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...

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Veröffentlicht in:Acta applicandae mathematicae Jg. 108; H. 2; S. 429 - 485
Hauptverfasser: Barbero-Liñán, M., Muñoz-Lecanda, M. C.
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Dordrecht Springer Netherlands 01.11.2009
Springer Nature B.V
Schlagworte:
34 Ordinary differential equations
34A General theory
49 Calculus of variations and optimal control; optimization
49J Existence theories
49K Necessary conditions and sufficient conditions for optimality
93 Systems Theory; Control
93C Control systems, guided systems
Aircraft
Applications of Mathematics
Applied mathematics
Calculus of Variations and Optimal Control; Optimization
Classificació AMS
Computational Mathematics and Numerical Analysis
Control
Control theory
Differential equations
Equacions diferencials ordinàries
Flow control
Geometry
Mathematical analysis
Mathematical optimization
Mathematics
Mathematics and Statistics
optimal control problems
Optimització matemàtica
Optimization
Partial Differential Equations
perturbation vectors
Pontryagin's Maximum Principle
Probability Theory and Stochastic Processes
Robotics
Sistemes de control
Studies
System theory
tangent perturbation cones
49J30
Optimal control problems
34A12
93C15
Pontryagin’s maximum principle
Perturbation vectors
49K05
49K15
Tangent perturbation cones
49J15
ISSN:0167-8019, 1572-9036, 1572-9036
Online-Zugang:Volltext
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Beschreibung
Zusammenfassung: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.
Bibliographie:SourceType-Scholarly Journals-1
ObjectType-Feature-1
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ISSN:0167-8019
1572-9036
1572-9036
DOI:10.1007/s10440-008-9320-5