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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  1. [1] E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987). Zbl0632.90038MR893179
  2. [2] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). Zbl0997.90505
  3. [3] L.C. Evans, Partial differential equations and Monge–Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu Zbl0954.35011
  4. [4] L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear). Zbl1032.37048MR1986317
  5. [5] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. Zbl0986.37056MR1822413
  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
  7. [7] A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. Zbl0943.37031MR1473840
  8. [8] A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001). 
  9. [9] J. Franklin, Methods of Mathematical Economics. SIAM, Classics in Appl. Math. 37 (2002). Zbl1075.90001MR1875314
  10. [10] D. Gomes, Numerical methods and Hamilton–Jacobi equations (to appear). 
  11. [11] P. Lax, Linear Algebra. John Wiley (1997). Zbl0904.15001MR1423602
  12. [12] P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton–Jacobi equations. CIRCA (1988) (unpublished). 
  13. [13] J. Mather, Minimal measures. Comment. Math Helvetici 64 (1989) 375-394. Zbl0689.58025MR998855
  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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