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), 513-528. 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, Springer 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., 4e série, 40 (2007), 445-452.  Google Scholar

[5]

G. Contreras, Action Potential and Weak KAM Solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. 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, Série I, 325 (1997), 649-652.  Google Scholar

[9]

A. Fathi, Théorème KAM faible et Théorie de Mather sur les systems Lagrangiens, C.R. Acad. Sci. Paris, Sér. I, 324 (1997), 1043-1046.  Google Scholar

[10]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, 327 (1998), 267-270.  Google Scholar

[11]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton Jacobi equation, Invent. Math., 155 (2004), 363-388. 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), 291-307. doi: 10.1515/ACV.2008.012.  Google Scholar

[13]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, in "International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé)" (F. Ledrappier, J. Lewowicz, S. Newhouse, eds.), Pitman Research Notes in Math., 362 (1996), 120-131; Reprinted in Bol. Soc. Bras. Mat., 28 (1997), 141-153. doi: 10.1007/BF01233389.  Google Scholar

[14]

J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. 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), 513-528. 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, Springer 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., 4e série, 40 (2007), 445-452.  Google Scholar

[5]

G. Contreras, Action Potential and Weak KAM Solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. 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, Série I, 325 (1997), 649-652.  Google Scholar

[9]

A. Fathi, Théorème KAM faible et Théorie de Mather sur les systems Lagrangiens, C.R. Acad. Sci. Paris, Sér. I, 324 (1997), 1043-1046.  Google Scholar

[10]

A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, 327 (1998), 267-270.  Google Scholar

[11]

A. Fathi and A. Siconolfi, Existence of $C^1$ critical subsolutions of the Hamilton Jacobi equation, Invent. Math., 155 (2004), 363-388. 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), 291-307. doi: 10.1515/ACV.2008.012.  Google Scholar

[13]

R. Mañé, Lagrangian flows: The dynamics of globally minimizing orbits, in "International Congress on Dynamical Systems in Montevideo (a tribute to Ricardo Mañé)" (F. Ledrappier, J. Lewowicz, S. Newhouse, eds.), Pitman Research Notes in Math., 362 (1996), 120-131; Reprinted in Bol. Soc. Bras. Mat., 28 (1997), 141-153. doi: 10.1007/BF01233389.  Google Scholar

[14]

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

[1]

Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159
[2]

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. The stochastic Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2009, 1 (3) : 295-315. doi: 10.3934/jgm.2009.1.295
[3]

Tomoki Ohsawa, Anthony M. Bloch. Nonholonomic Hamilton-Jacobi equation and integrability. Journal of Geometric Mechanics, 2009, 1 (4) : 461-481. doi: 10.3934/jgm.2009.1.461
[4]

Nalini Anantharaman, Renato Iturriaga, Pablo Padilla, Héctor Sánchez-Morgado. Physical solutions of the Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 513-528. doi: 10.3934/dcdsb.2005.5.513
[5]

María Barbero-Liñán, Manuel de León, David Martín de Diego, Juan C. Marrero, Miguel C. Muñoz-Lecanda. Kinematic reduction and the Hamilton-Jacobi equation. Journal of Geometric Mechanics, 2012, 4 (3) : 207-237. doi: 10.3934/jgm.2012.4.207
[6]

Larry M. Bates, Francesco Fassò, Nicola Sansonetto. The Hamilton-Jacobi equation, integrability, and nonholonomic systems. Journal of Geometric Mechanics, 2014, 6 (4) : 441-449. doi: 10.3934/jgm.2014.6.441
[7]

Yoshikazu Giga, Przemysław Górka, Piotr Rybka. Nonlocal spatially inhomogeneous Hamilton-Jacobi equation with unusual free boundary. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 493-519. doi: 10.3934/dcds.2010.26.493
[8]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[9]

Yuxiang Li. Stabilization towards the steady state for a viscous Hamilton-Jacobi equation. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1917-1924. doi: 10.3934/cpaa.2009.8.1917
[10]

Alexander Quaas, Andrei Rodríguez. Analysis of the attainment of boundary conditions for a nonlocal diffusive Hamilton-Jacobi equation. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 5221-5243. doi: 10.3934/dcds.2018231
[11]

Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223
[12]

Claudio Marchi. On the convergence of singular perturbations of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1363-1377. doi: 10.3934/cpaa.2010.9.1363
[13]

Isabeau Birindelli, J. Wigniolle. Homogenization of Hamilton-Jacobi equations in the Heisenberg group. Communications on Pure & Applied Analysis, 2003, 2 (4) : 461-479. doi: 10.3934/cpaa.2003.2.461
[14]

Manuel de León, David Martín de Diego, Miguel Vaquero. A Hamilton-Jacobi theory on Poisson manifolds. Journal of Geometric Mechanics, 2014, 6 (1) : 121-140. doi: 10.3934/jgm.2014.6.121
[15]

Laura Caravenna, Annalisa Cesaroni, Hung Vinh Tran. Preface: Recent developments related to conservation laws and Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (5) : i-iii. doi: 10.3934/dcdss.201805i
[16]

Fabio Camilli, Paola Loreti, Naoki Yamada. Systems of convex Hamilton-Jacobi equations with implicit obstacles and the obstacle problem. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1291-1302. doi: 10.3934/cpaa.2009.8.1291
[17]

Giuseppe Marmo, Giuseppe Morandi, Narasimhaiengar Mukunda. The Hamilton-Jacobi theory and the analogy between classical and quantum mechanics. Journal of Geometric Mechanics, 2009, 1 (3) : 317-355. doi: 10.3934/jgm.2009.1.317
[18]

Yasuhiro Fujita, Katsushi Ohmori. Inequalities and the Aubry-Mather theory of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 683-688. doi: 10.3934/cpaa.2009.8.683
[19]

Olga Bernardi, Franco Cardin. On $C^0$-variational solutions for Hamilton-Jacobi equations. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 385-406. doi: 10.3934/dcds.2011.31.385
[20]

Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (82)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences