Hamilton--Jacobi--Bellman equation for optimal control of stochastic Wasserstein--Hamiltonian system on graphs

Stochastic optimal control problems for Hamiltonian dynamics on graphs have wide-ranging applications in mechanics and quantum field theory, particularly in systems with graph-based structures. In this paper, we establish the existence and uniqueness of viscosity solutions for a new class of Hamilton--Jacobi--Bellman (HJB) equations arising from the optimal control of stochastic Wasserstein--Hamiltonian systems (SWHSs) on graphs. One distinctive feature of these HJB equations is the simultaneous involvement of the Wasserstein geometry on the Wasserstein space over graphs and the Euclidean geometry in physical space. The nonlinear geometric structure, along with the logarithmic potential induced by the graph-based state equation, adds further complexity to the analysis. To address these challenges, we introduce an energy-truncation technique within the doubling of variables framework, specifically designed to handle the interaction between the interiorly defined Wasserstein space on graphs and the unbounded Euclidean space. In particular, our findings demonstrate the well-posedness of HJB equations related to optimal control problems for both stochastic Schr\"odinger equation with polynomial nonlinearity and stochastic logarithmic Schr\"odinger equation on graphs. To the best of our knowledge, this work is the first to develop HJB equations for the optimal control of SWHSs on graphs.