Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives |
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| Institution: | 1. College of Information Science and Technology, Donghua University, Shanghai 201620, PR China;2. School of Computer Science, China University of Geosciences, Wuhan 430074, PR China;3. School of Engineering (MESA-Lab), University of California, Merced, CA 95343, USA;4. Department of Applied Mathematics, Donghua University, Shanghai 201620, PR China;1. Laboratoire Lmdan, Département de Mathématiques de la Décision, Université Cheikh Anta Diop de Dakar, Faculté des Sciences Economiques et Gestion, Dakar Fann, BP 5683, Senegal;2. Centre de Recherche Economique Appliquées/Laboratoire Ingénierie Financiére et Economique (LIFE)/Faculté des Sciences Économiques et de Gestion, Université Cheikh Anta Diop de Dakar, Dakar Fann, BP 5683, Senegal;1. Department of Mathematics, Guizhou University, Guizhou, Guiyang 550025, China;2. School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong, China |
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| Abstract: | We prove optimality conditions for different variational functionals containing left and right Caputo fractional derivatives. A sufficient condition of minimization under an appropriate convexity assumption is given. An Euler–Lagrange equation for functionals where the lower and upper bounds of the integral are distinct of the bounds of the Caputo derivative is also proved. Then, the fractional isoperimetric problem is formulated with an integral constraint also containing Caputo derivatives. Normal and abnormal extremals are considered. |
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