A computer algebra package, for the automatic computation of variational symmetries in optimal control, was recently developed by the authors [2,3]. Now we present a graphical user interface which permit to interact, in a point-and-click environment, with all the previous symbolical tools

Gouveia, Paulo D.F.

Torres, Delfim F.M.

Biblioteca Digital do IPB

A Maple interface for computing variationalsymmetries in optimal controlPaulo D. F. Gouveia*, Delfim F. M. TorresDepartment of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal[pgouveia@ipb.pt, delfim@mat.ua.pt]2000 Mathematics Subject Classification. 49-04, 49K15, 49S05The concept of variational symmetry entered into optimal control in the seventiesof the twentieth century. Variational symmetries, which keep an optimal controlproblem invariant, are very useful in optimal control, but unfortunately their studyis not easy, requiring lengthy and cumbersome calculations. Recently there hasbeen an interest in the application of Computer Algebra Systems to the study ofcontrol systems, and collections of symbolical tools are being developed to helpon the analysis and solution of complex problems. The first computer algebrapackage for computing the variational symmetries in the calculus of variations, andrespective Noether’s first integrals, was given by the authors in [1]; then extendedto the more general setting of optimal control [2] and, more recently, upgraded in [3]with the introduction of new capacities, by means of several optional parameters,and improvements of efficiency.Here we provide a graphical user interface to our computer algebra package[3]. This application is named octool and was created with Maplet technol-ogy, the graphical programming language of the Maple 10 system. With thisinterface users can, in a point-and-click environment, interact with all the sym-bolical tools of the package and deal with concrete problems of optimal control:(i) with the procedure Symmetry, to obtain the variational symmetries; (ii) withthe procedure Noether, to obtain the correspondent conservation laws; and (iii)with the PMP (Pontryagin Maximum Principle), to try to obtain the Pontryaginextremals or, alternatively, the equations of the Hamiltonian system, stationarycondition or the Hamiltonian. We refer the reader to [2, 3] for a general overviewon these Maple procedures. The Maplet octool allow us to quickly investigatethe problems, without learning all the optional parameters of the Maple proce-dures [1]–[3]. Moreover, it permits additional algebraic manipulations. The com-plete Maple package, with the new procedure octool, can be freely obtained fromhttp://www.mat.ua.pt/delfim/maple.htm together with many practical exam-ples and an online help database for the Maple system.[1] Gouveia, P.D.F., Torres, D.F.M., Computac¸a˜o Alge´brica no Ca´lculo das Variac¸o˜es:Determinac¸a˜o de Simetrias e Leis de Conservac¸a˜o (in Portuguese), TEMA Tend. Mat.Apl. Comput. 6 (2005), No. 1, 81–90.[2] Gouveia, P.D.F., Torres, D.F.M., Automatic Computation of Conservation Laws inthe Calculus of Variations and Optimal Control, Comput. Methods Appl. Math. 5(2005), No. 4, 387–409. [MR2194205] [Zbl 1079.49019][3] Gouveia, P.D.F., Torres, D.F.M., Rocha, E.A.M., Symbolic Computation of Varia-tional Symmetries in Optimal Control, Control & Cybernetics (Accepted. To appear,presumably, on No. 3/2006). E-Print: http://pam.pisharp.org/handle/2052/113.

A Maple interface for computing variational symmetries in optimal control

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A Maple interface for computing variational symmetries in optimal control

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