Ergodic Control of McKean–Vlasov Systems on the Wasserstein Space

We consider an optimal control problem with ergodic (long term average) reward for a McKean-Vlasov dynamics, where the coefficients of a controlled stochastic ifferential equation depend on the marginal law of the solution. Starting from the associated infinite time horizon expected discounted reward, we construct both the value λ of the ergodic problem and an associated function ϕ, which provide a viscosity solution to an ergodic Hamilton-Jacobi-Bellman (HJB) equation of elliptic type. In contrast to previous results, we consider the function ϕ and the HJB equation on the Wasserstein space, using concepts of derivatives with respect to probability measures. The pair (λ, ϕ) also provides information on limit behavior of related optimization problems, for instance, results of Abelian-Tauberian type or limits of value functions of control problems for finite time horizon when the latter tends to infinity. Many arguments are simplified by the use of a functional relation for ϕ in the form of a suitable dynamic programming principle.