Repositório Colecção:http://hdl.handle.net/10198/25162017-06-22T22:22:50Z2017-06-22T22:22:50ZAutomatic computation of conservation laws in the calculus of variations and optimal controlGouveia, Paulo D.F.Torres, Delfim F.M.http://hdl.handle.net/10198/19702014-05-07T15:55:06Z2006-01-01T00:00:00ZTítulo: Automatic computation of conservation laws in the calculus of variations and optimal control
Autor: Gouveia, Paulo D.F.; Torres, Delfim F.M.
Resumo: 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.
Descrição: Computer Application2006-01-01T00:00:00ZComputing ODE symmetries as abnormal variational symmetriesGouveia, Paulo D.F.Torres, Delfim F.M.http://hdl.handle.net/10198/19652014-05-07T15:55:17Z2008-01-01T00:00:00ZTítulo: Computing ODE symmetries as abnormal variational symmetries
Autor: Gouveia, Paulo D.F.; Torres, Delfim F.M.
Resumo: 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.
Descrição: Computer Application2008-01-01T00:00:00ZA double parabola with retroreflection propertiesGouveia, Paulo D.F.http://hdl.handle.net/10198/19642015-05-25T14:21:01Z2009-01-01T00:00:00ZTítulo: A double parabola with retroreflection properties
Autor: Gouveia, Paulo D.F.
Resumo: This Demonstration shows the reflective behavior of a two-dimensional shape formed by similar arcs of two parabolas. An interesting and unexpected reflective behavior arises when this shape takes a specific configuration, with two parabolas of unit focal length. In this particular configuration it is found that, except for the rays with absolute value of the entry angle less than 19.47º, a ray emerges from the cavity with a trajectory that is nearly opposite to its entry trajectory. This characterizes a cavity with retroreflective behavior. Even in the case of trajectories with entry angle below 19.47º, the exit direction does not appear to vary greatly from the entry direction.2009-01-01T00:00:00ZReflection of parallel rays by a two-dimensional body of nearly maximal resistanceGouveia, Paulo D.F.http://hdl.handle.net/10198/19622015-05-25T14:21:35Z2009-01-01T00:00:00ZTítulo: Reflection of parallel rays by a two-dimensional body of nearly maximal resistance
Autor: Gouveia, Paulo D.F.
Resumo: This Demonstration shows a class of nonconvex bodies that maximize resistance as they move forward while slowly rotating through a rarefied medium. This class of shapes yields a resistance very close to the theoretical maximum, improving all previous results found by the authors.2009-01-01T00:00:00Z