Linear programming interpretations of Mather’s variational principle

L. C. Evans; D. Gomes

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 8, page 693-702
  • ISSN: 1292-8119

Abstract

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We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an n -dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].

How to cite

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Evans, L. C., and Gomes, D.. "Linear programming interpretations of Mather’s variational principle." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 693-702. <http://eudml.org/doc/244650>.

@article{Evans2002,
abstract = {We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an $n$-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].},
author = {Evans, L. C., Gomes, D.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {linear programming; duality; weak KAM theory},
language = {eng},
pages = {693-702},
publisher = {EDP-Sciences},
title = {Linear programming interpretations of Mather’s variational principle},
url = {http://eudml.org/doc/244650},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Evans, L. C.
AU - Gomes, D.
TI - Linear programming interpretations of Mather’s variational principle
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 693
EP - 702
AB - We discuss some implications of linear programming for Mather theory [13, 14, 15] and its finite dimensional approximations. We find that the complementary slackness condition of duality theory formally implies that the Mather set lies in an $n$-dimensional graph and as well predicts the relevant nonlinear PDE for the “weak KAM” theory of Fathi [6, 7, 8, 5].
LA - eng
KW - linear programming; duality; weak KAM theory
UR - http://eudml.org/doc/244650
ER -

References

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  6. [6] A. Fathi, Théorème KAM faible et théorie de Mather sur les systèmes lagrangiens. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1043-1046. Zbl0885.58022MR1451248
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  10. [10] D. Gomes, Numerical methods and Hamilton–Jacobi equations (to appear). 
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  14. [14] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. Zbl0696.58027MR1109661
  15. [15] J. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems. Transition to Chaos in Classical and Quantum Mechanics, edited by S. Graffi. Sringer, Lecture Notes in Math. 1589 (1994). Zbl0822.70011MR1323222

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