Linear programming interpretations of Mather's variational principle
ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 693-702.

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].

DOI : 10.1051/cocv:2002030
Classification : 90C05, 35F20
Mots-clés : linear programming, duality, weak KAM theory 
@article{COCV_2002__8__693_0,
     author = {Evans, L. C. and Gomes, D.},
     title = {Linear programming interpretations of {Mather's} variational principle},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {693--702},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002030},
     zbl = {1090.90143},
     language = {en},
     url = {https://www.numdam.org/articles/10.1051/cocv:2002030/}
}
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
SP  - 693
EP  - 702
VL  - 8
PB  - EDP-Sciences
UR  - https://www.numdam.org/articles/10.1051/cocv:2002030/
DO  - 10.1051/cocv:2002030
LA  - en
ID  - COCV_2002__8__693_0
ER  - 
%0 Journal Article
%A Evans, L. C.
%A Gomes, D.
%T Linear programming interpretations of Mather's variational principle
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 693-702
%V 8
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv:2002030/
%R 10.1051/cocv:2002030
%G en
%F COCV_2002__8__693_0
Evans, L. C.; Gomes, D. Linear programming interpretations of Mather's variational principle. ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 693-702. doi : 10.1051/cocv:2002030. https://www.numdam.org/articles/10.1051/cocv:2002030/

[1] E.J. Anderson and P. Nash, Linear Programming in Infinite Dimensional Spaces. Wiley (1987). | MR | Zbl

[2] D. Bertsimas and J. Tsitsiklis, Introduction to Linear Optimization. Athena Scientific (1997). | Zbl

[3] L.C. Evans, Partial differential equations and Monge-Kantorovich mass transfer (survey paper). Available at the website of LCE, at math.berkeley.edu | Zbl

[4] L.C. Evans, Some new PDE methods for weak KAM theory. Calc. Var. Partial Differential Equations (to appear). | MR | Zbl

[5] L.C. Evans and D. Gomes, Effective Hamiltonians and averaging for Hamiltonian dynamics I. Arch. Rational Mech. Anal. 157 (2001) 1-33. | MR | Zbl

[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. | MR | Zbl

[7] A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 649-652. | MR | Zbl

[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). | MR | Zbl

[10] D. Gomes, Numerical methods and Hamilton-Jacobi equations (to appear).

[11] P. Lax, Linear Algebra. John Wiley (1997). | MR | Zbl

[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 Helvetici 64 (1989) 375-394. | MR | Zbl

[14] J. Mather, Action minimizing invariant measures for positive definite Lagrangian systems. Math. Z. 207 (1991) 169-207. | MR | Zbl

[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). | MR | Zbl

  • Lin, Zhiqi; Liu, Xuxun; Zhou, Huan; Wu, Jie Adaptive Time-Varying Routing for Energy Saving and Load Balancing in Wireless Body Area Networks, IEEE Transactions on Mobile Computing, Volume 23 (2024) no. 1, p. 90 | DOI:10.1109/tmc.2022.3213471
  • Kouhkouh, Hicham A Viscous Ergodic Problem with Unbounded and Measurable Ingredients, Part 1: HJB Equation, SIAM Journal on Control and Optimization, Volume 62 (2024) no. 1, p. 415 | DOI:10.1137/22m1478069
  • Lopes, A. O.; Thieullen, Ph. Transport and Large Deviations for Schrodinger Operators and Mather Measures, Modeling, Dynamics, Optimization and Bioeconomics III, Volume 224 (2018), p. 247 | DOI:10.1007/978-3-319-74086-7_11
  • Ishii, Hitoshi; Mitake, Hiroyoshi; Tran, Hung V. The vanishing discount problem and viscosity Mather measures. Part 1: The problem on a torus, Journal de Mathématiques Pures et Appliquées, Volume 108 (2017) no. 2, p. 125 | DOI:10.1016/j.matpur.2016.10.013
  • Gaitsgory, Vladimir; Rossomakhine, Sergey Averaging and Linear Programming in Some Singularly Perturbed Problems of Optimal Control, Applied Mathematics Optimization, Volume 71 (2015) no. 2, p. 195 | DOI:10.1007/s00245-014-9257-1
  • Lopes, A. O.; Oliveira, E. R.; Thieullen, Ph. The Dual Potential, the Involution Kernel and Transport in Ergodic Optimization, Dynamics, Games and Science, Volume 1 (2015), p. 357 | DOI:10.1007/978-3-319-16118-1_20
  • Gaitsgory, Vladimir; Manic, Ludmila; Rossomakhine, Sergey On Average Control Generating Families for Singularly Perturbed Optimal Control Problems with Long Run Average Optimality Criteria, Set-Valued and Variational Analysis, Volume 23 (2015) no. 1, p. 87 | DOI:10.1007/s11228-014-0306-3
  • Lopes, Artur O.; Mengue, Jairo K. Duality Theorems in Ergodic Transport, Journal of Statistical Physics, Volume 149 (2012) no. 5, p. 921 | DOI:10.1007/s10955-012-0626-3
  • Ahmetoğlu, Feyzullah Kuhn–Tucker Conditions for a Convex Programming Problem in Banach Spaces Partially Ordered by Cone with Empty Interior, Numerical Functional Analysis and Optimization, Volume 33 (2012) no. 4, p. 363 | DOI:10.1080/01630563.2012.657739
  • Ban, Liqun; Mordukhovich, Boris S.; Song, Wen Lipschitzian stability of parametric variational inequalities over generalized polyhedra in Banach spaces, Nonlinear Analysis: Theory, Methods Applications, Volume 74 (2011) no. 2, p. 441 | DOI:10.1016/j.na.2010.09.001
  • Gaitsgory, Vladimir; Quincampoix, Marc, 49th IEEE Conference on Decision and Control (CDC) (2010), p. 6680 | DOI:10.1109/cdc.2010.5717687
  • Granieri, Luca A finite dimensional linear programming approximation of Mather's variational problem, ESAIM: Control, Optimisation and Calculus of Variations, Volume 16 (2010) no. 4, p. 1094 | DOI:10.1051/cocv/2009039
  • Finlay, Luke; Gaitsgory, Vladimir; Lebedev, Ivan, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference (2009), p. 1207 | DOI:10.1109/cdc.2009.5400261
  • Gaitsgory, Vladimir; Quincampoix, Marc Linear Programming Approach to Deterministic Infinite Horizon Optimal Control Problems with Discounting, SIAM Journal on Control and Optimization, Volume 48 (2009) no. 4, p. 2480 | DOI:10.1137/070696209
  • Finlay, Luke; Gaitsgory, Vladimir; Lebedev, Ivan Duality in Linear Programming Problems Related to Deterministic Long Run Average Problems of Optimal Control, SIAM Journal on Control and Optimization, Volume 47 (2008) no. 4, p. 1667 | DOI:10.1137/060676398
  • De Pascale, Luigi; Gelli, Maria Stella; Granieri, Luca Minimal measures, one-dimensional currents and the Monge–Kantorovich problem, Calculus of Variations and Partial Differential Equations, Volume 27 (2006) no. 1, p. 1 | DOI:10.1007/s00526-006-0017-1
  • Gaitsgory, Vladimir; Rossomakhine, Sergey, Proceedings of the 45th IEEE Conference on Decision and Control (2006), p. 5012 | DOI:10.1109/cdc.2006.377568
  • Gaitsgory, Vladimir; Rossomakhine, Sergey Linear Programming Approach to Deterministic Long Run Average Problems of Optimal Control, SIAM Journal on Control and Optimization, Volume 44 (2006) no. 6, p. 2006 | DOI:10.1137/040616802
  • Evans, Lawrence C. A survey of partial differential equations methods in weak KAM theory, Communications on Pure and Applied Mathematics, Volume 57 (2004) no. 4, p. 445 | DOI:10.1002/cpa.20009

Cité par 19 documents. Sources : Crossref