Fractional Calculus in Analysis, Dynamics and Optimal Control

Jacky Cresson (Editor)
Professor, Laboratory of Applied Mathematics, University of Pau
Avenue University, Pau Cedex, France

Series: Mathematics Research Developments
BISAC: MAT027000

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Volume 10

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Special issue: Resilience in breaking the cycle of children’s environmental health disparities
Edited by I Leslie Rubin, Robert J Geller, Abby Mutic, Benjamin A Gitterman, Nathan Mutic, Wayne Garfinkel, Claire D Coles, Kurt Martinuzzi, and Joav Merrick

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This book is devoted to applications of fractional calculus in classical fields of mathematics like analysis, dynamics, partial differential equations and optimal control. The first chapter deals with the notion of local fractional derivatives and its applications to the study of regularity and geometry of curves. The second chapter develops the notion of fractional embedding and fractional assymetric calculus of variations in order to find fractional Lagrangian variational structures for classical dissipative partial differential equations.

In continuation of this chapter, a fractional analogue of the classical Pontryagin maximum principle is proved for discrete and continuous fractional optimal control problems. The fourth chapter gives a first mathematical model that allows a rigorous connection to be made between the dynamics of chaotic Hamiltonian systems and fractional dynamics, mixing the previous approaches of G. Zaslavsky and R. Hilfer. Finally, numerical methods to deal with fractional optimal control problems are discussed and implemented. All the chapters are self-contained and complete proofs are given. (Imprint: Nova)

Introduction

Chapter I. Local Fractional Derivatives
(N.C. Dias and J.N. Prata, University of Lisboa, Portugal)

Chapter II. Fractional Variational Embedding and Lagrangian Formulations of Dissipative Partial Differential Equations
(Jacky Cresson, University of Pau and Observatoire de Paris, Syrte, France)

Chapter III. A Class of Fractional Optimal Control Problems and Fractional Pontryagin’s Systems. Variational Integrator and Existence of Continuous/Discrete Noether’s Theorems
(Loic Bourdin, Laboratoire de Mathematiques et de leurs Applications de Pau, Universite de Pau et des Pays del' Adour, Pau Cedex, France)

Chapter IV. Fractal Traps and Fractional Dynamics
(Pierre Inizan, IMCCE, Observatoire de Paris, France)

Chapter V. Numerical Approximations to Fractional Problems of the Calculus of Variations and Optimal Control
(Shakoor Pooseh, Ricardo Almeida and Delfim F.M. Torres, University of Aveiro, Portugal)

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