Computing the effective Hamiltonian using a variation AL approach

SIAM Journal on Control and Optimization

Volumn 43, Issue 3, 2005, Pages 792-812

  Gomes, Diogo A ^a   Oberman, Adam M ^b,c  

^a INSTITUTO SUPERIOR TÉCNICO   (Portugal)

^b UNIVERSITY OF TEXAS AT AUSTIN   (United States)

^c SIMON FRASER UNIVERSITY   (Canada)

Author keywords

Calculus of variations; Hamilton Jacobi; Homogenization; Numerics

Indexed keywords

CALCULUS OF VARIATIONS; HAMILTON-JACOBI; HOMOGENIZATION; NUMERICS;

EIGENVALUES AND EIGENFUNCTIONS; NUMERICAL METHODS; OPTIMIZATION; PARTIAL DIFFERENTIAL EQUATIONS; PROBLEM SOLVING; SET THEORY;

HAMILTONIANS;

EID: 19944362518     PISSN: 03630129     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0363012902417620     Document Type: Article

References (42)

1. [AI01] O. ALVAREZ AND H. ISHII, Hamilton-Jacobi equations with partial gradient and application to homogenization, Comm. Partial Differential Equations, 26 (2001), pp. 983-1002.
2. [AKN97] V. I. ARNOLD, V. V. KOZLOV, AND A. I. NEISHTADT, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997 (translated from the 1985 Russian original by A. Iacob).
3. [Ari97] M. ARISAWA, Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Existence of the ergodic attractor, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), pp. 415-438.
4. [Ari98] M. ARISAWA, Ergodic problem for the Hamilton-Jacobi-Bellman equation. II. Ann. Inst. H. Poincaré Anal. Non Linéaire, 15 (1998), pp. 1-24.
5. x F(x, f(x), f′(x)), Ark. Mat., 6 (1965), pp. 33-53.
6. x F(x, f(x), f′(x)). II. Ark. Mat., 6 (1966), pp. 409-431.
7. ∞, in Optimal Control of Differential Equations (Athens, OH, 1993), Dekker, New York, 1994, pp. 39-47.
8. [BCD97] M. BARDI AND I. CAPUZZO-DOLCETTA, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Birkhäuser Boston, Boston, MA, 1997.
9. ∞ functionals, Arch. Ration. Mech. Anal., 157 (2001), pp. 255-283.
10. ∞ functionals, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), pp. 495-517.
11. [BS00] G. BARLES AND P. E. SOUGANIDIS, On the large time behavior of solutions of Hamilton-Jacobi equations, SIAM J. Math. Anal., 31 (2000), pp. 925-939.
12. [BS01] G. BARLES AND P. E. SOUGANIDIS, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), pp. 1311-1323.
13. [CDI01] I. CAPUZZO-DOLCETTA AND H. ISHII, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), pp. 1113-1129.
14. [CIL92] M. G. CRANDALL, H. ISHII, AND P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), pp. 1-67.
15. [CIPP98] G. CONTRERAS, R. ITURRIAGA, G. P. PATERNAIN, AND M. PATERNAIN, Lagrangian graphs, minimizing measures and Mañé's critical values, Geom. Funct. Anal., 8 (1998), pp. 788-809.
16. [Con95] M. CONCORDEL, Periodic Homogenization of Hamilton-Jacobi Equations, Ph.D. thesis, Department of Mathematics, University of California at Berkeley, Berkeley, CA, 1995.
17. [E99] WEINAN E., Aubry-Mather theory and periodic solutions of the forced Burgers equation, Comm. Pure Appl. Math., 52 (1999), pp. 811-828.
18. [EG01] L. C. EVANS AND D. GOMES, Effective Hamiltonians and averaging for Hamiltonian dynamics. I., Arch. Ration. Mech. Anal., 157 (2001), pp. 1-33.
19. [EG02] L. C. EVANS AND D. GOMES, Effective Hamiltonians and averaging for Hamiltonian dynamics. II., Arch. Ration. Mech. Anal., 161 (2002), pp. 271-305.
20. [EJ89] L. C. EVANS AND M. R. JAMES, The Hamilton-Jacobi-Bellman equation for time-optimal control, SIAM J. Control Optim., 27 (1989), pp. 1477-1489.
21. [EMS95] P. F. EMBID, A. J. MAJDA, AND P. E. SOUGANIDIS, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. Fluids, 7 (1995), pp. 2052-2060.
22. [Eva98] L. C. EVANS, Partial Differential Equations, AMS, Providence, RI, 1998.
23. [Eva03] L. C. EVANS, Some new PDE methods for weak KAM theory, Calc. Var. Partial Differential Equations, 17 (2003), pp. 159-177.
24. [Fat97a] A. FATHI, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 325 (1997), pp. 649-652.
25. [Fat97b] A. FATHI, Théorème KAM faible et théorie de Mather sur les systèmes Lagrangiens, C. R. Acad. Sci. Paris Sér. I Math., 324 (1997), pp. 1043-1046.
26. [Fat98a] A. FATHI, Orbite hétéroclines et ensemble de Peierls, C. R. Acad. Sci. Paris Sér. I Math., 326 (1998), pp. 1213-1216.
27. [Fat98b] A. FATHI, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris Sér. I Math., 327 (1998), pp. 267-270.
28. [Fle80] R. FLETCHER, Practical methods of optimization, Vol. 1, Unconstrained optimization, John Wiley & Sons, Chichester, UK, 1980.
29. [FS93] W. H. FLEMING AND H. M. SONER, Controlled Markov Processes and Viscosity Solutions, Springer-Verlag, New York, 1993.
30. 1 critical subsolutions of the Hamilton-Jacobi equation, Invent. Math., 155 (2004), pp. 363-388.
31. [Gom01a] D. GOMES, Hamilton-Jacobi Methods for Vakonomic Mechanics, preprint, 2001; available online at http://www.math.ist.utl.pt/~dgomes/.
32. [Gom01b] D. GOMES, Viscosity solutions of Hamilton-Jacobi equations, and asymptotics for Hamiltonian systems, Gal. Var. Partial Differential Equations, 14 (2002), pp. 345-357.
33. [Gom02] D. A. GOMES, A stochastic analogue of Aubry-Mather theory, Nonlinearity, 15 (2002), pp. 581-603.
34. [KBM01] B. KHOUIDER, A. BOURLIOUX, AND A. J. MAJDA, Parametrizing the burning speed enhancement by small-scale periodic flows, I., Unsteady shears, flame residence time and bending, Combust. Theory Model., 5 (2001), pp. 295-318.
35. [LPV88] P. L. LIONS, G. PAPANICOLAU, AND S. R. S. VARADHAN, Homogenization of Hamilton-Jacobi Equations, manuscript, 1988.
36. [LS03] P.-L. LIONS AND P. E. SOUGANIDIS, Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting, Comm. Pure Appl. Math., 56 (2003), pp. 1501-1524.
37. [Mat89a] J. N. MATHER, Minimal action measures for positive-definite Lagrangian systems, in Proceedings of the 9th International Congress on Mathematical Physics (Swansea, UK, 1988), Hilger, Bristol, UK, 1989, pp. 466-468.
38. [Mat89b] J. N. MATHER, Minimal measures, Comment. Math. Helv., 64 (1989), pp. 375-394.
39. [Mat91] J. N. MATHER, Action minimizing invariant measures for positive definite Lagrangian systems, Math. Z., 207 (1991), pp. 169-207.
40. [Mn92] R. MAÑÉ, On the minimizing measures of Lagrangian dynamical systems, Nonlinearity, 5 (1992), pp. 623-638.
41. [Mn96] R. MAÑÉ, Generic properties and problems of minimizing measures of Lagrangian systems, Nonlinearity, 9 (1996), pp. 273-310.
42. [Qia01] J. QIAN, A note on a numerical method for effective Hamiltonians, J. Comput. Appl. Math., submitted.