Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions

Recently, the method based on generating functions is proposed for nonlinear optimal control problems. For a finite time optimal control problem with given boundary condition, once a generating function for a fixed boundary condition is obtained, any optimal trajectory of the same system for different boundary conditions can be generated easily. An algorithm to compute an approximate solution to the Hamilton-Jacobi equation with respect to the generating function for a nonlinear optimal control problem is developed in this paper. Numerical examples illustrate the effectiveness of the proposed method.

[1]  Emiliano Cristiani,et al.  Initialization of the Shooting Method via the Hamilton-Jacobi-Bellman Approach , 2009, 0910.0521.

[2]  Kenji Fujimoto,et al.  Approximate Solutions to Hamilton-Jacobi Equations Based on Chebyshev Polynomials , 2008 .

[3]  Kenji Fujimoto,et al.  Approximate Solutions to the Hamilton-Jacobi Equations for Generating Functions with a Quadratic Cost Function with Respect to the Input , 2012 .

[4]  Hiroshi Yamakawa,et al.  A new optimal orbit control for two-point boundary-value problem using generating functions , 2009 .

[5]  Hans P. Geering,et al.  Optimal control with engineering applications , 2007 .

[6]  D. Scheeres,et al.  Solving Relative Two-Point Boundary Value Problems: Spacecraft Formation Flight Transfers Application , 2004 .

[7]  Randal W. Beard,et al.  Galerkin approximations of the generalized Hamilton-Jacobi-Bellman equation , 1997, Autom..

[8]  Arjan van der Schaft,et al.  Analytical Approximation Methods for the Stabilizing Solution of the Hamilton–Jacobi Equation , 2008, IEEE Transactions on Automatic Control.

[9]  E. G. Al'brekht.  On the optimal stabilization of nonlinear systems , 1961 .

[10]  Daniel J. Scheeres,et al.  Determination of optimal feedback terminal controllers for general boundary conditions using generating functions , 2006, Autom..

[11]  Arthur J. Krener,et al.  Solution of Hamilton Jacobi Bellman equations , 2000, Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187).

[12]  W. Marsden.  I and J , 2012 .

[13]  A. P. Willemstein,et al.  Optimal regulation of nonlinear dynamical systems on a finite interval , 1977 .

[14]  C. Poole,et al.  Classical Mechanics, 3rd ed. , 2002 .

[15]  D. Lukes.  Optimal Regulation of Nonlinear Dynamical Systems , 1969 .

[16]  S. Sieniutycz,et al.  Optimal Control: An Introduction , 2001 .