descriptionPublicationkeyboard_double_arrow_right Article , Preprint , Other literature type 09 Oct 2008Embargo end date: 01 Jan 2008 English Publisher:Springer Science and Business Media LLCJournal:Acta Applicandae Mathematicae, volume 108, pages 429-485 (issn: 0167-8019, eissn: 1572-9036,

Authors: Barbero Liñán, María; Muñoz Lecanda, Miguel Carlos;

doi: 10.1007/s10440-008-9320-5 , 10.48550/arxiv.0805.1169

arXiv: http://arxiv.org/abs/0805.1169

handle: 2117/1987

Geometric Approach to Pontryagin’s Maximum Principle

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Abstract

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\textendash{}contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.

Final version. Minors changes have been made. 56 pages

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Keywords

Differential equations, 34A12, 49J15, 49J30,49K05, 49K15,93C15, Control::93C Control systems, Optimització matemàtica, :93C Control systems, guided systems [Control], Pontryagin's Maximum Principle, Sistemes de control, :34 Ordinary differential equations::34A General theory [Classificació AMS], FOS: Mathematics, guided systems, Control::93C Control systems, guided systems, Equacions diferencials ordinàries, System theory, Mathematics - Optimization and Control, :49K Necessary conditions and sufficient conditions for optimality [optimization], :93 Systems Theory [Classificació AMS], Mathematical optimization, :49J Existence theories [optimization], optimization::49K Necessary conditions and sufficient conditions for optimality, :49 Calculus of variations and optimal control [Classificació AMS], perturbation vectors, tangent perturbation cones, Classificació AMS::93 Systems Theory, Optimization and Control (math.OC), Classificació AMS::49 Calculus of variations and optimal control, optimization::49J Existence theories, optimal control problems, Classificació AMS::34 Ordinary differential equations::34A General theory

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