Journal of Applied Mathematics
Volume 2009 (2009), Article ID 291849, 14 pages
Research Article

Convexity of the Set of Fixed Points Generated by Some Control Systems

Departamento de Control Automatico, Cinvestav, A.P. 14-740, Avenida Instituto Politecnico Nacional 2508, 07360 Mexico DF, Mexico

Received 20 May 2009; Accepted 26 August 2009

Academic Editor: Andrei Agrachev

Copyright © 2009 Vadim Azhmyakov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We deal with an application of the fixed point theorem for nonexpansive mappings to a class of control systems. We study closed-loop and open-loop controllable dynamical systems governed by ordinary differential equations (ODEs) and establish convexity of the set of trajectories. Solutions to the above ODEs are considered as fixed points of the associated system-operator. If convexity of the set of trajectories is established, this can be used to estimate and approximate the reachable set of dynamical systems under consideration. The estimations/approximations of the above type are important in various engineering applications as, for example, the verification of safety properties.