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Mittag-Leffler wavelets and their applications for solving fractional optimal control problems

JOURNAL OF VIBRATION AND control 2025年第5-6期31卷 753-767页

作者： Ghasempour, Arezoo Ordokhani, Yadollah Sabermahani, Sedigheh Alzahra Univ Fac Math Sci Dept Math Tehran Iran Alzahra Univ Fac Math Sci Dept Math Vanak St Tehran *** Iran

Herein, we design a new scheme for finding approximate solutions to fractional optimal control problems (OCPs) with and without delay. In this strategy, we introduce Mittag-Leffler wavelet functions and develop a new Riemann-Liouville fractional integral operator for these functions utilizing the hypergeometric function. The properties of the operational matrix have reflected well in the process of the numerical method and affect the accuracy of the proposed method directly. Employing the Riemann-Liouville fractional integral operator, delay operational matrix, and Galerkin method, the considered problems lead to systems of algebraic equations. An error analysis is proposed. Finally, some illustrative numerical tests are given to show the precision and validity of the suggested technique. The proposed method is very efficient for solving the OCPs with delay and without delay, and gives very accurate results.

关键词： Mittag-Leffler wavelet optimal control problems hypergeometric function numerical method operational matrix convergence analysis

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A New Finite Element Method for Elliptic optimal control problems with Pointwise State Constraints in Energy Spaces

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JOURNAL OF SCIENTIFIC COMPUTING 2025年第1期102卷 1-29页

作者： Gong, Wei Tan, Zhiyu Chinese Acad Sci Acad Math & Syst Sci Inst Computat Math NCMIS Beijing Peoples R China Chinese Acad Sci Inst Computat Math Acad Math & Syst Sci LSEC Beijing Peoples R China Xiamen Univ Sch Math Sci Xiamen 361005 Fujian Peoples R China Xiamen Univ Fujian Prov Key Lab Math Modeling & High Performan Xiamen 361005 Fujian Peoples R China

In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use energy space regularizations in the objective functional, while the equivalent representations of the energy space norms, i.e., the H-1(Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{-1}(\varOmega )$$\end{document}-norm for the distributed control, the H1/2(Gamma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{1/2}(\varGamma )$$\end{document}-norm for the Dirichlet control and the H-1/2(Gamma)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H<^>{-1/2}(\varGamma )$$\end{document}-norm for the Neumann control, enable us to transform the optimal control problem into an elliptic variational inequality involving only the state variable. The elliptic variational inequalities are second order for the three cases, and include additional equality constraints for Dirichlet or Neumann boundary control problems. Standard C0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C<^>0$$\end{document} finite element methods can be used to solve the resulted variational inequalities. We provide preliminary a priori error es

关键词： optimal control problems Distributed control Boundary control Pointwise state constraints Energy space method Variational inequalities Finite element method

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optimal-control problems INVOLVING 2ND BOUNDARY-VALUE-problems OF PARABOLIC TYPE

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SIAM JOURNAL ON control AND OPTIMIZATION 1983年第5期21卷 729-757页

作者： WU, ZS TEO, KL UNIV NEW S WALES DEPT APPL MATHKENSINGTONNSW 2033AUSTRALIA

Three classes of optimal control problems involving second boundary value problems of parabolic type are considered. The controls are assumed to act through the forcing terms and through the initial and boundary condi... 详细信息

关键词： parabolic differential equations second boundary value problems optimal control problems necessary and sufficient conditions existence theory algorithms convergence of algorithms

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optimal control problems of phase relaxation models

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 2001年第3期109卷 557-585页

作者： Barbu, V Bernardi, ML Colli, P Gilardi, G Univ Al I Cuza Dept Math Iasi Romania Univ Pavia Dept Math F I-27100 Pavia Italy

This work is concerned with optimal control problems with convex cost criterion governed by the relaxed Stefan problem with or without memory. The existence of an optimal control is proved and necessary conditions for a given function to be an optimal control are found. Moreover, an asymptotic analysis is performed as the time relaxation parameter tends to zero.

关键词： optimal control problems relaxed Stefan problem memory effects existence of optimal control asymptotic analysis in terms of relaxation coefficient necessary conditions of optimality

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optimal control problems of Sources in Distributed Systems on the Classes of Impulsive, Piecewise Constant and Heaviside Functions

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JOURNAL OF AUTOMATION AND INFORMATION SCIENCES 2011年第5期43卷 64-82页

作者： Aida-zade, K. R. Ashrafova, Ye R. Azerbaijan State Oil Acad Baku Azerbaijan Natl Acad Sci Azerbaijan Inst Cybernet Baku Azerbaijan

problems of optimal control of movable lumped sources in distributed systems when controls are determined by impulsive piecewise constant and Heaviside functions are considered. The optimal control problems are investigated for various cases according to the position of the sources. The necessary conditions of optimality are obtained with respect to all considered problems of optimal control on these classes, which consists in deriving constructive analytical formulas for the gradient of a functional in the space of optimized parameters. The results of some numerical experiments are also given.

关键词： optimal control problems distributed systems optimality condition piecewise constant function gradient of functional

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optimal control problems of parabolic equations with an infinite number of variables and with equality constraints

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IMA JOURNAL OF MATHEMATICAL control AND INFORMATION 2008年第1期25卷 37-48页

作者： Bahaa, G. M. Beni Suef Univ Fac Sci Dept Math Bani Suwayf Egypt

optimal control problems of systems governed by parabolic equations with an infinite number of variables and with additional equality constraints are considered. The extremum principle, as well as sufficient condition... 详细信息

关键词： optimal control problems parabolic equations with an infinite number of variables equality constraints Dubovitskii-Milyutin theorem conical approximations optimality conditions

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optimal control problems for Complex Heat Transfer Equations with Fresnel Matching Conditions

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2022年第3期62卷 372-381页

作者： Chebotarev, A. Yu Russian Acad Sci Inst Appl Math Far Eastern Branch Vladivostok 690041 Russia

A class of optimal control problems for a system of nonlinear elliptic equations simulating radiative heat transfer with Fresnel matching conditions on the surfaces of discontinuity of the refractive index is considered. Based on estimates for the solution of the boundary value problem, the solvability of the optimal control problems is proved. The existence and uniqueness of the solution of a linearized problem with the matching conditions is analyzed, and the nondegeneracy of the optimality conditions is proved. As an example, a control problem with boundary observation is considered and the relay-like character of the optimal control is shown.

关键词： stationary equations of radiative heat transfer Fresnel matching conditions optimal control problems optimality conditions relay control

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optimal control problems of mean-field forward-backward stochastic differential equations with partial information

Optimal control problems of mean-field forward-backward stoc...

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第25届中国控制与决策会议

作者： ZUO Shanshan MIN Hui School of Mathematics and Statistics Shandong University

ISBN: (纸本)9781467355339

This paper mainly works on an optimal control problem of mean-field forward-backward stochastic differential equations （MFFBSDEs） with partial information. But different from the general optimal control problems, this paper is concerned with the case of partial information and state equations are coupled at initial time. Meanwhile, we introduce the mean-field theory. By virtue of the classical convex variational technique, we establish a necessary maximum principle for the optimization problems.

关键词： MFFBSDEs partial information optimal control problems maximum principle.

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MESH INDEPENDENCE OF THE GRADIENT PROJECTION METHOD FOR optimal-control problems

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SIAM JOURNAL ON control AND OPTIMIZATION 1992年第2期30卷 477-493页

作者： KELLEY, CT SACHS, EW North Carolina State University Raleigh

The gradient projection method and its application to optimal control have been analyzed with regard to the rate of convergence quite extensively. Here another aspect of this method is considered. In finite dimension the set of active indices is identified after finitely many iterations under mild nondegeneracy assumptions. However, it is clear that this property is restricted to the finite-dimensional case. In this paper, for example, a sequence of discretized optimal control problems is considered, and it is observed that the number of steps to identify all active indices increases with the refinement of the discretization. A result analogous to the finite-dimensional result is valid in this situation if identification of active indices is understood in the correct light. This paper shows that if a different termination criterion is imposed, then the number of necessary steps for termination is indeed mesh-independent. Numerical observations illustrating this result are reported for various examples from optimal control.

关键词： GRADIENT PROJECTION METHOD FINITE IDENTIFICATION OF ACTIVE INDEXES MESH INDEPENDENCE optimal control problems

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High accuracy analysis of nonconforming MFEM for constrained optimal control problems governed by Stokes equations

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APPLIED MATHEMATICS LETTERS 2016年 53卷 17-24页

作者： Guan, Hongbo Shi, Dongyang Guan, Xiaofei Zhengzhou Univ Light Ind Coll Math & Informat Sci Zhengzhou 450002 Peoples R China Zhengzhou Univ Sch Math & Stat Zhengzhou 450001 Peoples R China Tongji Univ Dept Math Shanghai 200092 Peoples R China

In this paper, we propose a stable nonconforming mixed finite element method (MFEM) for the constrained optimal control problems (OCPs) governed by Stokes equations, in which the EQ(1)(rot)-constant scheme just satisfies the discrete inf-sup condition. The superclose and superconvergence results are obtained. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Nonconforming mixed finite element Supercloseness and superconvergence optimal control problems Stokes equations

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