The Herglotz variational problem on spheres and its optimal control approach

The main goal of this paper is to extend the generalized variational problem of Herglotz type to the more general context of the Euclidean sphere S^n. Motivated by classical results on Euclidean spaces, we derive the generalized Euler-Lagrange equation for the corresponding variational problem defined on the Riemannian manifold S^n. Moreover, the problem is formulated from an optimal control point of view and it is proved that the Euler-Lagrange equation can be obtained from the Hamiltonian equations. It is also highlighted the geodesic problem on spheres as a particular case of the generalized Herglotz problem

Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration

We consider a second-order variational problem depending on the covariant acceleration, which is related to the notion of Riemannian cubic polynomials. This problem and the corresponding optimal control problem are described in the context of higher order tangent bundles using geometric tools. The main tool, a presymplectic variant of Pontryagin’s maximum principle, allows us to study the dynamics of the control problem

Variational and optimal control approaches for the second-order Herglotz problem on spheres

The present paper extends the classical second–order variational problem of Herglotz type to the more general context of the Euclidean sphere Sn following variational and optimal control approaches. The relation between the Hamiltonian equations and the generalized Euler-Lagrange equations is established. This problem covers some classical variational problems posed on the Riemannian manifold Sn such as the problem of finding cubic polynomials on S^n. It also finds applicability on the dynamics of the simple pendulum in a resistive medium.publishe

Some applications of quasi-velocities in optimal control

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinematic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelerations. The formulation of the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently (Maruskin and Bloch, 2007). We also provide several examples to illustrate our construction.Comment: Revtex 4.1, 20 pages. To appear in Int. J. Geom. Meth. Modern Physic

Lagrangian Lie subalgebroids generating dynamics for second-order mechanical systems on Lie algebroids

The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems, and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew’s triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symplectic Lie algebroid, by building on the geometric formalism for mechanics on Lie algebroids developed by M. de León, J.C. Marrero and E. Martínez on “Lagrangian submanifolds and dynamics on Lie algebroids”publishe

An intrinsic version of the k-harmonic equation

The notion of k-harmonic curves is associated with the kth-order variational problem defined by the k-energy functional. The present paper gives a geometric formulation of this higher-order variational problem on a Riemannian manifold M and describes a generalized Legendre transformation defined from the kth-order tangent bundle $T^kM$ to the cotangent bundle $T^*T^{k-1}M$. The intrinsic version of the Euler–Lagrange equation and the corresponding Hamiltonian equation obtained via the Legendre transformation are achieved. Geodesic and cubic polynomial interpolation is covered by this study, being explored here as harmonic and biharmonic curves. The relationship of the variational problem with the optimal control problem is also presented for the case of biharmonic curves.publishe

The tax behavior of taxpayers: literature review

O desenvolvimento das sociedades, a crescente educação e consciencialização dos indivíduos acerca dos impostos e medidas fiscais, conduziu a alterações no comportamento dos contribuintes no que diz respeito à perceção e cumprimento fiscal. Recolhendo e sintetizando a literatura existente, efetuámos uma revisão de literatura de forma a perceber quais as variáveis que influenciam os comportamentos dos contribuintes no que respeita ao seu (in)cumprimento e de que forma se manifestam na sociedade de hoje. A revisão efetuada permitiu-nos concluir que existem diversos fatores influentes no comportamento de cumprimento fiscal, podendo-se destacar os fatores económicos, os fatores comportamentais, sociais e psicológicos e os fatores políticos ou institucionais. Alguns estudos permitiram concluir que tanto a auditoria fiscal como a penalização fiscal são dissuasoras de evasão fiscal, embora haja opiniões divergentes no que respeita a qual das duas medidas seja mais eficaz no combate à evasão fiscal. No que diz respeito à perceção de equidade do sistema fiscal, verificou-se que esta é influenciada pelo conhecimento fiscal e por fatores sociais dos contribuintes, sendo que este último afeta também o comportamento de cumprimento dos contribuintes quando acreditam que o incumprimento é consistente com as expectativas e normas do grupo social em que se inserem. Verificou-se que na presença de um nível elevado de educação, há uma maior aceitação dos impostos sobre o rendimento e das taxas corretivas, porém, os contribuintes são movidos pelo seu interesse próprio. Por fim, concluiu-se que os fatores políticos ou institucionais devem ter em conta sobretudo o conhecimento e perceção que os contribuintes têm acerca da atuação da administração fiscal e a opinião destes acerca das alterações de forma a não fomentar o comportamento de incumprimento fiscal.The development of society, the increasing education and consciousness of individuals about taxes and fiscal measures leads to changes in taxpayer’s behavior regarding tax compliance and tax perception. In this paper, we present a literature review in order to understand which variables have impact on these behaviors, using a synthesis methodology that allows an interpretation of social actions from previous studies. The results show that either the tax audit as the penalty rate are deterrents of tax evasion, although there are divergent opinions about which one is most effective. Regarding equity perception of tax system, it have been observed that the level of knowledge and social factors affect that equity perception, when the taxpayers believe that noncompliance is consistent with in-group expectation and norms. On the other hand, it was found evidence that in the presence of a high level of education, there are a great acceptability of income taxes and corrective taxes. Nevertheless, taxpayers are often moved by self-interest. We concluded also that it is interesting to consider political factors. In particular, the analyses of taxpayers’ perception concerning on the activities of the tax administration and the fiscal changes to not induce a noncompliance behavior

Polinómios cúbicos Riemannianos : abordagem Hamiltoniana e generalizações

Tese de doutoramento em Matemática (Matemática Pura) apresentada à Faculdade de Ciências e Tecnologia da Universidade de CoimbraEsta dissertação é dedicada ao estudo dos polinómios cúbicos Riemannianos e a algumas generalizações deste conceito e da teoria envolvente, no sentido a seguir ex- plicado. O trabalho contribui essencialmente para um novo formalismo Hamiltoniano e da ênfase µa situação em que temos como espaço de configuração um grupo de Lie conexo e compacto. O trabalho é iniciado com a exposição do problema variacional clássico de segunda ordem, que permite definir as curvas conhecidas como polinómios cúbicos Riemanni- anos e a análise de alguns dos invariantes ao longo destas curvas. No âmbito dos fibrados tangentes de ordem superior, apresentamos a versão intrínseca das equações de Euler-Lagrange e ainda a correspondente abordagem Hamiltoniana resultante da transformação de Legendre generalizada. Introduzimos o problema de controlo óptimo dos polinómios cúbicos Riemannianos, cujo sistema de controlo está associado ao pro- blema variacional destes polinómios. Para o efeito, é adaptada para ordem dois, a formulação geométrica de um sistema de controlo de primeira ordem. Prosseguimos depois para a descrição Hamiltoniana deste problema de controlo, através de uma vari- ante presimpléctica do princípio do máximo de Pontryagin e aplicamos o respectivo algoritmo de restrição. É discutida também a relação existente entre os formalismos Lagrangiano e de controlo óptimo apresentados. Os resultados são concretizados para os polinómios cúbicos em grupos de Lie conexos e compactos e esta situação é sim- plificada com a trivialização µa esquerda do sistema Hamiltoniano simpléctico obtido. Analisamos as simetrias do sistema, recorrendo ao método de redução simpléctica de Marsden-Weinstein, obtendo no final um sistema com menos graus de liberdade do que o inicial. Exemplificamos as abordagens expostas, com a apresentação do problema de controlo óptimo dinâmico do corpo rígido livre e esférico. Numa segunda etapa, estendemos o problema de controlo óptimo dos polinómios cúbicos em grupos de Lie a um problema com um sistema de controlo de conexão afim mais geral. Mais concretamente, e em paralelismo com o estudo feito para os polinómios cúbicos, é explorado o formalismo presimpléctico e a trivialização do sis- tema Hamiltoniano simpléctico, obtido para este problema mais geral. Relacionamos a dinâmica estudada para o referido problema de controlo óptimo, com a dinâmica de um problema variacional com restrições, que aparece na literatura como uma extensão do clássico problema variacional dos polinómios cúbicos. Apresentamos alguns exemplos elementares, que ilustram bem a abordagem apresentada. Concluímos o trabalho com o enquadramento dos problemas de controlo óptimo estudados ao longo da tese, na teoria mais geral dos algebróides de Lie. Sob este ponto de vista, enriquecemos o texto com alguns exemplos de problemas (cinemáticos e dinâmicos) relacionados com o corpo rígido. Alguns dos resultados desta tese foram já objecto de publicação em [2, 3, 4, 5, 6, 7, 8].This thesis is devoted to the study of Riemannian cubic polynomials and to some generalizations of this concept and the related theory, in the sense explained below. The work contributes essentially to a new Hamiltonian formalism and gives emphasis to the situation where the con¯guration space is a compact and connected Lie group. The work begins with an exposition of the classical second order variational prob- lem that de¯nes the curves known as Riemannian cubic polynomials and the analysis of some invariants along these curves. In the context of higher order tangent bundles, we present the intrinsic version of the Euler-Lagrange equations and the correspond- ing Hamiltonian approach resulting from the generalized Legendre transformation. We introduce the optimal control problem of Riemannian cubic polynomials, whose con- trol system is associated with the variational problem of these polynomials. For this purpose, the geometric formulation of a ¯rst order control system is adapted to order two. We proceed then to the Hamiltonian description of this control problem, using a presymplectic variant of the Pontryagin maximum principle, and apply the appropri- ate contraint algorithm. The relation between the introduced Lagrangian and optimal control formalims is also discussed. The results are implemented for the cubic polyno- mials on compact and connected Lie groups and this situation is simpli¯ed with the left trivialization of the obtained symplectic Hamiltonian system. We analyze the sym- metries of the system using the symplectic reduction procedure of Marsden-Weinstein, getting in the end a system with fewer degrees of freedom than the original one. We also exemplify our approaches with the dynamic optimal control problem of the free and spherical rigid body. In a second step, we extend the optimal control problem of cubic polynomials on Lie groups to a more general problem with an a±ne connection control system. More speci¯cally, in parallel with the study of cubic polynomials, we explore the presymplec- tic formalism and the trivialization of the symplectic Hamiltonian system obtained for this further general problem. We relate the dynamics studied for the above optimal control problem, with the dynamics of a variational problem with constraints, which appears in the literature as an extension of the classical variational problem of cubic polynomials. Some elementary examples that illustrate our approach are also given. We conclude with a framework in the more general theory of Lie algebroids of the optimal control problems studied throughout the thesis. From this point of view, we provide some examples of (kinematic and dynamic) problems related to the rigid body. Some of the results of the thesis can be found in [2, 3, 4, 5, 6, 7, 8].Este trabalho foi apoiado pelo programa PROTEC 2008, Programa de apoio à formação avançada de docentes do Ensino Superior Politécnico da Fundação para a Ciência e a Tecnologia (FCT). O trabalho teve ainda o apoio do projecto intitulado Generalized Lagrangian and Hamiltonian Techniques for Geometric Control Theory, no âmbito do programa Ações Integradas Luso-Espanholas 2007 do Conselho de Reitores das Universidades Portuguesas (Ação E-3/07)

Corrigendum: Cubic polynomials on Lie groups: reduction of the Hamiltonian system

The purpose of this corrigendum is to replace lemma 6 on page 13 of the paper to guarantee the accuracy of other results derived from it, in particular, the discussion after remark 4 on page 15. In the original version, the result we prove does not allow us to conclude, as we claim, that the set of constants of the motion we identify can be used with the Lie–Cartan theorem. The formulation of the lemma is misleading. Besides, we need the additional hypothesis that G is semisimple to be able to prove the correct statement. Therefore, both the statement and the proof should be replaced by the following.publishe

Optimal control and quasi-velocities

In this paper we study optimal control problems for nonholonomic systems defined on Lie algebroids by using quasi-velocities. We consider both kinemactic, i.e. systems whose cost functional depends only on position and velocities, and dynamic optimal control problems, i.e. systems whose cost functional depends also on accelarations. Formulating the problem directly at the level of Lie algebroids turns out to be the correct framework to explain in detail similar results appeared recently [48]. We also provide several examples to illustrate our constructio