Finite Dimensional Projections of HJB Equations in the Wasserstein Space

This paper continues the study of controlled interacting particle systems with common noise started in [W. Gangbo, S. Mayorga and A. {\'{S}}wi{ę}ch, \textit{SIAM J. Math. Anal.} 53 (2021), no. 2, 1320--1356] and [S. Mayorga and A. {\'{S}}wi{ę}ch, \textit{SIAM J. Control Optim.} 61 (2023), no. 2, 820--851]. First, we extend the following results of the previously mentioned works to the case of multiplicative noise: (i) We generalize 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; (ii) we prove, under certain additional assumptions, \(C^{1,1}\) regularity of \(V\) in the spatial variable. The second main contribution of the present work is the proof that if \(DV\) is continuous (which, in particular, includes the previously proven case of \(C^{1,1}\) regularity in the spatial variable), the value function \(V\) projects precisely onto the value functions \(u_n\). Using this projection property, we show that optimal controls of the finite dimensional problem correspond to optimal controls of the infinite dimensional problem and vice versa. In the case of a linear state equation, we are able to prove that \(V\) projects precisely onto the value functions \(u_n\) under relaxed assumptions on the coefficients of the cost functional by using approximation techniques in the Wasserstein space, thus covering cases where \(V\) may not be differentiable.