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Mathematical Problems in Engineering
Volume 2015, Article ID 198380, 8 pages
Research Article

Robust Nonlinear Control Design via Stable Manifold Method

1Department of Applied Analysis and Complex Dynamical Systems, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
2Department of Mechanical and Environmental Informatics, Graduate School of Information Science and Engineering, Tokyo Institute of Technology, 2-12-1 W8-1, O-okayama, Meguro-ku, Tokyo 152-8552, Japan
3Department of Mechatronics, Faculty of Science and Engineering, Nanzan University, 18 Yamazato-cho, Shyowa-ku, Nagoya 466-8673, Japan

Received 8 September 2015; Accepted 5 November 2015

Academic Editor: Wenguang Yu

Copyright © 2015 Yoshiki Abe et al. 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.


This paper proposes a systematic numerical method for designing robust nonlinear controllers without a priori lower-dimensional approximation with respect to solutions of the Hamilton-Jacobi equations. The method ensures the solutions are globally calculated with arbitrary accuracy in terms of the stable manifold method that is a solver of Hamilton-Jacobi equations in nonlinear optimal control problems. In this realization, the existence of stabilizing solutions of the Hamilton-Jacobi equations can be derived from some properties of the linearized system and the equivalent Hamiltonian system that is obtained from a transformation of the Hamilton-Jacobi equation. A numerical example is shown to validate the design method.