Cálculo das variações em escalas temporais e aplicações à economia

Doutoramento em MatemáticaWe consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.Consideramos alguns problemas do cálculo das variações em escalas temporais. Primeiramente, demonstramos equações do tipo de Euler-Lagrange e condições de transversalidade para problemas de horizonte infinito generalizados. De seguida, consideramos a composição de uma certa função escalar com os integrais delta e nabla de um campo vetorial. Presta-se atenção a problemas extremais inversos para funcionais variacionais em escalas de tempo arbitrárias. Começamos por demonstrar uma condição necessária para uma equação dinâmica integro-diferencial ser uma equação de Euler-Lagrange. Resultados novos e interessantes para o cálculo discreto e quantum são obtidos como casos particulares. Além disso, usando a equação de Euler-Lagrange e a condição de Legendre fortalecida, obtemos uma forma geral para uma funcional variacional atingir um mínimo local, num determinado ponto do espaço vetorial. No final, duas aplicações interessantes em termos económicos são apresentadas. No primeiro caso, consideramos uma empresa que quer programar as suas políticas de produção e de investimento para alcançar uma determinada taxa de produção e maximizar a sua competitividade no mercado futuro. O outro problema é mais complexo e relaciona a inflação e o desemprego, que inflige uma perda social. A perda social é escrita como uma função da taxa de inflação p e a taxa de desemprego u, com diferentes pesos. Em seguida, usando as relações conhecidas entre p, u, e a taxa de inflação esperada π, reescrevemos a função de perda social como uma função de π. A resposta é obtida através da aplicação do cálculo das variações, a fim de encontrar a curva ótima π que minimiza a perda social total ao longo de um determinado intervalo de tempo

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

The delta-nabla calculus of variations for composition functionals on time scales

Abstract We develop the calculus of variations on time scales for a functional that is the composition of a certain scalar function with the delta and nabla integrals of a vector valued field. Euler-Lagrange equations, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. Interesting corollaries and examples are presented