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A hybrid approach to parameter identification of linear delay differential equations involving multiple delays

INTERNATIONAL JOURNAL OF CONTROL 2018年第5期91卷 1034-1052页

作者： Marzban, Hamid Reza Isfahan Univ Technol Dept Math Sci Esfahan Iran

In this paper, we are concerned with the parameter identification of linear time-invariant systems containing multiple delays. The approach is based upon a hybrid of block-pulse functions and Legendre's polynomials. The convergence of the proposed procedure is established and an upper error bound with respect to the L-2-norm associated with the hybrid functions is derived. The problem under consideration is first transformed into a system of algebraic equations. The least squares technique is then employed for identification of the desired parameters. Several multi-delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method. It is shown that the proposed approach is also applicable to a class of nonlinear multi-delay systems. It is demonstrated that the suggested procedure provides accurate results for the desired parameters.

关键词： Parameter identification block-pulse functions Legendre's polynomials hybrid functions multi-delay systems least squares technique

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Optimal control of linear multi-delay systems with piecewise constant delays

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IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION 2018年第1期35卷 183-212页

作者： Marzban, Hamid Reza Hoseini, Sayyed Mohammad Isfahan Univ Technol Dept Math Sci POB *** Esfahan Iran

This article is concerned with the numerical solution of optimal control of linear multi-delay systems involving piecewise constant time delays. The proposed approach is based on direct method using a hybrid of block-pulse functions and Legendre polynomials. Combining block-pulse functions and Legendre polynomials allows one to simultaneously make use of the properties of the two mentioned bases. The operational matrices of delay and derivative corresponding to the proposed hybrid functions are constructed. Two error bounds related to hybrid functions are established. In addition, the necessary conditions of optimality are derived. These conditions are expressed in the form of a Pontryagin type maximum principle. The excellent properties of the hybrid functions together with the associated operational matrices are used to transform the delayed optimal control problem into a parameter optimization problem whose solution is much easier than the original one. An illustrative example is investigated to demonstrate the effectiveness of the suggested numerical scheme.

关键词： block-pulse functions Legendre polynomials hybrid functions piecewise constant delay optimal control multi-delay systems necessary optimality conditions

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A direct approach for the solution of nonlinear optimal control problems with multiple delays subject to mixed state-control constraints

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APPLIED MATHEMATICAL MODELLING 2018年 53卷 189-213页

作者： Marzban, H. R. Pirmoradian, H. Isfahan Univ Technol Dept Math Sci POB *** Esfahan Iran

This paper deals with the numerical investigation of nonlinear optimal control problems with multiple delays in which the state trajectory and control input are subject to mixed state-control constraints. A direct approach based on a hybrid of block-pulse functions and Lagrange interpolation is proposed. The constrained optimal control problem is first reformulated as an unconstrained optimization one using a penalty function technique. The resulting optimization problem is then solved by means of the Lagrange multipliers procedure. The proposed framework is an extension and also a modification of the conventional Lagrange interpolation. Combining block-pulse functions and Lagrange interpolation allows one to simultaneously make use the advantages of the two mentioned bases. The operational matrices of delay and derivative associated with the hybrid functions are presented. An upper error bound for the proposed hybrid functions with respect to the maximum norm is obtained. Simulation studies are provided to verify the validity and reliability of the developed procedure. (C) 2017 Elsevier Inc. All rights reserved.

关键词： Nonlinear optimal control Multiple delays block-pulse functions Lagrange interpolation Hybrid functions Operational matrix

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A novel approach for the numerical investigation of optimal control problems containing multiple delays

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OPTIMAL CONTROL APPLICATIONS & METHODS 2018年第1期39卷 302-325页

作者： Marzban, Hamid Reza Pirmoradian, Hoda Isfahan Univ Technol Dept Math Sci Esfahan Iran

This paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block-pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimizaion problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points Furthermore, 2 upper bounds on the error with respect to the L-2-norm and infinity norm are established. Several optimal control problems containing multiple delays are carried out to illustrate the various aspects of the proposed approach. The simulation results are compared with either analytical or numerical solutions available in the literature.

关键词： block-pulse functions hybrid functions Lagrange interpolating polynomials mathematical programming problem multidelay systems optimal control

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Solution of fractional Volterra-Fredholm integro-differential equations under mixed boundary conditions by using the HOBW method

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ADVANCES IN DIFFERENCE EQUATIONS 2019年第1期2019卷 115-115页

作者： Ali, Mohamed R. Hadhoud, Adel R. Srivastava, H. M. Benha Univ Benha Fac Engn Dept Math Banha Egypt Menoufia Univ Fac Sci Dept Math Shibin Al Kawm Egypt Univ Victoria Dept Math & Stat Victoria BC Canada China Med Univ China Med Univ Hosp Dept Med Res Taichung Taiwan

A new approximate technique is introduced to find a solution of FVFIDE with mixed boundary conditions. This paper started from the meaning of Caputo fractional differential operator. The fractional derivatives are replaced by the Caputo operator, and the solution is demonstrated by the hybrid orthonormal Bernstein and block-pulse functions wavelet method (HOBW). We demonstrate the convergence analysis for this technique to emphasize its reliability. The applicability of the HOBW is demonstrated using three examples. The approximate results of this technique are compared with the correct solutions, which shows that this technique has approval with the correct solutions to the problems.

关键词： Orthonormal Bernstein block-pulse functions Wavelet method Fractional integro-differential equations Fractional calculus Approximate solution

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block pulse算子矩阵法求解变系数时间分数阶对流扩散方程（英文）

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高等学校计算数学学报 2019年第4期41卷 346-357页

作者：王进斌马立峰解加全郭蓉太原科技大学机械工程学院太原科技大学重型机械教育部工程研究中心太原理工大学机械与运载工程学院太原科技大学应用科学学院工

The numerical technique based on two-dimensional block pulse functions(2D-BPFs) is proposed for solving the time fractional convection diffusion equations with variable coeficients(FCDEs).We introduce the block pulse operational matrices of the fractional order ***,we translate the original equation into a Sylvester equation by the proposed ***,some numerical examples are given and numerical results are shown to demonstrate the accuracy and reliability of the above-mentioned algorithm.

关键词： Time-fractional convection diffusion equations block-pulse functions Operational matrix Caputo derivative

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Application of the hybrid functions to solve neutral delay functional differential equations

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2017年第3期94卷 503-514页

作者： Sedaghat, S. Nemati, S. Ordokhani, Y. Buein Zahra Tech Univ Dept Math Buein Zahra Qazvin Iran Univ Mazandaran Fac Math Sci Dept Math Babol Sar Iran Alzahra Univ Fac Math Sci Dept Math Tehran Iran

This paper presents a computational technique for the solution of the neutral delay differential equations with state-dependent and time-dependant delays. The properties of the hybrid functions which consist of block-pulse functions plus Legendre polynomials are presented. The approach uses these properties together with the collocation points to reduce the main problems to systems of nonlinear algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.

关键词： neutral delay differential equation function approximation hybrid Legend repolynomials block-pulse functions

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A comparative study of numerical integration based on block-pulse and sinc functions and Chebyshev wavelet

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MATHEMATICAL SCIENCES 2012年第1期6卷 8-8页

作者： Rostami, Majid Hashemizadeh, Elham Heidari, Mohammd Islamic Azad Univ Dept Math Naragh Branch Naragh Iran Islamic Azad Univ Karaj Branch Young Res Club 2 Karaj Iran

In this paper, numerical integration rules based on block-pulse functions and Chebyshev wavelet are proposed to find approximate values of definite integrals. Errors of these numerical integrations are given. These numerical integrations are compared by sinc functions numerical integration method. Some numerical examples are provided to illustrate the accuracy of proposed rules and comparison between them. The main advantage of proposed numerical integration methods are their efficiency and simple applicability.

关键词： Numerical integration block-pulse functions Sinc functions Chebyshev wavelet

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Optimal control of linear multi-delay systems based on a multi-interval decomposition scheme

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OPTIMAL CONTROL APPLICATIONS & METHODS 2016年第1期37卷 190-211页

作者： Marzban, Hamid Reza Isfahan Univ Technol Dept Math Sci Esfahan Iran

This paper presents a novel approximation scheme to the numerical treatment of linear time-varying multi-delay systems with a quadratic performance index. A direct approach based on a hybrid of block-pulse functions and Chebyshev polynomials is successfully developed. The operational matrix of delay associated to multi-delay systems is constructed by an efficient manner. The excellent properties of hybrid functions together with the operational matrices of integration, delay, and product are then used to transform the optimal control problem into a mathematical optimization problem whose solution is much more easier than the original one. The procedure described in the current paper can be regarded as a multi-interval decomposition scheme. The convergence of the proposed method is verified numerically. A wide variety of multi-delay systems are investigated to demonstrate the effectiveness and computational efficiency of the proposed numerical scheme. The method has a simple structure, is easy to implement, and provides very accurate solutions. Copyright (C) 2015 John Wiley & Sons, Ltd.

关键词： optimal control multi-delay block-pulse functions Chebyshev polynomials hybrid functions operational matrix

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Numerical treatment of non-linear optimal control problems involving piecewise constant delay

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IMA JOURNAL OF MATHEMATICAL CONTROL AND INFORMATION 2016年第4期33卷 1103-1134页

作者： Marzban, Hamid Reza Hoseini, Sayyed Mohammad Isfahan Univ Technol Dept Math Sci Esfahan Iran

An efficient approximation scheme for solving non-linear optimal control problems with a piecewise constant delay function is introduced. The proposed method is based on a hybrid of block-pulse functions and Lagrange interpolating polynomials using the well-known Legendre-Gauss-Lobatto points. A direct approach is applied to transform the main problem into a finite-dimensional optimization problem whose solution is much easier than the original one. The operational matrix of derivative associated with the hybrid functions is constructed. The necessary conditions of optimality corresponding to non-linear piecewise constant delay systems are derived. Several numerical experiments are investigated to evaluate the performance and computational efficiency of the proposed procedure. The method is easy to implement and provides very accurate results.

关键词： non-linear optimal control piecewise constant delay systems block-pulse functions Lagrange interpolating polynomials hybrid functions

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