Optimal Control with Random Parameters: A Multiscale Approach

We model the parameters of a control problem as an ergodic diffusion process evolving at a faster time scale than the state variables. We study the asymptotics as the speed of the parameters gets large. We prove the convergence of the value function to the solution of a limit Cauchy problem for a Hamilton-Jacobi equation whose Hamiltonian is a suitable average of the initial one. We give several examples where the effective Hamiltonian allows to define a limit control problem whose dynamics and payoff are linear or nonlinear averages of the initial data. This is therefore a constant-parameter approximation of the control problem with random entries. Our results hold if the fast random parameters are the only disturbances acting on the system, and then the limit system is deterministic, but also for dynamics affected by a white noise, and then the limit is a controlled diffusion.