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This work concerns the optimal control problem for McKean-Vlasov SDEs. Inorder to characterize the value function, we develop the viscosity solutiontheory for Hamilton-Jacobi-Bellman (HJB) equations on the Wasserstein spaceusing Mortensen's derivative. In particular, a comparison principle forviscosity solution is established. Our approach is based on Borwein-Preissvariational principle to overcome the loss of compactness for bounded sets inthe Wasserstein space.

Viscosity solutions to HJB equations associated with optimal control problem for McKean-Vlasov SDEs