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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

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