We study a stochastic optimal control problem for a two scale system driven by an infinite dimensional stochastic differential equation which consists of “slow” and “fast” components. We use the theory of viscosity solutions in Hilbert spaces to show that as the speed of the fast component goes to infinity, the value function of the optimal control problem converges to the viscosity solution of a reduced effective equation. We consider a rather general case where the evolution is given by an abstract semilinear stochastic differential equation with nonlinear dependence on the controls. The results of this paper generalize to the infinite dimensional case the finite dimensional results of Alvarez and Bardi [SIAM J. Control Optim. 40 (2001/02) 1159–1188] and complement the results in Hilbert spaces obtained recently in Guatteri and Tessitore [To appear in: Appl. Math. Optim. (2019) https://doi.org/10.1007/s00245-019-09577-y].
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Keywords: Hamilton-Jacobi-Bellman equation, viscosity solution, stochastic optimal control, singular perturbation, two-scale system
@article{COCV_2021__27_1_A8_0,
author = {\'Swi\k{e}ch, Andrzej},
title = {Singular perturbations and optimal control of stochastic systems in infinite dimension: {HJB} equations and viscosity solutions},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021001},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021001/}
}
TY - JOUR AU - Święch, Andrzej TI - Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021001/ DO - 10.1051/cocv/2021001 LA - en ID - COCV_2021__27_1_A8_0 ER -
%0 Journal Article %A Święch, Andrzej %T Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021001/ %R 10.1051/cocv/2021001 %G en %F COCV_2021__27_1_A8_0
Święch, Andrzej. Singular perturbations and optimal control of stochastic systems in infinite dimension: HJB equations and viscosity solutions. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 6. doi: 10.1051/cocv/2021001
[1] and , Viscosity solutions methods for singular perturbations in deterministic and stochastic control. SIAM J. Control Optim. 40 (2002) 1159–1188.
[2] and , Singular perturbations of nonlinear degenerate parabolic PDEs: a general convergence result. Arch. Ration. Mech. Anal. 170 (2003) 17–61.
[3] and , Ergodicity, stabilization, and singular perturbations for Bellman-Isaacs equations. Mem. Amer. Math. Soc. 204 (2010) 1–88.
[4] and , On ergodic stochastic control. Commun. Partial Differ. Equ. 23 (1998) 2187–2217.
[5] and , Optimal control with random parameters: a multiscale approach. Eur. J. Control 17 (2011) 30–45.
[6] , and , Convergence by viscosity methods in multiscale financial models with stochastic volatility. SIAM J. Financial Math. 1 (2010) 230–65.
[7] , Perturbation methods in optimal control. Translated from the French by . Wiley/Gauthier-Villars Series in Modern Applied Mathematics. John Wiley & Sons, Ltd., Chichester; Gauthier-Villars, Montrouge (1988).
[8] and Singular perturbations in risk-sensitive stochastic control. SIAM J. Control Optim. 48 (2010) 3675–3697.
[9] and , Singular perturbations in ergodic control of diffusions. SIAM J. Control Optim. 46 (2007) 1562–1577.
[10] , and , User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27 (1992) 1–67.
[11] and , Viscosity solutions of Hamilton-Jacobi equations in infinite dimensions. IV. Hamiltonians with unbounded linear terms. J. Funct. Anal. 90 (1990) 237–283.
[12] and , Stochastic equations in infinite dimensions. Second edition. Encyclopedia of Mathematics and its Applications, 152. Cambridge University Press, Cambridge (2014).
[13] and , Singular perturbations in infinite-dimensional control systems. SIAM J. Control Optim. 42 (2003) 1795–1812.
[14] , and , Stochastic optimal control in infinite dimension. Dynamic programming and HJB equations. With a contribution by Marco Fuhrman and Gianmario Tessitore. Probability Theory and Stochastic Modelling, 82. Springer, Cham (2017).
[15] , Viscosity methods for large deviations estimates of multiscale stochastic processes. ESAIM: COCV 24 (2018) 605–637.
[16] and , Singular limit of BSDEs and optimal control of two scale stochastic systems in infinite dimensional spaces. To appear in: Appl. Math. Optim. (2019) . | DOI
[17] and , Two-scale stochastic systems. Asymptotic analysis and control. Applications of Mathematics (New York), 49. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin (2003).
[18] and , On control of two-scale stochastic systems with linear dynamics in the fast variables. Math. Control Signals Syst. 9 (1996) 107–122.
[19] and , Perron’s method and the method of relaxed limits for “unbounded” PDE in Hilbert spaces. Studia Math. 176 (2006) 249–277.
[20] , Weak convergence methods and singularly perturbed stochastic control and filtering problems. Systems & Control: Foundations & Applications, 3. Birkhäuser Boston, Inc., Boston, MA (1990).
[21] , Une inégalité pour les opérateurs elliptiques du second ordre. Ann. Mat. Pura Appl. 127 (1981) 1–11.
[22] , and , Singular perturbations for a subelliptic operator. ESAIM: COCV 24 (2018) 1429–1451.
[23] , Singular perturbations and time scales in control theory and applications: an overview. Singularly perturbed dynamic systems in control technology. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algor. 9 (2002) 233–278.
[24] Singular perturbations in stochastic ergodic control problems. SIAM J. Control Optim. 50 (2012) 3203–3223.
[25] , Singular perturbation of stochastic control and differential games. Ph.D. thesis, The University of Sydney (2020).
[26] , , and , Singular perturbations and time scales in control theories and applications: an overview 2002–2012. Int. J. Inf. Syst. Sci. 9 (2014) 1–36.
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