An algorithm for construction of ε-value functions for the Bolza control problem

Edyta Jacewicz

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 2, page 391-428
  • ISSN: 1641-876X

Abstract

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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.

How to cite

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Jacewicz, Edyta. "An algorithm for construction of ε-value functions for the Bolza control problem." International Journal of Applied Mathematics and Computer Science 11.2 (2001): 391-428. <http://eudml.org/doc/207513>.

@article{Jacewicz2001,
abstract = {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.},
author = {Jacewicz, Edyta},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {ε-value function; Hamilton-Jacobi equation; non-linear optimisation; approximate minimum; dynamic programming; Bolza problem; optimal control; nonlinear optimization; -value function; approximation minimum},
language = {eng},
number = {2},
pages = {391-428},
title = {An algorithm for construction of ε-value functions for the Bolza control problem},
url = {http://eudml.org/doc/207513},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Jacewicz, Edyta
TI - An algorithm for construction of ε-value functions for the Bolza control problem
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 2
SP - 391
EP - 428
AB - 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.
LA - eng
KW - ε-value function; Hamilton-Jacobi equation; non-linear optimisation; approximate minimum; dynamic programming; Bolza problem; optimal control; nonlinear optimization; -value function; approximation minimum
UR - http://eudml.org/doc/207513
ER -

References

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  2. Bellman R. (1957): Dynamic Programming. — New York: Princeton Univ. Press. Zbl0077.13605
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  6. Fleming W. and Rishel R. (1975): Deterministic and Stochastic Optimal Control. — Berlin: Springer. Zbl0323.49001
  7. 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. Zbl0334.49028
  8. Jacewicz E. and Nowakowski A. (1995): Stability of approximations in optimal non-linear control. — Optimization, Vol.34, No.2, pp.173–184. Zbl0853.49016
  9. Nowakowski A. (1988): Sufficient condition for ε-optimality. — Control Cybern., Vol.17, pp.29–43. Zbl0667.49013
  10. Nowakowski A. (1990): Characterizations of an approximate minimum in optimal control. — J. Optim. Theory Appl., Vol.66, pp.95–12. Zbl0682.49025
  11. Polak E. (1997): Optimization. Algorithms and Consistent Approximations. — New York: Springer. Zbl0899.90148

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