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January  2013, 12(1): 207-236. doi: 10.3934/cpaa.2013.12.207

On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations

Martino Bardi 1, and  Gabriele Terrone 2,

1. 

Dipartimento di Matematica, Università di Padova, via Trieste, 63; I-35121 Padova
2. 

Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa

Received  April 2011 Revised  September 2011 Published  September 2012

We study the homogenization of Hamilton-Jacobi equations with oscillating initial data and non-coercive Hamiltonian, mostly of the Bellman-Isaacs form arising in optimal control and differential games. We describe classes of equations for which pointwise homogenization fails for some data. We prove locally uniform homogenization for various Hamiltonians with some partial coercivity and some related restrictions on the oscillating variables, mostly motivated by the applications to differential games, in particular of pursuit-evasion type. The effective initial data are computed under some assumptions of asymptotic controllability of the underlying control system with two competing players.
Keywords: oscillating initial data., Homogenization, Isaacs equations, viscosity solutions, Hamilton-Jacobi equations, differential games.
Mathematics Subject Classification: Primary: 35B27, 35F21, 49N70; Secondary: 49L2.
Citation: Martino Bardi, Gabriele Terrone. On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 207-236. doi: 10.3934/cpaa.2013.12.207
References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Homogenization of Hamilton-Jacobi equations: numerical methods, Math. Models Methods Appl. Sci., 18 (2008), 1115-1143. doi: 10.1142/S0218202508002978.  Google Scholar

[2]

O. Alvarez, Homogenization of Hamilton-Jacobi equations in perforated sets, J. Differential Equations, 159 (1999), 543-577. doi: 10.1006/jdeq.1999.3665.  Google Scholar

[3]

O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61. doi: 10.1007/s00205-003-0266-5.  Google Scholar

[4]

O. Alvarez and M. Bardi, Ergodic problems in differential games, In "Advances in Dynamic Game Theory," volume 9 of Ann. Internat. Soc. Dynam. Games, pages 131-152. Birkhäuser Boston, Boston, MA, 2007. doi: 10.1007/978-0-8176-4553-3_7.  Google Scholar

[5]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010). doi: 10.1090/S0065-9266-09-00588-2.  Google Scholar

[6]

O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027.  Google Scholar

[7]

O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, In "Geometric Control and Nonsmooth Analysis," volume 76 of Ser. Adv. Math. Appl. Sci., pages 1-27. World Sci. Publ., Hackensack, NJ, 2008.  Google Scholar

[8]

M. Arisawa, P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413.  Google Scholar

[9]

M. Bardi, On differential games with long-time-average cost, In "Advances in Dynamic Games and Their Applications," volume 10 of Ann. Internat. Soc. Dynam. Games, pages 3-18. Birkhäuser Boston Inc., Boston, MA, 2009.  Google Scholar

[10]

M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Birkhäuser Boston Inc., Boston, MA, 1997.  Google Scholar

[11]

G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466. doi: 10.1007/s00526-007-0097-6.  Google Scholar

[12]

G. Barles and P. E. Souganidis, Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations, C. R. Acad. Sci. Paris Sect. I Math., 330 (2000), 963-968. Google Scholar

[13]

K. Bhattacharya and B. Craciun, Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773-805. doi: 10.1017/S0308210500002675.  Google Scholar

[14]

I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group, Commun. Pure Appl. Anal., 2 (2003), 461-479. doi: 10.3934/cpaa.2003.2.461.  Google Scholar

[15]

A. Braides and A. Defranceschi, Homogenization of multiple integrals, in volume 12 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press Oxford University Press, New York, 1998.  Google Scholar

[16]

F. Camilli and A. Siconolfi, Effective Hamiltonian and homogenization of measurable eikonal equations, Arch. Ration. Mech. Anal., 183 (2007), 1-20. doi: 10.1007/s00205-006-0001-0.  Google Scholar

[17]

I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), 1113-1129. doi: 10.1512/iumj.2001.50.1933.  Google Scholar

[18]

P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $R^2/Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856. doi: 10.1016/j.anihpc.2009.11.015.  Google Scholar

[19]

P. Cardaliaguet, P.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl., 91 (2009), 339-363. doi: 10.1016/j.matpur.2009.01.014.  Google Scholar

[20]

P. Cardaliaguet, J. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561. doi: 10.1007/s00205-010-0332-8.  Google Scholar

[21]

A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., 345 (2009), 749-782. doi: 10.1007/s00208-009-0372-2.  Google Scholar

[22]

W. E., A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759. doi: 10.1002/cpa.3160440702.  Google Scholar

[23]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265.  Google Scholar

[24]

L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57 (2004), 445-480. doi: 10.1002/cpa.20009.  Google Scholar

[25]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[26]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Lecture Notes, ().   Google Scholar

[27]

D. A. Gomes, Hamilton-Jacobi methods for vakonomic mechanics, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 233-257. doi: 10.1007/s00030-007-5012-5.  Google Scholar

[28]

K. Horie and H. Ishii, Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure, Indiana Univ. Math. J., 47 (1998), 1011-1058. doi: 10.1512/iumj.1998.47.1385.  Google Scholar

[29]

C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/\epsilon)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89. doi: 10.1007/s00205-007-0074-4.  Google Scholar

[30]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000.  Google Scholar

[31]

P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. Google Scholar

[32]

C. Marchi, On the convergence of singular perturbations of Hamilton-Jacobi equations, Commun. Pure Appl. Anal., 9 (2010), 1363-1377. doi: 10.3934/cpaa.2010.9.1363.  Google Scholar

[33]

F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309. doi: 10.1007/s002050050198.  Google Scholar

[34]

P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control Optim., 31 (1993), 604-623. doi: 10.1137/0331027.  Google Scholar

[35]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11.  Google Scholar

[36]

B. Stroffolini, Homogenization of Hamilton-Jacobi equations in Carnot groups, ESAIM Control Optim. Calc. Var., 13 (2007), 107-119 (electronic). doi: 10.1051/cocv:2007005.  Google Scholar

[37]

G. Terrone, "Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and Fully Nonlinear Partial Differential Equations," PhD thesis, University of Padova, 2008. Google Scholar

[38]

C. Viterbo, Symplectic homogenization, Preprint, 2008, arXiv:0801.0206v1">arXiv:0801.0206v1" target="_blank">arXiv:0801.0206v1. Google Scholar

[39]

J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2.  Google Scholar

show all references

References:
[1]

Y. Achdou, F. Camilli and I. Capuzzo Dolcetta, Homogenization of Hamilton-Jacobi equations: numerical methods, Math. Models Methods Appl. Sci., 18 (2008), 1115-1143. doi: 10.1142/S0218202508002978.  Google Scholar

[2]

O. Alvarez, Homogenization of Hamilton-Jacobi equations in perforated sets, J. Differential Equations, 159 (1999), 543-577. doi: 10.1006/jdeq.1999.3665.  Google Scholar

[3]

O. Alvarez and M. Bardi, Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result, Arch. Ration. Mech. Anal., 170 (2003), 17-61. doi: 10.1007/s00205-003-0266-5.  Google Scholar

[4]

O. Alvarez and M. Bardi, Ergodic problems in differential games, In "Advances in Dynamic Game Theory," volume 9 of Ann. Internat. Soc. Dynam. Games, pages 131-152. Birkhäuser Boston, Boston, MA, 2007. doi: 10.1007/978-0-8176-4553-3_7.  Google Scholar

[5]

O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations, Mem. Amer. Math. Soc., 204 (2010). doi: 10.1090/S0065-9266-09-00588-2.  Google Scholar

[6]

O. Alvarez, M. Bardi and C. Marchi, Multiscale problems and homogenization for second-order Hamilton-Jacobi equations, J. Differential Equations, 243 (2007), 349-387. doi: 10.1016/j.jde.2007.05.027.  Google Scholar

[7]

O. Alvarez, M. Bardi and C. Marchi, Multiscale singular perturbations and homogenization of optimal control problems, In "Geometric Control and Nonsmooth Analysis," volume 76 of Ser. Adv. Math. Appl. Sci., pages 1-27. World Sci. Publ., Hackensack, NJ, 2008.  Google Scholar

[8]

M. Arisawa, P.-L. Lions, On ergodic stochastic control, Comm. Partial Differential Equations, 23 (1998), 2187-2217. doi: 10.1080/03605309808821413.  Google Scholar

[9]

M. Bardi, On differential games with long-time-average cost, In "Advances in Dynamic Games and Their Applications," volume 10 of Ann. Internat. Soc. Dynam. Games, pages 3-18. Birkhäuser Boston Inc., Boston, MA, 2009.  Google Scholar

[10]

M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations," Birkhäuser Boston Inc., Boston, MA, 1997.  Google Scholar

[11]

G. Barles, Some homogenization results for non-coercive Hamilton-Jacobi equations, Calc. Var. Partial Differential Equations, 30 (2007), 449-466. doi: 10.1007/s00526-007-0097-6.  Google Scholar

[12]

G. Barles and P. E. Souganidis, Some counterexamples on the asymptotic behavior of the solutions of Hamilton-Jacobi equations, C. R. Acad. Sci. Paris Sect. I Math., 330 (2000), 963-968. Google Scholar

[13]

K. Bhattacharya and B. Craciun, Homogenization of a Hamilton-Jacobi equation associated with the geometric motion of an interface, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 773-805. doi: 10.1017/S0308210500002675.  Google Scholar

[14]

I. Birindelli and J. Wigniolle, Homogenization of Hamilton-Jacobi equations in the Heisenberg group, Commun. Pure Appl. Anal., 2 (2003), 461-479. doi: 10.3934/cpaa.2003.2.461.  Google Scholar

[15]

A. Braides and A. Defranceschi, Homogenization of multiple integrals, in volume 12 of "Oxford Lecture Series in Mathematics and its Applications," The Clarendon Press Oxford University Press, New York, 1998.  Google Scholar

[16]

F. Camilli and A. Siconolfi, Effective Hamiltonian and homogenization of measurable eikonal equations, Arch. Ration. Mech. Anal., 183 (2007), 1-20. doi: 10.1007/s00205-006-0001-0.  Google Scholar

[17]

I. Capuzzo Dolcetta and H. Ishii, On the rate of convergence in homogenization of Hamilton-Jacobi equations, Indiana Univ. Math. J., 50 (2001), 1113-1129. doi: 10.1512/iumj.2001.50.1933.  Google Scholar

[18]

P. Cardaliaguet, Ergodicity of Hamilton-Jacobi equations with a noncoercive nonconvex Hamiltonian in $R^2/Z^2$, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 837-856. doi: 10.1016/j.anihpc.2009.11.015.  Google Scholar

[19]

P. Cardaliaguet, P.-L. Lions and P. E. Souganidis, A discussion about the homogenization of moving interfaces, J. Math. Pures Appl., 91 (2009), 339-363. doi: 10.1016/j.matpur.2009.01.014.  Google Scholar

[20]

P. Cardaliaguet, J. Nolen and P. E. Souganidis, Homogenization and enhancement for the G-equation, Arch. Ration. Mech. Anal., 199 (2011), 527-561. doi: 10.1007/s00205-010-0332-8.  Google Scholar

[21]

A. Davini and A. Siconolfi, Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case, Math. Ann., 345 (2009), 749-782. doi: 10.1007/s00208-009-0372-2.  Google Scholar

[22]

W. E., A class of homogenization problems in the calculus of variations, Comm. Pure Appl. Math., 44 (1991), 733-759. doi: 10.1002/cpa.3160440702.  Google Scholar

[23]

L. C. Evans, Periodic homogenisation of certain fully nonlinear partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), 245-265.  Google Scholar

[24]

L. C. Evans, A survey of partial differential equations methods in weak KAM theory, Comm. Pure Appl. Math., 57 (2004), 445-480. doi: 10.1002/cpa.20009.  Google Scholar

[25]

L. C. Evans and P. E. Souganidis, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Univ. Math. J., 33 (1984), 773-797. doi: 10.1512/iumj.1984.33.33040.  Google Scholar

[26]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics,", Lecture Notes, ().   Google Scholar

[27]

D. A. Gomes, Hamilton-Jacobi methods for vakonomic mechanics, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 233-257. doi: 10.1007/s00030-007-5012-5.  Google Scholar

[28]

K. Horie and H. Ishii, Homogenization of Hamilton-Jacobi equations on domains with small scale periodic structure, Indiana Univ. Math. J., 47 (1998), 1011-1058. doi: 10.1512/iumj.1998.47.1385.  Google Scholar

[29]

C. Imbert and R. Monneau, Homogenization of first-order equations with $(u/\epsilon)$-periodic Hamiltonians. I. Local equations, Arch. Ration. Mech. Anal., 187 (2008), 49-89. doi: 10.1007/s00205-007-0074-4.  Google Scholar

[30]

H. Ishii, Almost periodic homogenization of Hamilton-Jacobi equations, In "International Conference on Differential Equations," Vol. 1, 2 (Berlin, 1999), pages 600-605. World Sci. Publ., River Edge, NJ, 2000.  Google Scholar

[31]

P.-L. Lions, G. Papanicolau and S. R. S. Varadhan, Homogeneization of Hamilton-Jacobi equations, Unpublished, 1986. Google Scholar

[32]

C. Marchi, On the convergence of singular perturbations of Hamilton-Jacobi equations, Commun. Pure Appl. Anal., 9 (2010), 1363-1377. doi: 10.3934/cpaa.2010.9.1363.  Google Scholar

[33]

F. Rezakhanlou and J. E. Tarver, Homogenization for stochastic Hamilton-Jacobi equations, Arch. Ration. Mech. Anal., 151 (2000), 277-309. doi: 10.1007/s002050050198.  Google Scholar

[34]

P. Soravia, Pursuit-evasion problems and viscosity solutions of Isaacs equations, SIAM J. Control Optim., 31 (1993), 604-623. doi: 10.1137/0331027.  Google Scholar

[35]

P. E. Souganidis, Stochastic homogenization of Hamilton-Jacobi equations and some applications, Asymptot. Anal., 20 (1999), 1-11.  Google Scholar

[36]

B. Stroffolini, Homogenization of Hamilton-Jacobi equations in Carnot groups, ESAIM Control Optim. Calc. Var., 13 (2007), 107-119 (electronic). doi: 10.1051/cocv:2007005.  Google Scholar

[37]

G. Terrone, "Singular Perturbation and Homogenization Problems in Control Theory, Differential Games and Fully Nonlinear Partial Differential Equations," PhD thesis, University of Padova, 2008. Google Scholar

[38]

C. Viterbo, Symplectic homogenization, Preprint, 2008, arXiv:0801.0206v1">arXiv:0801.0206v1" target="_blank">arXiv:0801.0206v1. Google Scholar

[39]

J. Xin, "An Introduction to Fronts in Random Media," Springer, New York, 2009. doi: 10.1007/978-0-387-87683-2.  Google Scholar

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