Fractional Euler-Lagrange differential equations via Caputo derivatives

We review some recent results of the fractional variational calculus. Necessary optimality conditions of Euler-Lagrange type for functionals with a Lagrangian containing left and right Caputo derivatives are given. Several problems are considered: with fixed or free boundary conditions, and in presence of integral constraints that also depend on Caputo derivatives.Comment: This is a preprint of a paper whose final and definite form will appear as Chapter 9 of the book Fractional Dynamics and Control, D. Baleanu et al. (eds.), Springer New York, 2012, DOI:10.1007/978-1-4614-0457-6_9, in pres

Universality in Systems with Power-Law Memory and Fractional Dynamics

There are a few different ways to extend regular nonlinear dynamical systems by introducing power-law memory or considering fractional differential/difference equations instead of integer ones. This extension allows the introduction of families of nonlinear dynamical systems converging to regular systems in the case of an integer power-law memory or an integer order of derivatives/differences. The examples considered in this review include the logistic family of maps (converging in the case of the first order difference to the regular logistic map), the universal family of maps, and the standard family of maps (the latter two converging, in the case of the second difference, to the regular universal and standard maps). Correspondingly, the phenomenon of transition to chaos through a period doubling cascade of bifurcations in regular nonlinear systems, known as "universality", can be extended to fractional maps, which are maps with power-/asymptotically power-law memory. The new features of universality, including cascades of bifurcations on single trajectories, which appear in fractional (with memory) nonlinear dynamical systems are the main subject of this review.Comment: 23 pages 7 Figures, to appear Oct 28 201

Fractional variational calculus of variable order

We study the fundamental problem of the calculus of variations with variable order fractional operators. Fractional integrals are considered in the sense of Riemann-Liouville while derivatives are of Caputo type.Comment: Submitted 26-Sept-2011; accepted 18-Oct-2011; withdrawn by the authors 21-Dec-2011; resubmitted 27-Dec-2011; revised 20-March-2012; accepted 13-April-2012; to 'Advances in Harmonic Analysis and Operator Theory', The Stefan Samko Anniversary Volume (Eds: A. Almeida, L. Castro, F.-O. Speck), Operator Theory: Advances and Applications, Birkh\"auser Verlag (http://www.springer.com/series/4850

Time-Fractional Optimal Control of Initial Value Problems on Time Scales

We investigate Optimal Control Problems (OCP) for fractional systems involving fractional-time derivatives on time scales. The fractional-time derivatives and integrals are considered, on time scales, in the Riemann--Liouville sense. By using the Banach fixed point theorem, sufficient conditions for existence and uniqueness of solution to initial value problems described by fractional order differential equations on time scales are known. Here we consider a fractional OCP with a performance index given as a delta-integral function of both state and control variables, with time evolving on an arbitrarily given time scale. Interpreting the Euler--Lagrange first order optimality condition with an adjoint problem, defined by means of right Riemann--Liouville fractional delta derivatives, we obtain an optimality system for the considered fractional OCP. For that, we first prove new fractional integration by parts formulas on time scales.Comment: This is a preprint of a paper accepted for publication as a book chapter with Springer International Publishing AG. Submitted 23/Jan/2019; revised 27-March-2019; accepted 12-April-2019. arXiv admin note: substantial text overlap with arXiv:1508.0075

Fractional order optimal control problems with free terminal time

We consider fractional order optimal control problems in which the dynamic control system involves integer and fractional order derivatives and the terminal time is free. Necessary conditions for a state/control/terminal- time triplet to be optimal are obtained. Situations with constraints present at the end time are also considered. Under appropriate assumptions, it is shown that the obtained necessary optimality conditions become sufficient. Numer- ical methods to solve the problems are presented, and some computational simulations are discussed in detail

ДИСТАНЦІЙНЕ НАВЧАННЯ ЯК ІНФОРМАЦІЙНО-КОМУНІКАТИВНА ТЕХНОЛОГІЯ ДОДИПЛОМНОЇ ПІДГОТОВКИ ЛІКАРІВ З ПЕДІАТРІЇ В УМОВАХ СУЧАСНИХ ВИКЛИКІВ

For the implementation and adequate virtual provision of distance education at O. Bohomolets National Medical University, the NMU Neuron distance educational platform is used, which in the current situation can become an effective component of the educational process, while having a number of advantages. It not only provides the student with the most convenient place and time, access to electronic knowledge bases, but also significantly enhances his/her active role in the self-study process, including during the quarantined process, and requires the teacher to improve the teaching in terms of content, and in its form. The main types of distance learning courses are traditional forms of educational training: lectures, practical classes, seminars, labs, consultations and other relevant legislation. In general, distance learning technologies as a complex of modern educational technologies in the higher medical education system should be constantly improved, in particular through the use of modern educational platforms, qualitative methodological filling of web resources, introduction of the system of quality control of distance learning in order to meet the challenges of time.Для реалізації та адекватного віртуального забезпечення дистанційної освіти у Національному медичному університеті імені О. О. Богомольця використовується  дистанційна освітня платформа NMU Neuron, що  в нинішній ситуації може стати дієвою складовою освітнього процесу, маючи при цьому низку переваг. Воно не лише забезпечує навчання студента у найбільш зручному місці та часі, доступ до електронних баз знань, але й суттєво підвищує його активну роль у самому процесі самостійного навчання, зокрема й під час вимушеного внаслідок карантину, а від викладача вимагає вдосконалення викладання як за змістом, так і за його формою. Основними видами занять за дистанційною формою навчання є традиційні форми освітньої підготовки: лекція, практичні заняття, семінар, лабораторні заняття, консультації та інші, що відповідають чинному законодавству. У цілому, технології дистанційного навчання як комплекс сучасних освітніх технологій у системі вищої медичної освіти має постійно вдосконалюватися, зокрема шляхом використання сучасних освітніх платформ, якісного методичного наповнення веб-ресурсів, запровадження системи контролю якості дистанційного навчання, аби відповідати викликам часу

Fractional variational problems depending on indefinite integrals and with delay

The aim of this paper is to exhibit a necessary and sufficient condition of optimality for functionals depending on fractional integrals and derivatives, on indefinite integrals and on presence of time delay. We exemplify with one example, where we nd analytically the minimizer