Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control
A 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. Thi...
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MDPI AG
2023-06-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/15/6/1285 |
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| author | Luis Blanco Díaz Cristina Sardón Fernando Jiménez Alburquerque Javier de Lucas |
| author_facet | Luis Blanco Díaz Cristina Sardón Fernando Jiménez Alburquerque Javier de Lucas |
| author_sort | Luis Blanco Díaz |
| collection | DOAJ |
| description | A 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<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, 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. |
| first_indexed | 2024-03-11T01:52:05Z |
| format | Article |
| id | doaj.art-6aaf7fef0afb43e38f391bc902116162 |
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| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-03-11T01:52:05Z |
| publishDate | 2023-06-01 |
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| series | Symmetry |
| spelling | doaj.art-6aaf7fef0afb43e38f391bc9021161622023-11-18T12:52:04ZengMDPI AGSymmetry2073-89942023-06-01156128510.3390/sym15061285Geometric Numerical Methods for Lie Systems and Their Application in Optimal ControlLuis Blanco Díaz0Cristina Sardón1Fernando Jiménez Alburquerque2Javier de Lucas3Department of Applied Mathematics, Universidad Politécnica de Madrid (UPM), c. José Gutiérrez Abascal 2, 28006 Madrid, SpainDepartment of Applied Mathematics, Universidad Politécnica de Madrid (UPM), c. José Gutiérrez Abascal 2, 28006 Madrid, SpainDepartment of Applied Mathematics, Universidad Politécnica de Madrid (UPM), c. José Gutiérrez Abascal 2, 28006 Madrid, SpainDepartment of Mathematical Methods in Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warsaw, PolandA 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<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi mathvariant="double-struck">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>, 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.https://www.mdpi.com/2073-8994/15/6/1285Lie group integrationgeometric numerical methodsnumerical methods for Lie systems |
| spellingShingle | Luis Blanco Díaz Cristina Sardón Fernando Jiménez Alburquerque Javier de Lucas Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control Symmetry Lie group integration geometric numerical methods numerical methods for Lie systems |
| title | Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control |
| title_full | Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control |
| title_fullStr | Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control |
| title_full_unstemmed | Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control |
| title_short | Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control |
| title_sort | geometric numerical methods for lie systems and their application in optimal control |
| topic | Lie group integration geometric numerical methods numerical methods for Lie systems |
| url | https://www.mdpi.com/2073-8994/15/6/1285 |
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