Linear programming interpretations of Mather's variational principle

L. C. Evans; D. Gomes

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

top
We discuss some implications of linear programming for Mather theory [13-15] 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 [5-8].

How to cite

top

Evans, L. C., and Gomes, D.. "Linear programming interpretations of Mather's variational principle." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 693-702. <http://eudml.org/doc/90665>.

@article{Evans2010,
abstract = { We discuss some implications of linear programming for Mather theory [13-15] 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 [5-8]. },
author = {Evans, L. C., Gomes, D.},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Linear programming; duality; weak KAM theory.; linear programming; weak KAM theory},
language = {eng},
month = {3},
pages = {693-702},
publisher = {EDP Sciences},
title = {Linear programming interpretations of Mather's variational principle},
url = {http://eudml.org/doc/90665},
volume = {8},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 693
EP - 702
AB - We discuss some implications of linear programming for Mather theory [13-15] 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 [5-8].
LA - eng
KW - Linear programming; duality; weak KAM theory.; linear programming; weak KAM theory
UR - http://eudml.org/doc/90665
ER -

References

top
  1. E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987).  Zbl0632.90038
  2. D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997).  
  3. L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu  
  4. L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear).  Zbl1032.37048
  5. L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal.157 (2001) 1-33.  Zbl0986.37056
  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.  
  7. A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math.325 (1997) 649-652.  Zbl0943.37031
  8. A. Fathi, Weak KAM theory in Lagrangian Dynamics, Preliminary Version. Lecture Notes (2001).  
  9. J. Franklin, Methods of Mathematical Economics. SIAM, Classics in Appl. Math.37 (2002).  
  10. D. Gomes, Numerical methods and Hamilton-Jacobi equations (to appear).  
  11. P. Lax, Linear Algebra. John Wiley (1997).  
  12. P.-L. Lions, G. Papanicolaou and S.R.S. Varadhan, Homogenization of Hamilton-Jacobi equations. CIRCA (1988) (unpublished).  
  13. J. Mather, Minimal measures. Comment. Math Helvetici64 (1989) 375-394.  Zbl0689.58025
  14. J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z.207 (1991) 169-207.  Zbl0696.58027
  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).  Zbl0822.70011

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.