American Institute of Mathematical Sciences

Advanced
December  2011, 15(3): 623-635. doi: 10.3934/dcdsb.2011.15.623

Limit of the infinite horizon discounted Hamilton-Jacobi equation

Renato Iturriaga 1, and  Héctor Sánchez-Morgado 2,

1. 

CIMAT, A.P. 402, 3600, Guanajuato. Gto
2. 

I. de Matemáticas, UNAM, Cd. Universitaria, México, D.F. 04510

Received  October 2007 Revised  March 2010 Published  February 2011

Motivated by the infinite horizon discounted problem, we study the convergence of solutions of the Hamilton Jacobi equation when the discount vanishes. If the Aubry set consists in a finite number of hyperbolic critical points, we give an explicit expression for the limit. Additionaly, we give a new characterization of Mañé's critical value as for wich the set of viscosity solutions is equibounded.
Keywords: Hamilton-Jacobi equation..
Mathematics Subject Classification: Primary: 37J50, 49L25; Secondary: 70H2.
Citation: Renato Iturriaga, Héctor Sánchez-Morgado. Limit of the infinite horizon discounted Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 623-635. doi: 10.3934/dcdsb.2011.15.623
References:
[1]

N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation,, Disc. Cont. Dyn. Sys. Series B, 5 (2005), 513.  doi: 10.3934/dcdsb.2005.5.513.  Google Scholar

[2]

M. Bardi and I. C. Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhausser, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

G. Barles, "Solutions de Viscosité des Équations de Hamilton Jacobi,", Mathématiques et Applications, 17 (1994).   Google Scholar

[4]

P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds,, Ann. Scient. Éc. Norm. Sup., 40 (2007), 445.   Google Scholar

[5]

G. Contreras, Action Potential and Weak KAM Solutions,, Calc. Var. Partial Differential Equations, 13 (2001), 427.  doi: 10.1007/s005260100081.  Google Scholar

[6]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians,", 22 Colóquio Brasileiro de Matemática, (1999).   Google Scholar

[7]

A. Fathi, "The Weak KAM Theorem in Lagrangian Dynamics,", Cambridge Studies in Advanced Mathematics, (2010).   Google Scholar

[8]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris, 325 (1997), 649.   Google Scholar

[9]

A. Fathi, Théorème KAM faible et Théorie de Mather sur les systems Lagrangiens,, C.R. Acad. Sci. Paris, 324 (1997), 1043.   Google Scholar

[10]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik,, C. R. Acad. Sci. Paris, 327 (1998), 267.   Google Scholar

[11]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar

[12]

D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures,, Advances in Calculus of Variations, 1 (2008), 291.  doi: 10.1515/ACV.2008.012.  Google Scholar

[13]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits,, in, 362 (1996), 120.  doi: 10.1007/BF01233389.  Google Scholar

[14]

J. Mather, Action minimizing measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

show all references

References:
[1]

N. Anantharaman, R. Iturriaga, P. Padilla and H. Sánchez-Morgado, Physical solutions of the Hamilton-Jacobi equation,, Disc. Cont. Dyn. Sys. Series B, 5 (2005), 513.  doi: 10.3934/dcdsb.2005.5.513.  Google Scholar

[2]

M. Bardi and I. C. Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations,", Birkhausser, (1997).  doi: 10.1007/978-0-8176-4755-1.  Google Scholar

[3]

G. Barles, "Solutions de Viscosité des Équations de Hamilton Jacobi,", Mathématiques et Applications, 17 (1994).   Google Scholar

[4]

P. Bernard, Existence of $C^{1,1}$ critical sub-solutions of the Hamilton-Jacobi equation on compact manifolds,, Ann. Scient. Éc. Norm. Sup., 40 (2007), 445.   Google Scholar

[5]

G. Contreras, Action Potential and Weak KAM Solutions,, Calc. Var. Partial Differential Equations, 13 (2001), 427.  doi: 10.1007/s005260100081.  Google Scholar

[6]

G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians,", 22 Colóquio Brasileiro de Matemática, (1999).   Google Scholar

[7]

A. Fathi, "The Weak KAM Theorem in Lagrangian Dynamics,", Cambridge Studies in Advanced Mathematics, (2010).   Google Scholar

[8]

A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls,, C. R. Acad. Sci. Paris, 325 (1997), 649.   Google Scholar

[9]

A. Fathi, Théorème KAM faible et Théorie de Mather sur les systems Lagrangiens,, C.R. Acad. Sci. Paris, 324 (1997), 1043.   Google Scholar

[10]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik,, C. R. Acad. Sci. Paris, 327 (1998), 267.   Google Scholar

[11]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton Jacobi equation,, Invent. Math., 155 (2004), 363.  doi: 10.1007/s00222-003-0323-6.  Google Scholar

[12]

D. Gomes, Generalized Mather problem and selection principles for viscosity solutions and Mather measures,, Advances in Calculus of Variations, 1 (2008), 291.  doi: 10.1515/ACV.2008.012.  Google Scholar

[13]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits,, in, 362 (1996), 120.  doi: 10.1007/BF01233389.  Google Scholar

[14]

J. Mather, Action minimizing measures for positive definite Lagrangian systems,, Math. Z., 207 (1991), 169.  doi: 10.1007/BF02571383.  Google Scholar

[1]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024
[2]

Ahmad El Hajj, Hassan Ibrahim, Vivian Rizik. $ BV $ solution for a non-linear Hamilton-Jacobi system. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020405
[3]

Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021007
[4]

Ágota P. Horváth. Discrete diffusion semigroups associated with Jacobi-Dunkl and exceptional Jacobi polynomials. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021002
[5]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002
[6]

Wenmeng Geng, Kai Tao. Large deviation theorems for dirichlet determinants of analytic quasi-periodic jacobi operators with Brjuno-Rüssmann frequency. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5305-5335. doi: 10.3934/cpaa.2020240
[7]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375
[8]

Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303
[9]

Julian Tugaut. Captivity of the solution to the granular media equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021002
[10]

Bilel Elbetch, Tounsia Benzekri, Daniel Massart, Tewfik Sari. The multi-patch logistic equation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021025
[11]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136
[12]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345
[13]

Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384
[14]

Anh Tuan Duong, Phuong Le, Nhu Thang Nguyen. Symmetry and nonexistence results for a fractional Choquard equation with weights. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 489-505. doi: 10.3934/dcds.2020265
[15]

Maicon Sônego. Stable transition layers in an unbalanced bistable equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020370
[16]

François Dubois. Third order equivalent equation of lattice Boltzmann scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 221-248. doi: 10.3934/dcds.2009.23.221
[17]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299
[18]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015
[19]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[20]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (50)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences