Advanced Methods in the Fractional Calculus of Variations

1. Introduction.- 2. Fractional Calculus.- 2.1. One-dimensional Fractional Calculus.- 2.2. Multidimensional Fractional Calculus.- 3. Fractional Calculus of Variations.- 3.1. Fractional Euler-Lagrange Equations.- 3.2. Fractional Embedding of Euler-Lagrange Equations.- 4. Standard Methods in Fractional Variational Calculus.- 4.1. Properties of Generalized Fractional Integrals.- 4.2. Fundamental Problem.- 4.3. Free Initial Boundary.- 4.4. Isoperimetric Problem.- 4.5. Noether's Theorem.- 4.6. Variational Calculus in Terms of a Generalized Integral.- 4.7. Generalized Variational Calculus of Several Variables.- 4.8. Conclusion.- 5. Direct Methods in Fractional Calculus of Variations.- 5.1. Existence of a Minimizer for a Generalized Functional.- 5.2. Necessary Optimality Condition for a Minimizer.- 5.3. Some Improvements.- 5.4. Conclusion.- 6. Application to the Sturm-Liouville Problem.- 6.1. Useful Lemmas.- 6.2. The Fractional Sturm-Liouville Problem.- 7. Conclusion.- Appendix - Two Convergence Lemmas.- Index.

Agnieszka B. Malinowska is an Assistant Professor in the Bialystok University of Technology, Poland, she is affiliated with this university since 1995. From 2008 to 2011 she was a Senior Researcher at the University of Aveiro, Portugal. She obtained her M.Sc. in Mathematics from the Warsaw University, Ph.D. and Habilitation  in Technical Sciences (Automation and Robotics) from the Systems Research Institute of the Polish Academy of Sciences. Her research interests include topics in the areas of Multiobjective Optimization, Calculus of Variations, Time Scale Theory, and Fractional Calculus. She is co-author of a book about the Fractional Calculus of Variations with Imperial College Press and World Scientific Publishing, 2012, a book on Quantum Variational Calculus with Springer, 2014, and  author and co-author of 38 papers in international journals ranked by ISI Web of Science.

Tatiana Odzijewicz is a Post-doc Researcher at the Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, Portugal. She received M.Sc. from University of Bialystok, Poland (2009), and PhD in Mathematics and Applications from University of Aveiro, Portugal (2013). Her research is mainly in fractional calculus, generalized operators, the calculus of variations and optimal control. Tatiana Odzijewicz has been recipient, in 2012, of the celebrated Grunvald-Letnikov Award.

Delfim F. M. Torres was born August 16, 1971, in Nampula, Mozambique. He is currently an Associate Professor with Habilitation at the Department of Mathematics, University of Aveiro (UA), Portugal; Coordinator of the group on Mathematical Theory of Systems and Control of the Center for Research and Development in Mathematics and Applications (CIDMA); Director of The FCT Doctoral Program in Mathematics and Applications (FCT MAP-PDMA) of the Universities of Minho, Aveiro,  Porto and UBI, Portugal; Editor-in-Chief of the Int. J. Appl. Math. Stat. (IJAMAS) since 2008; Editor-in-Chief

"The authors study variational problems containing generalized fractional integrals and derivatives, following Euler-Lagrange type methods and also the direct methods of Tonelli and others. ... Each chapter has its own reference list, which adds-up to a generous bibliography. The book will be attractive to graduate students wishing to enter this research area, as well as to researchers interested in control theory, optimization and other applications." (Ubiratan D'Ambrosio, zbMATH 1330.49001, 2016)