State-constrained optimal control on Wasserstein spaces over Riemannian manifolds

We study a state-constrained optimal control problem in Mayer form on the Wasserstein space P 2 (M ) of a complete (possibly non-compact) Riemannian manifold M . The controlled dynamics is given by a nonlocal continuity equation, where the velocity field depends on both the space variable and the evolving probability measure. In the presence of state constraints, the associated value function may fail to be continuous, which prevents a direct characterization through Hamilton-Jacobi-Bellman equations (HJB). Following a level-set approach, we introduce an auxiliary value function defined on an extended space and prove that its zero-sublevel set recovers the epigraph of the original value function.Our main result shows that this auxiliary function is the unique viscosity solution of a suitable HJB equation on P 2 (M ). To prove uniqueness, we develop a comparison principle based on directional differentiability properties of the squared Wasserstein distance. These properties are shown to hold under a geometric assumption on the underlying manifold, satisfied in particular by Cartan-Hadamard manifolds and spaces with sectional curvature bounded from below. This extends previous results obtained in the unconstrained case and in Euclidean or compact settings to the state-constrained framework on general complete Riemannian manifolds.