# The linear programming approach to deterministic optimal control problems

Daniel Hernández-Hernández; Onésimo Hernández-Lerma; Michael Taksar

Applicationes Mathematicae (1996)

- Volume: 24, Issue: 1, page 17-33
- ISSN: 1233-7234

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topHernández-Hernández, Daniel, Hernández-Lerma, Onésimo, and Taksar, Michael. "The linear programming approach to deterministic optimal control problems." Applicationes Mathematicae 24.1 (1996): 17-33. <http://eudml.org/doc/219148>.

@article{Hernández1996,

abstract = {Given a deterministic optimal control problem (OCP) with value function, say $J^*$, we introduce a linear program $(P)$ and its dual $(P^*)$ whose values satisfy $\sup (P^*) \le \inf (P)\le J^*(t,x)$. Then we give conditions under which (i) there is no duality gap},

author = {Hernández-Hernández, Daniel, Hernández-Lerma, Onésimo, Taksar, Michael},

journal = {Applicationes Mathematicae},

keywords = {linear programming (in infinite-dimensional spaces); duality theory; optimal control; optimal problem in infinite-dimensional spaces; linear programming},

language = {eng},

number = {1},

pages = {17-33},

title = {The linear programming approach to deterministic optimal control problems},

url = {http://eudml.org/doc/219148},

volume = {24},

year = {1996},

}

TY - JOUR

AU - Hernández-Hernández, Daniel

AU - Hernández-Lerma, Onésimo

AU - Taksar, Michael

TI - The linear programming approach to deterministic optimal control problems

JO - Applicationes Mathematicae

PY - 1996

VL - 24

IS - 1

SP - 17

EP - 33

AB - Given a deterministic optimal control problem (OCP) with value function, say $J^*$, we introduce a linear program $(P)$ and its dual $(P^*)$ whose values satisfy $\sup (P^*) \le \inf (P)\le J^*(t,x)$. Then we give conditions under which (i) there is no duality gap

LA - eng

KW - linear programming (in infinite-dimensional spaces); duality theory; optimal control; optimal problem in infinite-dimensional spaces; linear programming

UR - http://eudml.org/doc/219148

ER -

## References

top- [1] E. J. Anderson and P. Nash, Linear Programming in Infinite-Dimensional Spaces, Wiley, Chichester, 1989.
- [2] W. H. Fleming, Generalized solutions and convex duality in optimal control, in: Partial Differential Equations and the Calculus of Variations, Vol. I, F. Colombini et al. (eds.), Birkhäuser, Boston, 1989, 461-471.
- [3] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, Springer, New York, 1975.
- [4] W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity Solutions, Springer, New York, 1992. Zbl0773.60070
- [5] W. H. Fleming and D. Vermes, Generalized solutions in the optimal control of diffusions, IMA Vol. Math. Appl. 10, W. H. Fleming and P. L. Lions (eds.), Springer, New York, 1988, 119-127.
- [6] W. H. Fleming, Convex duality approach to the optimal control of diffusions, SIAM J. Control Optim. 27 (1989), 1136-1155. Zbl0693.93082
- [7] O. Hernández-Lerma, Existence of average optimal policies in Markov control processes with strictly unbounded costs, Kybernetika (Prague) 29 (1993), 1-17. Zbl0792.93120
- [8] O. Hernández-Lerma and D. Hernández-Hernández, Discounted cost Markov decision processes on Borel spaces: The linear programming formulation, J. Math. Anal. Appl. 183 (1994), 335-351. Zbl0820.90124
- [9] O. Hernández-Lerma and J. B. Lasserre, Linear programming and average optimality of Markov control processes on Borel spaces-unbounded costs, SIAM J. Control Optim. 32 (1994), 480-500. Zbl0799.90120
- [10] J. L. Kelley, General Topology, Van Nostrand, New York, 1957.
- [11] R. M. Lewis and R. B. Vinter, Relaxation of optimal control problems to equivalent convex programs, J. Math. Anal. Appl. 74 (1980), 475-493. Zbl0443.49015
- [12] J. E. Rubio, Control and Optimization, Manchester Univ. Press, Manchester, 1986. Zbl1095.49500
- [13] R. H. Stockbridge, Time-average control of martingale problems: a linear programming formulation, Ann. Probab. 18 (1990), 206-217. Zbl0699.49019

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