Blanco Díaz, LuisSardón, CristinaJiménez Alburquerque, FernandoLucas, Javier de2026-01-192026-01-192023-06-19Blanco Díaz, L., Sardón, C., Jiménez Alburquerque, F., & de Lucas, J. (2023). Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control. Symmetry, 15(6). https://doi.org/10.3390/SYM15061285.https://doi.org/10.3390/sym15061285https://hdl.handle.net/20.500.14468/31468The registered version of this article, first published in Symmetry, is available online at the publisher's website: MDPI, https://doi.org/10.3390/sym15061285La versión registrada de este artículo, publicado por primera vez en Symmetry, está disponible en línea en el sitio web del editor: MDPI, https://doi.org/10.3390/sym15061285A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL(𝑛,ℝ) , which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given.eninfo:eu-repo/semantics/openAccess12 MatemáticasartículoLie group integrationgeometric numerical methodsnumerical methods for Lie systems2073-8994