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Convergence of an iterative algorithm for solving Hamilton-Jacobi type equations

Authors: Jerry Markman and I. Norman Katz
Journal: Math. Comp. 71 (2002), 77-103
MSC (2000): Primary 93B40, 49N35, 65P10
Published electronically: March 9, 2001
MathSciNet review: 1862989
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Abstract | References | Similar Articles | Additional Information


Solutions of the optimal control and $H_\infty$-control problems for nonlinear affine systems can be found by solving Hamilton-Jacobi equations. However, these first order nonlinear partial differential equations can, in general, not be solved analytically. This paper studies the rate of convergence of an iterative algorithm which solves these equations numerically for points near the origin. It is shown that the procedure converges to the stabilizing solution exponentially with respect to the iteration variable. Illustrative examples are presented which confirm the theoretical rate of convergence.

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Additional Information

Jerry Markman
Affiliation: Department of Systems Science and Mathematics, Washington University, Campus Box 1040, One Brookings Drive, St. Louis, Missouri 63130

I. Norman Katz
Affiliation: Department of Systems Science and Mathematics, Washington University, Campus Box 1040, One Brookings Drive, St. Louis, Missouri 63130

Keywords: Hamilton-Jacobi equations, convergence, optimal control
Received by editor(s): December 1, 1998
Received by editor(s) in revised form: February 17, 2000
Published electronically: March 9, 2001
Additional Notes: The results reported here are part of the doctoral dissertation of the first author.
This work was supported in part by the National Science Foundation under grant number DMS-9626202 and in part by the Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory, Air Force Materiel Command, USAF, under agreement number F30602-99-2-0551. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Defense Advanced Research Projects Agency (DARPA), the Air Force Research Laboratory, or the U.S. Government.
Article copyright: © Copyright 2001 American Mathematical Society