Limit of the infinite horizon discounted Hamilton-Jacobi equation

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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

Abstract
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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 and 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.
[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.
[3]	G. Barles, "Solutions de Viscosité des Équations de Hamilton Jacobi," Mathématiques et Applications, 17, Springer 1994.
[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.
[5]	G. Contreras, Action Potential and Weak KAM Solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.
[6]	G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians," 22 Colóquio Brasileiro de Matemática, (1999).
[7]	A. Fathi, "The Weak KAM Theorem in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, 2010.
[8]	A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris, Série I, 325 (1997), 649-652.
[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.
[10]	A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, 327 (1998), 267-270.
[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.
[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.
[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.
[14]	J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.

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.
[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.
[3]	G. Barles, "Solutions de Viscosité des Équations de Hamilton Jacobi," Mathématiques et Applications, 17, Springer 1994.
[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.
[5]	G. Contreras, Action Potential and Weak KAM Solutions, Calc. Var. Partial Differential Equations, 13 (2001), 427-458. doi: 10.1007/s005260100081.
[6]	G. Contreras and R. Iturriaga, "Global Minimizers of Autonomous Lagrangians," 22 Colóquio Brasileiro de Matemática, (1999).
[7]	A. Fathi, "The Weak KAM Theorem in Lagrangian Dynamics," Cambridge Studies in Advanced Mathematics, 2010.
[8]	A. Fathi, Solutions KAM faibles conjuguées et barrières de Peierls, C. R. Acad. Sci. Paris, Série I, 325 (1997), 649-652.
[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.
[10]	A. Fathi, Sur la convergence du semi-groupe de Lax-Oleinik, C. R. Acad. Sci. Paris, Série I, 327 (1998), 267-270.
[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.
[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.
[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.
[14]	J. Mather, Action minimizing measures for positive definite Lagrangian systems, Math. Z., 207 (1991), 169-207. doi: 10.1007/BF02571383.

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