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Mathematical Problems in Engineering
Volume 2005 (2005), Issue 4, Pages 465-475

On the optimal control of affine nonlinear systems

Statistics and Applied Mathematics Institute of the Romanian Academy, P.O. Box 1-24, Bucharest 010145, Romania

Received 20 January 2004

Copyright © 2005 Hindawi Publishing Corporation. 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.


The minimization control problem of quadratic functionals for the class of affine nonlinear systems with the hypothesis of nilpotent associated Lie algebra is analyzed. The optimal control corresponding to the first-, second-, and third-order nilpotent operators is determined. In this paper, we have considered the minimum fuel problem for the multi-input nilpotent control and for a scalar input bilinear system for such systems. For the multi-input system, usually an analytic closed-form solution for the optimal control ui(t) is not possible and it is necessary to use numerical integration for the set of m nonlinear coupled second-order differential equations. The optimal control of bilinear systems is obtained by considering the Lie algebra generated by the system matrices. It should be noted that we have obtained an open-loop control depending on the initial value of the state x0.