March  2010, 3(1): 1-17. doi: 10.3934/dcdss.2010.3.1

Strict abnormal extremals in nonholonomic and kinematic control systems


INRIA (Projet CORIDA), Institut Élie Cartan de Nancy, Université Nancy 1, BP 239, Vandoeuvre-lès-Nancy 54506, France


Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona, Spain

Received  June 2008 Revised  January 2009 Published  December 2009

In optimal control problems, there exist different kinds of extremals; that is, curves candidates to be solution: abnormal, normal and strictly abnormal. The key point for this classification is how those extremals depend on the cost function. We focus on control systems such as nonholonomic control mechanical systems and the associated kinematic control systems as long as they are equivalent.
   With all this in mind, first we study conditions to relate an optimal control problem for the mechanical system with another one for the associated kinematic system. Then, Pontryagin's Maximum Principle will be used to connect the abnormal extremals of both optimal control problems.
   An example is given to glimpse what the abnormal solutions for kinematic systems become when they are considered as extremals to the optimal control problem for the corresponding nonholonomic mechanical systems.
Citation: María Barbero-Liñán, Miguel C. Muñoz-Lecanda. Strict abnormal extremals in nonholonomic and kinematic control systems. Discrete & Continuous Dynamical Systems - S, 2010, 3 (1) : 1-17. doi: 10.3934/dcdss.2010.3.1

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