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Mathematical Problems in Engineering
Volume 8 (2002), Issue 4-5, Pages 413-438

Higher-order techniques for some problems of nonlinear control

1Dipartimento di Matematica per le Decisioni, Universita di Firenze, V. C. Lombroso 6/17, Firenze 50134, Italy
2Institute for Control Sciences (IPU), Russian Academy of Sciences, Moscow, Russia

Received 7 February 2002; Revised 18 February 2002

Copyright © 2002 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.


A natural first step when dealing with a nonlinear problem is an application of some version of linearization principle. This includes the well known linearization principles for controllability, observability and stability and also first-order optimality conditions such as Lagrange multipliers rule or Pontryagin's maximum principle. In many interesting and important problems of nonlinear control the linearization principle fails to provide a solution. In the present paper we provide some examples of how higher-order methods of differential geometric control theory can be used for the study nonlinear control systems in such cases. The presentation includes: nonlinear systems with impulsive and distribution-like inputs; second-order optimality conditions for bang–bang extremals of optimal control problems; methods of high-order averaging for studying stability and stabilization of time-variant control systems.