The problem considered is that of approximate minimisation of the Bolza problem of optimal control. Starting from Bellman's method of dynamic programming, we define the ε-value function to be an approximation to the value function being a solution to the Hamilton-Jacobi equation. The paper shows an approach that can be used to construct an algorithm for calculating the values of an ε-value function at given points, thus approximating the respective values of the value function.
@article{bwmeta1.element.bwnjournal-article-amcv15i2p177bwm,
author = {Pustelnik, Jan},
title = {A method for constructing $\epsilon$-value functions for the Bolza problem of optimal control},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {15},
year = {2005},
pages = {177-186},
zbl = {1087.49014},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv15i2p177bwm}
}
Pustelnik, Jan. A method for constructing ε-value functions for the Bolza problem of optimal control. International Journal of Applied Mathematics and Computer Science, Tome 15 (2005) pp. 177-186. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv15i2p177bwm/
[000] Adams R.A. (1975): Sobolev spaces. - New York: Academic Press. | Zbl 0314.46030
[001] Bryson S. and Levy D. (2001): Central schemes for multi-dimensional Hamilton-Jacobi Equations. - NASA Techn. Rep., NAS-01-014.
[002] Cesari L. (1983): Optimization - Theory and Applications. - New York: Springer. | Zbl 0506.49001
[003] Fleming W.H. and Rishel R.W. (1975): Deterministic and Stochastic Optimal Control. - New York: Springer.
[004] Jacewicz E. (2001): An algorithm for construction of ε-value functions for the Bolza control problem. - Int. J. Appl. Math. Comput. Sci., Vol. 11, No. 2, pp. 391-428. | Zbl 0974.49016
[005] Karlsen K.H. and Risebro N.H. (2002): Unconditionally stable methods for Hamilton-Jacobi equations. - J. Comput. Phys., Vol. 180, No. 2, pp. 710-735. | Zbl 1143.65365
[006] Kurganov A. and Tadmor E. (2000): New high-resolution semi-discrete central schemes for Hamilton-Jacobi equations. - J. Comput. Phys., Vol. 160,No. 2, pp. 720-742. | Zbl 0961.65077
[007] Szpiro A. and Dupuis P. (2002): Second order numerical methods for first order Hamilton-Jacobi equations. - SIAM J. Numer. Anal.,Vol. 40, No. 3, pp. 1136-1183. | Zbl 1031.49027
[008] Tang H.Z., Tang T. and Zhang P. (2003): An adaptive mesh redistribution method for nonlinear Hamilton-Jacobi equations in two- and three-dimensions. - J. Comput. Phys., Vol. 188, No. 2, pp. 543-572. | Zbl 1037.65091