Discrete and Continuous Dynamical Systems - Series S (DCDS-S)

Strict abnormal extremals in nonholonomic and kinematic control systems

Pages: 1 - 17, Volume 3, Issue 1, March 2010      doi:10.3934/dcdss.2010.3.1

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María Barbero-Liñán - INRIA (Projet CORIDA), Institut Élie Cartan de Nancy, Université Nancy 1, BP 239, Vandoeuvre-lès-Nancy 54506, France (email)
Miguel C. Muñoz-Lecanda - Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech., Edificio C-3, Campus Norte UPC. C/ Jordi Girona 1, E-08034 Barcelona, Spain (email)

Abstract: 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.

Keywords:  nonholonomic control mechanical systems, kinematic control systems, Pontryagin's Maximum Principle, extremals, abnormality.
Mathematics Subject Classification:  34A26, 49J15, 49K15, 70G45, 70H05.

Received: June 2008;      Revised: January 2009;      Available Online: December 2009.