We study some classes of singular perturbation problems where the dynamics of the fast variables evolve in the whole space obeying to an infinitesimal operator which is subelliptic and ergodic. We prove that the corresponding ergodic problem admits a solution which is globally Lipschitz continuous and it has at most a logarithmic growth at infinity. The main result of this paper establishes that, as ϵ → 0, the value functions of the singular perturbation problems converge locally uniformly to the solution of an effective problem whose operator and terminal data are explicitly given in terms of the invariant measure for the ergodic operator.

Accepted:

DOI: 10.1051/cocv/2017063

Keywords: Subelliptic equations, Heisenberg group, invariant measure, singular perturbations, viscosity solutions, degenerate elliptic equations

^{1}; Marchi, Claudio

^{1}; Tchou, Nicoletta

^{1}

@article{COCV_2018__24_4_1429_0, author = {Mannucci, Paola and Marchi, Claudio and Tchou, Nicoletta}, title = {Singular perturbations for a subelliptic operator}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1429--1451}, publisher = {EDP-Sciences}, volume = {24}, number = {4}, year = {2018}, doi = {10.1051/cocv/2017063}, zbl = {1414.35019}, mrnumber = {3922437}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2017063/} }

TY - JOUR AU - Mannucci, Paola AU - Marchi, Claudio AU - Tchou, Nicoletta TI - Singular perturbations for a subelliptic operator JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1429 EP - 1451 VL - 24 IS - 4 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017063/ DO - 10.1051/cocv/2017063 LA - en ID - COCV_2018__24_4_1429_0 ER -

%0 Journal Article %A Mannucci, Paola %A Marchi, Claudio %A Tchou, Nicoletta %T Singular perturbations for a subelliptic operator %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1429-1451 %V 24 %N 4 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017063/ %R 10.1051/cocv/2017063 %G en %F COCV_2018__24_4_1429_0

Mannucci, Paola; Marchi, Claudio; Tchou, Nicoletta. Singular perturbations for a subelliptic operator. ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 4, pp. 1429-1451. doi : 10.1051/cocv/2017063. http://archive.numdam.org/articles/10.1051/cocv/2017063/

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