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.
Provides the reader with a unifying approach to the fractional calculus of variations
The first treatment of the fractional calculus of variations and optimal control with general kernels
Broadens accessibility of a recent and very active research area to a wider community
Describes many open problems and challenges to motivate the work of other control theorists
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 Multi objective 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 of the Int. J. Math. Stat. (IJMS) from 2007 to 2011; and Associate Editor for several other international journals. Delfim F. M. Torres received his Diploma ("Licenciatura") honors degree in 1994 on Computer Science Engineering (a 5 years full time course) from University of Coimbra, Portugal. In 1998 he received his M.Sc. on Optimization and Control Theory (2 years full time course) and in 2002 his Ph.D. on Mathematics (3 years full time research), both degrees from UA. He has done his Habilitation in Mathematics in 2011. Several M.Sc. dissertations, 9 Ph.D. theses, and 7 post-docs were successfully finished under his supervision; 7 more Ph.D. theses and 2 post-docs are ongoing.
Delfim F. M. Torres has written about 300 scientific and pedagogical publications, including research papers in first class international journals, refereed conference proceedings, chapters in books and books (as author and also as an editor). In particular, he has co-authored 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 56 papers in international journals ranked in the 1st quartile by ISI Web of Science, according with the Journal Citation Reports Science Edition as released in 2013. Delfim F. M. Torres has 20 years of excellent teaching experience both to undergraduate and postgraduate students on Mathematics and Science and Engineering, in Portugal and abroad. He has held several positions of Invited Professor in Europe, Africa, and USA. He is a member of Scientific Committees and invited plenary speaker at several international conferences. His current research interests include several topics in the areas of the Calculus of Variations and Optimal Control, Fractional Calculus and Time Scales and its connections with the Mathematical Theory of Control, Optimization, Epidemiology and Mathematical Physics.
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.