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 & 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

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