Finite dimensional projections of Hamilton-Jacobi-Bellman equations in spaces of probability measures and stochastic optimal control of particle systems
ABSTRACT
In this talk we will present recent results about optimal control of large particle systems with common noise, interacting through their empirical measures. One way of analyzing the problem is by studying what happens in the limit as the number of particles $n$ goes to infinity. We will discuss how to prove the convergence of the value functions $u_n$ corresponding to control problems of $n$ particles to the value function $V$ corresponding to an appropriately defined infinite dimensional control problem, which is the unique viscosity solution of the limiting HJB equation in the Wasserstein space. The proofs of the convergence of $u_n$ to $V$ use PDE viscosity solution techniques. We will show that under certain additional assumptions, $V$ is $C^{1,1}$ in the spatial variable. We will then explain that if $DV$ is continuous, the value function $V$ projects precisely onto the value functions $u_n$. We will discuss how the $C^{1,1}$ regularity of $V$ allows to construct optimal feedback controls and how optimal controls for the finite dimensional problems correspond to optimal controls of the infinite dimensional problem and vice versa. We will also discuss how to relax assumptions on the coefficients of the cost functional by using approximation techniques in the Wasserstein space to prove that $V$ projects precisely onto the value functions $u_n$ when $V$ may not be differentiable.