHOR KEYWORDS: Approximate solution; Discretization; Evolutionary algorithm; Optimal control problem

DOI: 10.1080/01630563.2011.587676

In this article, a numerical scheme on the basis of the measure theoretical approach for extracting approximate solutions of optimal control problems governed by nonlinear Fredholm integral equations is presented. The problem is converted to a linear programming in which its solution leads to construction of approximate solutions of the original problem. Finally, some numerical examples are given to demonstrate the efficiency of the approach. Copyright © Taylor & Francis Group, LLC.

AUTHOR KEYWORDS: 49J22; 49M99; 93C23

INDEX KEYWORDS: 49J22; 49M99; 93C23; Approximate solution; Nonlinear Fredholm integral equation; Numerical example; Numerical scheme; Optimal control problem; Optimal controls; Theoretical approach, Control; Integral equations; Optimal control systems; Optimization, Nonlinear equations

This paper presents an investigation on applying the methods of solving nonlinear least square problems (NLSP)'s for detecting approximate solutions of optimal control problems (OCP)'s under an iterative hybrid process. Levenberg- Marquardt method as a successful classic approach is applied for solving the created NLSP's. Results of implementing the proposed algorithm for solving some linear and nonlinear OCP's are given. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Approximation; Nonlinear least square problem; Optimal control; Penalty method

In this study, a numerical scheme for solving a class of optimal control problems governed by linear Volterra integral equations is presented. The approach is based on the homotopy perturbation theory. The numerical results reveal that the given scheme is extremely effective. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Approx- imation; Homotopy perturbation; Numerical method; Optimal control prblem; Volterra integral equation

The current research attempts to offer an approximate-analytical scheme to solve nonlinear fuzzy differential equations under generalized differentiability. In comparison with existing numerical methods, one may find the better capability and e±ciency of the given scheme. The proposed method is illustrated by some numerical examples. © 2010 Institute of Advanced Scientific Research.

AUTHOR KEYWORDS: Generalized differentiability; Local variational iteration method; Nonlinear fuzzy differential equation

This paper presents the investigations on the application of evolutionary algorithms particle swarm optimization (PSO) and invasive weed optimization (IWO) to find approximate solutions of optimal control problems. For this purpose, discrete form of the problem is converted to a quasi assignment problem, considering a partition of the time-control space. The approximate optimal control function is obtained as a piecewise constant function. Finally the numerical examples are given and by defining some tests, the results of applying evolutionary algorithms are compared. © 2010, INSInet Publication.

AUTHOR KEYWORDS: Approximation; Discretization; Evolutionary algorithm; Optimal control problem

This paper presents a measure-theoretical approach for finding approximate solution of complex optimal control problems. The general continuous optimal control problem for a class of hybrid dynamical systems is considered, and it is detailed that how to replace the problem by one in measure space and solve the equivalent finite-dimensional linear problem in a straightforward manner. Computational experiments show that the proposed algorithm could substantially reduce the solving time, compared with the other techniques. © ICIC International 2010.

AUTHOR KEYWORDS: Linear programming; Measure theory; Nonlinear hybrid system; Optimal control

INDEX KEYWORDS: Approximate solution; Computational experiment; Hybrid dynamical systems; Linear problems; Measure theory; Nonlinear hybrid system; Numerical scheme; Optimal control problem; Optimal controls, Algorithms; Equivalence classes; Hybrid computers; Hybrid systems; Optimization, Control

This paper presents a hybrid algorithm for solving complex optimal control problems based on decomposition. The general finite-time optimal control problem for a class of hybrid dynamical systems is considered, which has not been solved in a decomposed way by existing methods. The problem is first decomposed into a master problem and a subproblem, and then the two are linked via logic-based Benders decomposition. Computational experiments have been carried out for the considered problem. The results show that the proposed algorithm could substantially reduce the solving time, compared with directly solving by mixed integer solvers.

INDEX KEYWORDS: Benders decomposition; Computational experiment; Finite-time optimal control; Hybrid algorithms; Hybrid dynamical systems; Mixed integer; Optimal control problem; Optimal controls, Algorithms; Control; Hybrid computers; Optimization, Hybrid systems

DOI: 10.1016/j.cam.2009.06.038

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach. © 2009 Elsevier B.V. All rights reserved.

AUTHOR KEYWORDS: Approximation; Discretization; Iterative methods; Nonlinear Fredholm integral equations

INDEX KEYWORDS: Approximate solution; Approximation; Discretization; Exact solution; Iterative approach; Nonlinear Fredholm integral equation; Nonlinear Fredholm integral equations; Numerical scheme, Integral equations; Iterative methods, Nonlinear equations

DOI: 10.1016/j.amc.2007.01.096

In this paper, a measure-theoretical approach to find the approximate solutions for a class of first order nonlinear difference equations is introduced. In this method the problem is transformed to an equivalent optimization problem. Then, by considering it as a calculus of variations problem, some concepts in measure theory are used to approximate the solution. The procedure of constructing approximate solution in form of an algorithm is shown. Finally a numerical example is given. © 2007 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Calculus of variations; Difference equation; Linear programming; Measure theory

INDEX KEYWORDS: Approximation algorithms; Linear programming; Measurement theory; Nonlinear equations; Optimization; Problem solving, Approximate solutions; Calculus of variations; Nonlinear difference equations, Difference equations

DOI: 10.1016/j.amc.2005.04.009

In this paper, a new approach for finding an approximate solution for discrete optimal control problems is introduced. In this method the problem is transformed to a continuous optimal control problem whose solution may give rise to a good approximate solution for the original problem. Then, a measure-theoretical approach is applied to solve the new problem. The method is extended to solve time-optimal problems which is governed by a nonlinear discrete system. Finally, some numerical examples are proposed. © 2005 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Discrete optimal control; Linear programming; Measure theory

INDEX KEYWORDS: Approximation theory; Linear programming; Nonlinear systems, Discrete optimal control; Measure theory, Optimal control systems

DOI: 10.1016/j.amc.2005.04.008

In this paper, we proposed a method for finding a approximate solution for the nonlinear Fredholm integral equations of the second kind. In this approach, the nonlinearity of the kernel has no serious effect on the convergence of the solution. First, we convert the problem to an optimal control problem by introducing an artificial control function. Then by using some concepts of measure theory, we make a metamorphosis in the space of the problem and finally we will get a linear programming whose solution gives rise to the approximate solution of the integral equation. © 2005 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Fredholm integral equation; Linear programming; Measure theory; Optimal control problem

INDEX KEYWORDS: Convergence of numerical methods; Linear programming, Fredholm integral equations; Measure theory; Optimal control problem, Nonlinear equations

DOI: 10.1007/BF02831939

This paper describes a numerical scheme for optimal control of a time-dependent linear system to a moving final state. Discretization of the corresponding differential equations gives rise to a linear algebraic system. Defining some binary variables, we approximate the original problem by a mixed integer linear programming (MILP) problem. Numerical examples show that the resulting method is highly efficient. © 2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

AUTHOR KEYWORDS: Discretization; Mixed integer programming; Time-optimal control problem; Time-varying linear systems

PUBLISHER: Springer Verlag

DOI: 10.1016/j.amc.2004.08.002

In this paper we propose a method for finding the best linearization of nonlinear ODE problems by L2-norm. For this means we try to obtain the best approximation of a nonlinear system as a linear time-varying system. Our method for obtaining the above approximation is on the basis of analytic and numerical approaches. Finally we examine our methods to some numerical examples. © 2004 Elsevier Inc. All rights reserved.

AUTHOR KEYWORDS: Approximation theory; Discretization; Linear programming; Measure theory; Nonlinear systems

INDEX KEYWORDS: Approximation theory; Linear programming; Linearization; Time varying systems, Discretization; L2-norm; Linear time-varying system; Measure theory, Nonlinear systems

DOI: 10.1007/BF02936058

In this paper we present a new method for designing a nozzle. In fact the problem is to find the optimal domain for the solution of a linear or nonlinear boundary value PDE, where the boundary condition is defined over an unspecified domain. By an embedding process, the problem is first transformed to a new shape-measure problem, and then this new problem is replaced by another in which we seek to minimize a linear form over a subset of linear equalities. This minimization is global, and the theory allows us to develop a computational method to find the solution by a finite-dimensional linear programming problem.

AUTHOR KEYWORDS: Approximation theory; Linear programming; Measure theory; Nozzle problem; Optimal control; Optimal shape

PUBLISHER: Journal of Applied Mathemathics

DOI: 10.1007/BF02935750

In this paper we consider a heat flow in an inhomogeneous body without internal source. There exists special initial and boundary conditions in this system and we intend to find a convenient coefficient of heat conduction for this body so that body cool off as much as possible after definite time. We consider this problem in a general form as an optimal control problem which coefficient of heat conduction is optimal function. Then we replace this problem by another in which we seek to minimize a linear form over a subset of the product of two measures space defined by linear equalities. Then we construct an approximately optimal control.

AUTHOR KEYWORDS: Approximation theory; Heat equation; Linear programming; Measure theory

PUBLISHER: Journal of Applied Mathemathics

In this paper we consider an optimal control system de- scribed by n-dimensional heat equation with a thermal source. Thus problem is to find an optimal control which puts the system in a finite time T, into a stationary regime and to minimize a general objective func- tion. Here we assume there is no constraints on control. This problem is reduced to a moment problem.We modify the moment problem into one consisting of the minimization of a positive linear functional over a set of Radon measures and we show that there is an optimal measure corresponding to the optimal control. The above optimal measure ap- proximated by a finite combination of atomic measures. This construc- tion gives rise to a finite dimensional linear programming problem, where its solution can be used to determine the optimal combination of atomic measures. Then by using the solution of the above linear programming problem we find a piecewise-constant optimal control function which is an approximate control for the original optimal control problem. Finally we obtain piecewise-constant optimal control for two examples of heat equations with a thermal source in one-dimensional. © 2000 Korean Society for Computational & Applied Mathematics and Korean SIGCAM.

AUTHOR KEYWORDS: Approximation; Controllability; Heat equation; Linear programming; Measure theory; Moment problem

INDEX KEYWORDS: Controllability; Heat transfer; Linear programming; Optimal control systems, Approximation; Finite dimensional linear programming; Heat equation; Linear programming problem; Measure theory; Moment problems; Optimal control function; Positive linear functionals, Partial differential equations

PUBLISHER: Springer Verlag