Hostname: page-component-7bb8b95d7b-cx56b Total loading time: 0 Render date: 2024-09-21T05:14:36.405Z Has data issue: false hasContentIssue false
  • English
  • Français

Continuous dependence estimates for the ergodic problem of Bellman-Isaacs operators via the parabolic Cauchy problem∗

Published online by Cambridge University Press:  16 January 2012

Claudio Marchi
Claudio Marchi*
Affiliation:
Dip. di Matematica Pura ed Applicata, Università di Padova, via Trieste 63, 35121 Padova, Italy. marchi@math.unipd.it
Get access

Abstract

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.

Keywords

Continuous dependence estimatesparabolic Hamilton-Jacobi equationsviscosity solutionsergodic problemsdifferential gamessingular perturbations
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 4 , October 2012 , pp. 954 - 968
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

Alvarez, O. and Bardi, M., Singular perturbations of nonlinear degenerate parabolic PDEs : a general convergence result. Arch. Rational Mech. Anal. 170 (2003) 17–61. Google Scholar
O. Alvarez and M. Bardi, Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equation. Mem. Amer. Math. Soc. 204 (2010).
Arisawa, M. and Lions, P.L., On ergodic stochastic control. Comm. Partial Differential Equations 23 (1998) 2187–2217. Google Scholar
V.I. Arnold and A. Avez, Problèmes ergodiques de la mècanique classique. Gauthiers-Villars, Paris (1967).
Barles, G. and Lio, F. Da, 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. Google Scholar
Barles, G., Lio, F. Da, Lions, P.L. and Souganidis, P.E., 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. Google Scholar
Barles, G. and Jakobsen, E.R., Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Math. Comp. 76 (2007) 1861–1893. Google Scholar
G. Barles, O. Ley and H. Mitake, Short time uniqueness results for solutions of nonlocal and non-monotone geometric equations. arXiv:1005.5597.
Barles, G. and Souganidis, P.E., Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations. SIAM J. Math. Anal. 32 (2001) 1311–1326. Google Scholar
A. Bensoussan, Perturbation Methods in Optimal Control. Wiley/Gauthiers-Villars, Chichester (1988).
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for periodic Structures. North-Holland, Amsterdam (1978).
Biswas, I.H., Jakobsen, E.R. and Karlsen, K.H., 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. Google Scholar
Bourgoing, M., C 1, β regularity of viscosity solutions via a continuous-dependence result. Adv. Differential Equations 9 (2004) 447–480. Google Scholar
Caffarelli, L., Souganidis, P. and Wang, L., Homogenization of fully nonlinear, uniformly elliptic and parabolic partial differential equations in stationary ergodic media. Comm. Pure Appl. Math. 58 (2005) 319–361. Google Scholar
Cockburn, B., Gripenberg, G. and Londen, S.-O., Continuous dependence on the nonlinearity of viscosity solutions of parabolic equations. J. Differential Equations 170 (2001) 180–187. Google Scholar
I.P. Cornfeld, S.V. Fomin and Y.G. Sinai, Ergodic theory. Springer-Verlag, Berlin (1982).
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. Google Scholar
Crandall, M.G., Kocan, M. and Świech, A., L p-theory for fully nonlinear uniformly parabolic equations. Comm. Partial Differential Equations 25 (2000) 1997–2053. Google Scholar
Dong, H. and Krylov, N.V., 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. Google Scholar
A. Dontchev and T. Zolezzi, Well-posed Optimization Problems, Lecture Notes in Math. 1543. Berlin (1993).
Evans, L., Periodic homogenisation of certain fully nonlinear partial differential equations. Proc. Roy. Soc. Edinb. Sect. A 120 (1992) 245–265. Google Scholar
Fleming, W.H. and Souganidis, P.E., On the existence of value functions of two-players zero-sum stochastic differential games. Indiana Univ. Math. J. 38 (1989) 293–314. Google Scholar
Gripenberg, G., Estimates for viscosity solutions of parabolic equations with Dirichlet boundary conditions. Proc. Am. Math. Soc. 130 (2002) 3651–3660. Google Scholar
Ishii, H., On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDE’s. Comm. Pure Appl. Math. 42 (1989) 15–45. Google Scholar
Ishii, H. and Lions, P.L., Viscosity solutions of fully nonlinear second-order elliptic partial differential equations. J. Differential Equations 83 (1990) 26–78. Google Scholar
Jakobsen, E.R. and Georgelin, C.A., Continuous dependence results for non-linear Neumann type boundary value problems. J. Differential Equations 245 (2008) 2368–2396. Google Scholar
Jakobsen, E.R. and Karlsen, K.H., Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate parabolic equations. J. Differential Equations 183 (2002) 497–525. Google Scholar
Jakobsen, E.R. and Karlsen, K.H., Continuous dependence estimates for viscosity solutions of fully nonlinear degenerate elliptic equations. Electron. J. Differential Equations 39 (2002) 1–10. Google Scholar
Jakobsen, E.R. and Karlsen, K.H., Continuous dependence estimates for viscosity solutions of integro-PDEs. J. Differential Equations 212 (2005) 278–318. Google Scholar
V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994).
P.V. Kokotović, H.K. Khalil and J. O’Reilly, Singular perturbation methods in control : analysis and design. Academic Press, London (1986).
Lions, P.L. and Souganidis, P., 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. Google Scholar
B. Simon, Functional integration and quantum physics. Academic Press, New York (1979).
Souganidis, P.E., Existence of viscosity solutions of Hamilton-Jacobi equations. J. Differential Equations 56 (1985) 345–390. Google Scholar
Wang, L., On the regularity theory of fully nonlinear parabolic equations : I. Comm. Pure Appl. Math. 45 (1992) 27–76. Google Scholar
Wang, L., On the regularity theory of fully nonlinear parabolic equations : II. Comm. Pure Appl. Math. 45 (1992) 141–178. Google Scholar