Geometric approach to Pontryagin's Maximum Principle
Since the second half of the 20th century, Pontryagin’s Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geo...
| Autores: | , |
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| Formato: | artículo |
| Fecha de publicación: | 2008 |
| País: | España |
| Recursos: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/1987 |
| Acesso em linha: | https://hdl.handle.net/2117/1987 https://dx.doi.org/10.1007/s10440-008-9320-5 |
| Access Level: | acceso abierto |
| Palavra-chave: | Differential equations Mathematical optimization System theory Pontryagin's Maximum Principle perturbation vectors tangent perturbation cones optimal control problems Equacions diferencials ordinàries Optimització matemàtica Sistemes de control Classificació AMS::34 Ordinary differential equations::34A General theory Classificació AMS::49 Calculus of variations and optimal control optimization::49J Existence theories optimization::49K Necessary conditions and sufficient conditions for optimality Classificació AMS::93 Systems Theory Control::93C Control systems, guided systems |
| Resumo: | Since the second half of the 20th century, Pontryagin’s Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self–contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry. |
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