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This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacs operators. We establish such an estimate for the parabolic Cauchy problem in the whole space [0, +∞) × ℝn and, under some periodicity and either ellipticity or controllability assumptions, we deduce a similar estimate for the ergodic constant associated to the operator. An interesting byproduct of the latter result will be the local uniform convergence for some classes of singular perturbation problems.
@article{COCV_2012__18_4_954_0,
author = {Marchi, Claudio},
title = {Continuous dependence estimates for the ergodic problem of {Bellman-Isaacs} operators via the parabolic {Cauchy} problem},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {954--968},
publisher = {EDP-Sciences},
volume = {18},
number = {4},
year = {2012},
doi = {10.1051/cocv/2011203},
mrnumber = {3019467},
zbl = {1262.35030},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011203/}
}
TY - JOUR AU - Marchi, Claudio TI - Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 SP - 954 EP - 968 VL - 18 IS - 4 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011203/ DO - 10.1051/cocv/2011203 LA - en ID - COCV_2012__18_4_954_0 ER -
%0 Journal Article %A Marchi, Claudio %T Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 954-968 %V 18 %N 4 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011203/ %R 10.1051/cocv/2011203 %G en %F COCV_2012__18_4_954_0
Marchi, Claudio. Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 4, pp. 954-968. doi : 10.1051/cocv/2011203. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2011203/
[1] and , Singular perturbations of nonlinear degenerate parabolic PDEs : a general convergence result. Arch. Rational Mech. Anal. 170 (2003) 17-61. | Zbl | MR
[2] and , Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation. Mem. Amer. Math. Soc. 204 (2010). | Zbl | MR
[3] and , On ergodic stochastic control. Comm. Partial Differential Equations 23 (1998) 2187-2217. | Zbl | MR
[4] and , Problèmes ergodiques de la mècanique classique. Gauthiers-Villars, Paris (1967). | Zbl | MR
[5] and , On the boundary ergodic problem for fully nonlinear equations in bounded domains with general nonlinear Neumann boundary conditions. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 22 (2005) 521-541. | Zbl | MR | mathdoc-id
[6] , , and , Ergodic problems and periodic homogenization for fully nonlinear equations in half-space type domains with Neumann boundary conditions. Indiana Univ. Math. J. 57 (2008) 2355-2375. | Zbl | MR
[7] and , Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comp. 76 (2007) 1861-1893. | Zbl | MR
[8] , and , Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations. arXiv:1005.5597. | Zbl | MR
[9] and , Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32 (2001) 1311-1326. | Zbl | MR
[10] , Perturbation Methods in Optimal Control. Wiley/Gauthiers-Villars, Chichester (1988). | Zbl | MR
[11] , and , Asymptotic Analysis for periodic Structures. North-Holland, Amsterdam (1978). | Zbl | MR
[12] , and , Viscosity solutions for a system of integro-PDEs and connections to optimal switching and control of jump-diffusion processes. Appl. Math. Optim. 62 (2010) 47-80. | Zbl | MR
[13] , C1, β regularity of viscosity solutions via a continuous-dependence result. Adv. Differential Equations 9 (2004) 447-480. | Zbl | MR
[14] , and , Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58 (2005) 319-361. | Zbl | MR
[15] , and , Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations. J. Differential Equations 170 (2001) 180-187. | Zbl | MR
[16] , and , Ergodic theory. Springer-Verlag, Berlin (1982). | Zbl | MR
[17] , and , User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | Zbl | MR
[18] , and , Lp-theory for fully nonlinear uniformly parabolic equations. Comm. Partial Differential Equations 25 (2000) 1997-2053. | Zbl | MR
[19] and , The rate of convergence of finite-difference approximations for parabolic Bellman equations with Lipschitz coefficients in cylindrical domains. Appl. Math. Optim. 56 (2007) 37-66. | Zbl | MR
[20] and , Well-posed Optimization Problems, Lecture Notes in Math. 1543. Berlin (1993). | Zbl | MR
[21] , Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinb. Sect. A 120 (1992) 245-265. | Zbl | MR
[22] and , On the existence of value functions of two-players zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293-314. | Zbl | MR
[23] , Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions. Proc. Am. Math. Soc. 130 (2002) 3651-3660. | Zbl | MR
[24] , On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE's. Comm. Pure Appl. Math. 42 (1989) 15-45. | Zbl | MR
[25] and , Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990) 26-78. | Zbl | MR
[26] and , Continuous dependence results for non-linear Neumann type boundary value problems. J. Differential Equations 245 (2008) 2368-2396. | Zbl | MR
[27] and , Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. J. Differential Equations 183 (2002) 497-525. | Zbl | MR
[28] and , Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations. Electron. J. Differential Equations 39 (2002) 1-10. | Zbl | MR
[29] and , Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differential Equations 212 (2005) 278-318. | Zbl | MR
[30] , and , Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). | Zbl | MR
[31] , and , Singular perturbation methods in control : analysis and design. Academic Press, London (1986). | Zbl
[32] and , Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications. Ann. Inst. Henti Poincaré, Anal. Non Linéaire 22 (2005) 667-677. | Zbl | MR | mathdoc-id
[33] , Functional integration and quantum physics. Academic Press, New York (1979). | Zbl | MR
[34] , Existence of viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 56 (1985) 345-390. | Zbl | MR
[35] , On the regularity theory of fully nonlinear parabolic equations : I. Comm. Pure Appl. Math. 45 (1992) 27-76. | Zbl | MR
[36] , On the regularity theory of fully nonlinear parabolic equations : II. Comm. Pure Appl. Math. 45 (1992) 141-178. | Zbl | MR
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