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A New Approach to the Approximate Solutions of Hamilton-Jacobi Equations

Authors: Joe Imae, Kenjiro Shinagawa, Tomoaki Kobayashi, Guisheng Zhai

Abstract:

We propose a new approach on how to obtain the approximate solutions of Hamilton-Jacobi (HJ) equations. The process of the approximation consists of two steps. The first step is to transform the HJ equations into the virtual time based HJ equations (VT-HJ) by introducing a new idea of ‘virtual-time’. The second step is to construct the approximate solutions of the HJ equations through a computationally iterative procedure based on the VT-HJ equations. It should be noted that the approximate feedback solutions evolve by themselves as the virtual-time goes by. Finally, we demonstrate the effectiveness of our approximation approach by means of simulations with linear and nonlinear control problems.

Keywords: Nonlinear Control, Optimal Control, Hamilton-Jacobi Equation, Virtual-Time

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1083027

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References:


[1] D.L. Lukes, "Optimal regulation of nonlinear dynamical systems," SIAM J. Control, 7, 1969, pp. 75-100
[2] R. Beard, "Improving the closed-loop performance of nonlinear systems," PhD Thesis, Rensselaer Polytechnic Institute, Troy, New York, 1995
[3] R. Beard, G. Saridis and J. Wen, "Galerkin approximation of the generalized Hamilton-Jacobi-Bellman equation," Automatica, Vol.33, No.12, 1997, pp. 2159-2177.
[4] J. Imae, Y. Kikuchi, N. Ohtsuki, T. Kobayashi and G. Zhai, "Design of nonlinear control systems by means of differential genetic programming," Proceedings of the 43rd IEEE Conference on Decision and Control, 2004, pp. 2734-2739.
[5] A.K. Murad and F.L. Lewis, "Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach," Automatica, Vol. 41, No. 5, 2005, pp. 779-791.
[6] S.T. Glad, "Robustness of nonlinear state feedback: a survey,"
[7] Automatica, Vol. 23, No.4, 1987, pp. 425-435.
[8] G.Q.Xu, and S. P. Yung, "Lyapunov stability of abstract nonlinear dynamic system in Banach space," IMA Journal of Mathematical Control and Information, 20, 2003, pp.105-127.
[9] J. Imae, K. Shinagawa, A. Ueda, T. Kobayashi and G. Zhai, "An approach to Hamilton-Jacobi-Bellman equations using virtual time," Transaction of the Japan Society for Simulation Technology, Vol. 3, No. 1, 2011, pp.11-17, to be published.