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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 review on "Numerical solution of nonlinear Volterra-Fredholm integral equations using hybrid of . . ." [Alexandria Eng. J. 52 (2013) 551-555]

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ALEXANDRIA ENGINEERING JOURNAL 2019年第3期58卷 1099-1102页

作者： Sabzevari, Mehdi Univ Kashan Dept Appl Math Fac Math Sci Kashan Iran

This is a review on the published paper (Mirzaee and Hoseini, 2013), where there are some major scientific errors. After considering these errors, we attempt to correct them by presenting an accurate formula. In addition, to illustrate how the method works in practice after our correction, we introduce some numerical examples and present the corresponding results. Finally, we have a short review on the paper (Mirzaee and Hoseini, 2014) by the same authors. The same scientific errors are occurred in this paper, motivating us to correct them, as well. (C) 2019 The Authors. Published by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University.

关键词： Integral equations block-pulse functions Taylor polynomials

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Hybrid functions for nonlinear initial-value problems with applications to Lane-Emden type equations

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PHYSICS LETTERS A 2008年第37期372卷 5883-5886页

作者： Marzban, H. R. Tabrizidou, H. R. Razzaghi, M. Mississippi State Univ Dept Math & Stat Mississippi State MS 39762 USA Isfahan Univ Technol Dept Math Esfahan Iran Amir Kabir Univ Technol Dept Appl Math Tehran Iran

A numerical method for solving nonlinear initial-value problems is proposed. The Lane-Emden type equations which have many applications in mathematical physics are then considered. The method is based upon hybrid function approximations. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials are presented and are utilized to reduce the computation of nonlinear initial-value problems to a system of non-algebraic equations. The method is easy to implement and yields very accurate results. (C) 2008 Published by Elsevier B.V.

关键词： block-pulse functions Lagrange interpolating polynomials nonlinear initial-value problems Lane-Emden equation

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A hybrid approximation method for solving Hutchinson's equation

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COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION 2012年第1期17卷 100-109页

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

The hybrid function approximation method for solving Hutchinson's equation which is a nonlinear delay partial differential equation, is investigated. The properties of hybrid of block-pulse functions and Lagrange interpolating polynomials based on Legendre-Gauss-type points are presented and are utilized to replace the system of nonlinear delay differential equations resulting from the application of Legendre pseudospectral method, by a system of nonlinear algebraic equations. The validity and applicability of the proposed method are demonstrated through two illustrative examples on Hutchinson's equation. (C) 2011 Elsevier B.V. All rights reserved.

关键词： Nonlinear delay partial differential equations Hutchinson's equation Pseudospectral methods block-pulse functions Lagrange interpolating polynomials

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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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Solving linear integro-differential equation system by Galerkin methods with hybrid functions

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APPLIED MATHEMATICS AND COMPUTATION 2004年第3期159卷 603-612页

作者： Maleknejad, K Kajani, MT Islamic Azad Univ Fac Sci Dept Math Karaj Unit Rajae Shahr Karaj Iran Azad Univ Dept Math Khorasgan Esfahan Iran

In this paper, we use the hybrid Legendre and block-pulse functions on interval (0, 1) to solve the linear integro-differential equation system, and construct the quadrature formulae for the calculation of inner products of any functions, which are required in the Galerkin methods for integro-differential equation system. (C) 2003 Elsevier Inc. All rights reserved.

关键词： integro-differential equation operational matrix product operation legendre polynomials block-pulse functions quadrature formulae

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New sinusoidal basis functions and a neural network approach to solve nonlinear Volterra-Fredholm integral equations

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NEURAL COMPUTING & APPLICATIONS 2019年第9期31卷 4865-4878页

作者： Tomasiello, Stefania Macias-Diaz, Jorge E. Khastan, Alireza Alijani, Zahra Univ Salerno Consorzio Ric Sistemi ad Agenti CORISA Fisciano SA Italy Univ Autonoma Aguascalientes Dept Matemat & Fis Aguascalientes Mexico Inst Basic Sci Inst Adv Studies Dept Math Zanjan Iran Univ Tartu Dept Math & Stat Tartu Estonia

In this paper, we present and investigate the analytical properties of a new set of orthogonal basis functions derived from the block-pulse functions. Also, we present a numerical method based on this new class of functions to solve nonlinear Volterra-Fredholm integral equations. In particular, an alternative and efficient method based on the formalism of artificial neural networks is discussed. The efficiency of the mentioned approach is theoretically justified and illustrated through several qualitative and quantitative examples.

关键词： Nonlinear integral equations block-pulse functions sinusoidal basis functions learning rule

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Numerical solution of optimal control problems governed by integro-differential equations

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ASIAN JOURNAL OF CONTROL 2020年第3期22卷 1138-1146页

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

In this paper, an efficient hybrid approximation scheme for solving optimal control problems governed by integro-differential equations is proposed. The current approach is based on a generalization of the hybrid of block-pulse functions and Legendre's polynomials. An upper bound for the generalized hybrid functions with respect to the maximum norm is acquired and its convergence is demonstrated. The optimal control problem under study is transcribed to a mathematical programming one. Two illustrative examples are considered to verify the capability and reliability of the proposed procedure.

关键词： block-pulse functions generalized hybrid functions integro-differential equation legendre's polynomials optimal control

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THE GENERALIZED GEGENBAUER-HUMBERTS WAVELET FOR SOLVING FRACTIONAL DIFFERENTIAL EQUATIONS

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THERMAL SCIENCE 2020年第Sup1期24卷 S107-S118页

作者： Alkhalissi, Jumana H. S. Emiroglu, Ibrahim Secer, Aydin Bayram, Mustafa Yildiz Tech Univ Dept Math Engn Istanbul Turkey Biruni Univ Dept Comp Engn Istanbul Turkey

In this paper we present a new method of wavelets, based on generalized Gegenbauer-Humberts polynomials, named generalized Gegenbauer-Humberts wavelets. The operational matrix of integration are derived. By using the proposed method converted linear and non-linear fractional differential equation a system of algebraic equations. In addition, discussed some examples to explain the efficiency and accuracy of the presented method.

关键词： block-pulse functions operational matrix or integration the generalized Gegenbauer-Humberts polynomial fractional calculus orthogonal polynomials

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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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