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Torres D.f.m. - One of the best experts on this subject based on the ideXlab platform.
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Discrete direct methods in the fractional Calculus of Variations
Elsevier, 2024Co-Authors: Pooseh S., Almeida R., Torres D.f.m.Abstract:Finite differences, as a subclass of direct methods in the Calculus of Variations, consist in discretizing the objective functional using appropriate approximations for derivatives that appear in the problem. This article generalizes the same idea for fractional variational problems. We consider a minimization problem with a Lagrangian that depends on the left Riemann-Liouville fractional derivative. Using the Grünwald-Letnikov definition, we approximate the objective functional in an equispaced grid as a multi-variable function of the values of the unknown function on mesh points. The problem is then transformed to an ordinary static optimization problem. The solution to the latter problem gives an approximation to the original fractional problem on mesh points
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Leitmann's direct method of optimization for absolute extrema of certain problems of the Calculus of Variations on time scales
Elsevier, 2024Co-Authors: Malinowska A.b., Torres D.f.m.Abstract:The fundamental problem of the Calculus of Variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical Calculus of Variations as particular cases. In this note we follow Leitmann's direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale. © 2010 Elsevier Inc. All rights reserved
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Euler-Lagrange equations for composition functionals in Calculus of Variations on time scales
'American Institute of Mathematical Sciences (AIMS)', 2024Co-Authors: Malinowska A.b., Torres D.f.m.Abstract:In this paper we consider the problem of the Calculus of Variations for a functional which is the composition of a certain scalar function H with the delta integral of a vector valued field f, i.e., of the form H (int;ba f(t,xσ(t); δ(t))δt). Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail
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Generalized fractional Calculus with applications to the Calculus of Variations
'Elsevier BV', 2024Co-Authors: Odzijewicz T., Malinowska A.b., Torres D.f.m.Abstract:We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the Calculus of Variations with general fractional operators. Necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann-Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered
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Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the Calculus of Variations
'American Institute of Mathematical Sciences (AIMS)', 2024Co-Authors: Torres D.f.m.Abstract:For nonsmooth Euler-Lagrange extremals, Noether's conservation laws cease to be valid. We show that Emmy Noether's theorem of the Calculus of Variations is still valid in the wider class of Lipschitz functions, as long as one restrict the Euler-Lagrange extremals to those which satisfy the DuBoisReymond necessary condition. In the smooth case all Euler-Lagrange extremals are DuBois-Reymond extremals, and the result gives a proper extension of the classical Noether's theorem. This is in contrast with the recent developments of Noether's symmetry theorems to the optimal control setting, which give rise to non-proper extensions when specified for the problems of the Calculus of Variations. Results are also obtained for variational problems with higher-order derivatives
Delfim F. M. Torres - One of the best experts on this subject based on the ideXlab platform.
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the variable order fractional Calculus of Variations
arXiv: Optimization and Control, 2019Co-Authors: Ricardo Almeida, Dina Tavares, Delfim F. M. TorresAbstract:This book intends to deepen the study of the fractional Calculus, giving special emphasis to variable-order operators. It is organized in two parts, as follows. In the first part, we review the basic concepts of fractional Calculus (Chapter 1) and of the fractional Calculus of Variations (Chapter 2). In Chapter 1, we start with a brief overview about fractional Calculus and an introduction to the theory of some special functions in fractional Calculus. Then, we recall several fractional operators (integrals and derivatives) definitions and some properties of the considered fractional derivatives and integrals are introduced. In the end of this chapter, we review integration by parts formulas for different operators. Chapter 2 presents a short introduction to the classical Calculus of Variations and review different variational problems, like the isoperimetric problems or problems with variable endpoints. In the end of this chapter, we introduce the theory of the fractional Calculus of Variations and some fractional variational problems with variable-order. In the second part, we systematize some new recent results on variable-order fractional Calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018). In Chapter 3, considering three types of fractional Caputo derivatives of variable-order, we present new approximation formulas for those fractional derivatives and prove upper bound formulas for the errors. In Chapter 4, we introduce the combined Caputo fractional derivative of variable-order and corresponding higher-order operators. Some properties are also given. Then, we prove fractional Euler-Lagrange equations for several types of fractional problems of the Calculus of Variations, with or without constraints.
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advanced methods in the fractional Calculus of Variations
2015Co-Authors: Agnieszka B Malinowska, Tatiana Odzijewicz, Delfim F. M. TorresAbstract:This brief presents a general unifying perspective on the fractional Calculus. It brings together results of several recent approaches in generalizing the least action principle and the EulerLagrange equations to include fractional derivatives.The dependence of Lagrangians on generalized fractional operators as well as on classical derivatives is considered along with still more general problems in which integer-order integrals are replaced by fractional integrals. General theorems are obtained for several types of variational problems for which recent results developed in the literature can be obtained as special cases. In particular, the authors offer necessary optimality conditions of EulerLagrange type for the fundamental and isoperimetric problems, transversality conditions, and Noether symmetry theorems. The existence of solutions is demonstrated under Tonelli type conditions. The results are used to prove the existence of eigenvalues and corresponding orthogonal eigenfunctions of fractional SturmLiouville problems.Advanced Methods in the Fractional Calculus of Variations is a self-contained text which will be useful for graduate students wishing to learn about fractional-order systems. The detailed explanations will interest researchers with backgrounds in applied mathematics, control and optimization as well as in certain areas of physics and engineering.
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the dubois reymond fundamental lemma of the fractional Calculus of Variations and an euler lagrange equation involving only derivatives of caputo
Journal of Optimization Theory and Applications, 2013Co-Authors: Matheus J. Lazo, Delfim F. M. TorresAbstract:Derivatives and integrals of noninteger order were introduced more than three centuries ago but only recently gained more attention due to their application on nonlocal phenomena. In this context, the Caputo derivatives are the most popular approach to fractional Calculus among physicists, since differential equations involving Caputo derivatives require regular boundary conditions. Motivated by several applications in physics and other sciences, the fractional Calculus of Variations is currently in fast development. However, all current formulations for the fractional variational Calculus fail to give an Euler–Lagrange equation with only Caputo derivatives. In this work, we propose a new approach to the fractional Calculus of Variations by generalizing the DuBois–Reymond lemma and showing how Euler–Lagrange equations involving only Caputo derivatives can be obtained.
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Isoperimetric problems of the Calculus of Variations with fractional derivatives
Acta Mathematica Scientia, 2012Co-Authors: Ricardo Almeida, Rui A C Ferreira, Delfim F. M. TorresAbstract:Abstract In this article, we study isoperimetric problems of the Calculus of Variations with left and right Riemann-Liouville fractional derivatives. Both situations when the lower bound of the variational integrals coincide and do not coincide with the lower bound of the fractional derivatives are considered.
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noether s symmetry theorem for nabla problems of the Calculus of Variations
Applied Mathematics Letters, 2010Co-Authors: Natalia Martins, Delfim F. M. TorresAbstract:Abstract We prove a Noether-type symmetry theorem and a DuBois–Reymond necessary optimality condition for nabla problems of the Calculus of Variations on time scales.
Agnieszka B Malinowska - One of the best experts on this subject based on the ideXlab platform.
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advanced methods in the fractional Calculus of Variations
2015Co-Authors: Agnieszka B Malinowska, Tatiana Odzijewicz, Delfim F. M. TorresAbstract:This brief presents a general unifying perspective on the fractional Calculus. It brings together results of several recent approaches in generalizing the least action principle and the EulerLagrange equations to include fractional derivatives.The dependence of Lagrangians on generalized fractional operators as well as on classical derivatives is considered along with still more general problems in which integer-order integrals are replaced by fractional integrals. General theorems are obtained for several types of variational problems for which recent results developed in the literature can be obtained as special cases. In particular, the authors offer necessary optimality conditions of EulerLagrange type for the fundamental and isoperimetric problems, transversality conditions, and Noether symmetry theorems. The existence of solutions is demonstrated under Tonelli type conditions. The results are used to prove the existence of eigenvalues and corresponding orthogonal eigenfunctions of fractional SturmLiouville problems.Advanced Methods in the Fractional Calculus of Variations is a self-contained text which will be useful for graduate students wishing to learn about fractional-order systems. The detailed explanations will interest researchers with backgrounds in applied mathematics, control and optimization as well as in certain areas of physics and engineering.
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introduction to the fractional Calculus of Variations
2012Co-Authors: Agnieszka B MalinowskaAbstract:The Classical Calculus of Variations Fractional Calculus of Variations via Riemann - Liouville Operators Fractional Calculus of Variations via Caputo Operators Other Approaches to the Fractional Calculus of Variations Towards a Combined Fractional Mechanics and Quantization.
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leitmann s direct method of optimization for absolute extrema of certain problems of the Calculus of Variations on time scales
Applied Mathematics and Computation, 2010Co-Authors: Agnieszka B Malinowska, Delfim F. M. TorresAbstract:Abstract The fundamental problem of the Calculus of Variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical Calculus of Variations as particular cases. In this note we follow Leitmann’s direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale.
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natural boundary conditions in the Calculus of Variations
Mathematical Methods in The Applied Sciences, 2010Co-Authors: Agnieszka B MalinowskaAbstract:We prove necessary optimality conditions for problems of the Calculus of Variations on time scales with a Lagrangian depending on the free end-point. Copyright © 2010 John Wiley & Sons, Ltd.
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a fractional Calculus of Variations for multiple integrals with application to vibrating string
Journal of Mathematical Physics, 2010Co-Authors: Ricardo Almeida, Agnieszka B Malinowska, Delfim F. M. TorresAbstract:We introduce a fractional theory of the Calculus of Variations for multiple integrals. Our approach uses the recent notions of Riemann–Liouville fractional derivatives and integrals in the sense of Jumarie. The main results provide fractional versions of the theorems of Green and Gauss, fractional Euler–Lagrange equations, and fractional natural boundary conditions. As an application we discuss the fractional equation of motion of a vibrating string.
Malinowska A.b. - One of the best experts on this subject based on the ideXlab platform.
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Leitmann's direct method of optimization for absolute extrema of certain problems of the Calculus of Variations on time scales
Elsevier, 2024Co-Authors: Malinowska A.b., Torres D.f.m.Abstract:The fundamental problem of the Calculus of Variations on time scales concerns the minimization of a delta-integral over all trajectories satisfying given boundary conditions. This includes the discrete-time, the quantum, and the continuous/classical Calculus of Variations as particular cases. In this note we follow Leitmann's direct method to give explicit solutions for some concrete optimal control problems on an arbitrary time scale. © 2010 Elsevier Inc. All rights reserved
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Euler-Lagrange equations for composition functionals in Calculus of Variations on time scales
'American Institute of Mathematical Sciences (AIMS)', 2024Co-Authors: Malinowska A.b., Torres D.f.m.Abstract:In this paper we consider the problem of the Calculus of Variations for a functional which is the composition of a certain scalar function H with the delta integral of a vector valued field f, i.e., of the form H (int;ba f(t,xσ(t); δ(t))δt). Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail
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Generalized fractional Calculus with applications to the Calculus of Variations
'Elsevier BV', 2024Co-Authors: Odzijewicz T., Malinowska A.b., Torres D.f.m.Abstract:We study operators that are generalizations of the classical Riemann-Liouville fractional integral, and of the Riemann-Liouville and Caputo fractional derivatives. A useful formula relating the generalized fractional derivatives is proved, as well as three relations of fractional integration by parts that change the parameter set of the given operator into its dual. Such results are explored in the context of dynamic optimization, by considering problems of the Calculus of Variations with general fractional operators. Necessary optimality conditions of Euler-Lagrange type and natural boundary conditions for unconstrained and constrained problems are investigated. Interesting results are obtained even in the particular case when the generalized operators are reduced to be the standard fractional derivatives in the sense of Riemann-Liouville or Caputo. As an application we provide a class of variational problems with an arbitrary kernel that give answer to the important coherence embedding problem. Illustrative optimization problems are considered
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The contingent epiderivative and the Calculus of Variations on time scales
'Informa UK Limited', 2024Co-Authors: Girejko E., Malinowska A.b., Torres D.f.m.Abstract:The Calculus of Variations on time scales is considered. We propose a new approach to the subject that consists of applying a differentiation tool called the contingent epiderivative. It is shown that the contingent epiderivative applied to the Calculus of Variations on time scales is very useful: it allows to unify the delta and nabla approaches previously considered in the literature. Generalized versions of the Euler-Lagrange necessary optimality conditions are obtained, both for the basic problem of the Calculus of Variations and isoperimetric problems. As particular cases one gets the recent delta and nabla results. © 2012 Taylor and Francis Group, LLC
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Natural boundary conditions in the Calculus of Variations
John Wiley and Sons, 2024Co-Authors: Malinowska A.b., Torres D.f.m.Abstract:We prove necessary optimality conditions for problems of the Calculus of Variations on time scales with a Lagrangian depending on the free end-point. Copyright © 2010 John Wiley & Sons, Ltd.CEOCFCTFEDER/ POCI 201
Y. Imura - One of the best experts on this subject based on the ideXlab platform.
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Unified approach for Euler-Lagrange equation arising in Calculus of Variations
Proceedings of the 2003 American Control Conference 2003., 2024Co-Authors: Desineni Subbaram Naidu, Y. ImuraAbstract:We address the development of a unified approach for the necessary conditions for optimization of a functional arising in Calculus of Variations. In particular, we develop a unified approach for the Euler-Lagrange equation that is simultaneously applicable to both shift (q)-operator-based discrete-time systems and the derivative (d/dt)-operator-based continuous-time systems. It is shown that the Euler-Lagrange results are now obtained separately for continuous-time and discrete-time systems can be easily obtained form the unified approach. An illustrative example is given.
