Euler-Lagrange equations for composition functionals in calculus of variations on time scales

In this paper we consider the problem of the calculus of variations for a functional which is the composition of a certain scalar function $H$ with the delta integral of a vector valued field $f$, i.e., of the form $H(\int_{a}^{b}f(t,x^{\sigma}(t),x^{\Delta}(t))\Delta t)$. Euler-Lagrange equations, natural boundary conditions for such problems as well as a necessary optimality condition for isoperimetric problems, on a general time scale, are given. A number of corollaries are obtained, and several examples illustrating the new results are discussed in detail.Comment: Submitted 10-May-2009 to Discrete and Continuous Dynamical Systems (DCDS-B); revised 10-March-2010; accepted 04-July-201

A General Backwards Calculus of Variations via Duality

We prove Euler-Lagrange and natural boundary necessary optimality conditions for problems of the calculus of variations which are given by a composition of nabla integrals on an arbitrary time scale. As an application, we get optimality conditions for the product and the quotient of nabla variational functionals.Comment: Submitted to Optimization Letters 03-June-2010; revised 01-July-2010; accepted for publication 08-July-201

Direct and Inverse Variational Problems on Time Scales: A Survey

We deal with direct and inverse problems of the calculus of variations on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we give a general form for a variational functional to attain a local minimum at a given point of the vector space. Furthermore, we provide a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation (Helmholtz's problem of the calculus of variations on time scales). New and interesting results for the discrete and quantum settings are obtained as particular cases. Finally, we consider very general problems of the calculus of variations given by the composition of a certain scalar function with delta and nabla integrals of a vector valued field.Comment: This is a preprint of a paper whose final and definite form will be published in the Springer Volume 'Modeling, Dynamics, Optimization and Bioeconomics II', Edited by A. A. Pinto and D. Zilberman (Eds.), Springer Proceedings in Mathematics & Statistics. Submitted 03/Sept/2014; Accepted, after a revision, 19/Jan/201

Transversality Conditions for Infinite Horizon Variational Problems on Time Scales

We consider problems of the calculus of variations on unbounded time scales. We prove the validity of the Euler-Lagrange equation on time scales for infinite horizon problems, and a new transversality condition.Comment: Submitted 6-October-2009; Accepted 19-March-2010 in revised form; for publication in "Optimization Letters"

The Persistence and Interaction of Multi-Ethnic Settlement Remnants in The Cultural Landscape

The paper deals with remnants of multi-ethnic settlement: their form, interaction and persistence. In the past, the Podlasie region, situated in northeastern Poland, was an area of multi-ethnic settlement. The interaction of cultures brought the emergence of a new, borderland culture. As the years have passed, the memory of the sources of regional and local traditions has disappeared. Elements of ethnic and religious traditions have spread and survived in the material structures of the rural landscape. The most significant traces of cultural interactions and at the same time the remnants of past landscape are high roadside wooden crosses with an additional small iron cross on their top, decorated with the crescent moon and sunbeams. The cross with half-moon has its beginnings in old Christian symbolism, regional history and tradition. The crescent was always accompanied by sunbeams and they meant sun and moon, day and night, Christ and Our Lady. Its material durability appears to be greater than the collective memory of the locals. The roadside wooden crosses embellished with iron crescent cross are an interesting part of regional heritage. The symbol of the crescent was common here for all Christian inhabitants and Tatars, unifying all Podlasie people. This uniting symbol is the most valuable remnant of the interaction of multi-ethnic settlement in the cultural landscape of the Podlasie. These days, the 300 years of tradition falls into oblivion, but regional cultural heritage can be saved through tourism-related product and marketing. In peripheral, economically neglected areas like the study case, the remnants may become an impetus to develop the local economy through recreation and tourism. Furthermore, making new tourism products based on natural and cultural values can be a good opportunity to restore precious elements of the historical landscape

Investigation of the thermal stability of Mg/Co periodic multilayers for EUV applications

We present the results of the characterization of Mg/Co periodic multilayers and their thermal stability for the EUV range. The annealing study is performed up to a temperature of 400\degree C. Images obtained by scanning transmission electron microscopy and electron energy loss spectroscopy clearly show the good quality of the multilayer structure. The measurements of the EUV reflectivity around 25 nm (~49 eV) indicate that the reflectivity decreases when the annealing temperature increases above 300\degreeC. X-ray emission spectroscopy is performed to determine the chemical state of the Mg atoms within the Mg/Co multilayer. Nuclear magnetic resonance used to determine the chemical state of the Co atoms and scanning electron microscopy images of cross sections of the Mg/Co multilayers reveal changes in the morphology of the stack from an annealing temperature of 305\degreee;C. This explains the observed reflectivity loss.Comment: Published in Applied Physics A: Materials Science \& Processing Published at http://www.springerlink.com.chimie.gate.inist.fr/content/6v396j6m56771r61/ 21 page

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

On consensus in the Cucker--Smale type model on isolated time scales

This article addresses a consensus phenomenon in a Cucker-Smale model where the magnitude of the step size is not necessarily a constant but it is a function of time. In the considered model the weights of mutual influences in the group of agents do not change. A sufficient condition under which the proposed model tends to a consensus is obtained. This condition strikingly demonstrates the importance of the graininess function in a consensus phenomenon. The results are illustrated by numerical simulations.publishe