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- Computing ODE symmetries as abnormal variational symmetriesPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.
- Automatic computation of conservation laws in the calculus of variations and optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.
- Automatic computation of conservation laws in the calculus of variations and optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We present analytical computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in ¯nding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples are given.
- Computation of conservation laws in optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.Making use of a computer algebra system, we define computational tools to identify symmetries and conservation laws in optimal control.
- Computing ODE symmetries as abnormal variational symmetriesPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387 409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.
- Scientific computation of conservation laws in the calculus of variations and optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether’s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.
- Symbolic computation of variational symmetries in optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.; Rocha, Eugénio A.M.We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether’s first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8).
- Computação de simetrias variacionais e optimização da resistência aerodinâmica newtonianaPublication . Gouveia, Paulo D.F.Neste trabalho exploram-se alguns dos actuais recursos de computação científica no contexto da optimização estática e dinâmica. Começa-se por propor um conjunto de procedimentos computacionais algébricos que permitem automatizar todo o processo de obtenção de simetrias e leis de conservação, quer no contexto clássico do cálculo das variações, quer no contexto mais abrangente do controlo óptimo. A utilidade do package de funções desenvolvido é demonstrada com a identificação de novas leis de conservação para alguns problemas do controlo óptimo conhecidos na literatura. Estabelece-se depois uma relação entre as simetrias variacionais do controlo óptimo e as simetrias de equações diferenciais ordinárias. A partir dessa relação, deduz-se um método construtivo, alternativo aos já existentes, para obtenção de simetrias nesta segunda classe de problemas. Numa segunda parte do trabalho, investigam-se, com recurso a simulações computacionais, formas de corpos não convexos que maximizem a sua resistência aerodinâmica quando se desloquem em meios rarefeitos e, simultaneamente, exibam um ligeiro movimento rotacional. É obtido um importante resultado original para o caso bidimensional. Trata-se de uma forma geométrica que confere ao corpo uma resistência muito próxima do seu limite teórico (R=1.4965<1.5). In this thesis some of the scientific computational resources are explored in the context of static and dynamic optimization. A set of analytical computational tools is proposed in order to allow the identification, in an automatic way, of variational symmetries and conservation laws in the calculus of variations and optimal control. The usefulness of the developed routines is showed with the identification of new conservation laws to concrete optimal control problems found in the literature. A relationship between the variational symmetries of optimal control and the symmetries of ordinary differential equations is established. Based in this relationship, a constructive method is created for the purpose of getting the symmetries in this second class of problems. Finally, we investigate, by means of computational simulations, shapes of nonconvex bodies that maximize resistance to its motion on a rarefied medium, considering that bodies are moving forward and at the same time slowly rotating. An important result is obtained for the two-dimensional case which consists of a geometric shape that confers to the body a resistance very close to the supremum value (R = 1.4965 < 1.5). Some results of the thesis are available in the English language in the following references: the research reports [29, 35, 37, 79], the poster [36], the conference proceedings with referee [34] and the refereed journals [31, 32, 38, 80].