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

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

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.
Citation: Martino Bardi, Gabriele Terrone. On the homogenization of some non-coercive Hamilton--Jacobi--Isaacs equations. Communications on Pure and 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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Lecture Notes, to appear.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[38]

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

[39]

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

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.

[2]

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

[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.

[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.

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[26]

A. Fathi, "Weak KAM Theorem in Lagrangian Dynamics," Lecture Notes, to appear.

[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.

[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.

[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.

[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.

[31]

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

[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.

[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.

[34]

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

[35]

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

[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.

[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.

[38]

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

[39]

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

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