The problem considered is that of approximate numerical minimisation of the non-linear control problem of Bolza. Starting from the classical dynamic programming method of Bellman, an ε-value function is defined as an approximation for the value function being a solution to the Hamilton-Jacobi equation. The paper shows how an ε-value function which maintains suitable properties analogous to the original Hamilton-Jacobi value function can be constructed using a stable numerical algorithm. The paper shows the numerical closeness of the approximate minimum to the infimum of the Bolza functional.
@article{bwmeta1.element.bwnjournal-article-amcv11i2p391bwm,
author = {Jacewicz, Edyta},
title = {An algorithm for construction of $\epsilon$-value functions for the Bolza control problem},
journal = {International Journal of Applied Mathematics and Computer Science},
volume = {11},
year = {2001},
pages = {391-428},
zbl = {0974.49016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-amcv11i2p391bwm}
}
Jacewicz, Edyta. An algorithm for construction of ε-value functions for the Bolza control problem. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) pp. 391-428. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-amcv11i2p391bwm/
[000] Adams R.A. (1975): Sobolev Spaces. — New York, San Francisco, London: Academic Press. | Zbl 0314.46030
[001] Bellman R. (1957): Dynamic Programming. — New York: Princeton Univ. Press. | Zbl 0077.13605
[002] Cesari L. (1983): Optimization—Theory and Applications, Problems with Ordinary Differential Equations. — New York: Springer. | Zbl 0506.49001
[003] Ekeland I. (1974): On the variational principle. — J. Math. Anal. Appl., Vol.47, pp.324–353. | Zbl 0286.49015
[004] Ekeland I. (1979): Non-convex minimization problems. — Bull. Amer. Math. Soc., Vol.1, pp.443–474. | Zbl 0441.49011
[005] Fleming W. and Rishel R. (1975): Deterministic and Stochastic Optimal Control. — Berlin: Springer. | Zbl 0323.49001
[006] Gonzales R. (1976): Sur l’existence d’une solution maximale de l’equation de Hamilton-Jacobi. — C. R. Acad. Sc. Paris, Vol.282, pp.1287–1290. | Zbl 0334.49028
[007] Jacewicz E. and Nowakowski A. (1995): Stability of approximations in optimal non-linear control. — Optimization, Vol.34, No.2, pp.173–184. | Zbl 0853.49016
[008] Nowakowski A. (1988): Sufficient condition for ε-optimality. — Control Cybern., Vol.17, pp.29–43. | Zbl 0667.49013
[009] Nowakowski A. (1990): Characterizations of an approximate minimum in optimal control. — J. Optim. Theory Appl., Vol.66, pp.95–12. | Zbl 0682.49025
[010] Polak E. (1997): Optimization. Algorithms and Consistent Approximations. — New York: Springer. | Zbl 0899.90148