Hamilton--Jacobi--Bellman equation for optimal control of stochastic Wasserstein--Hamiltonian system on graphs
Jianbo Cui, Tonghe Dang
TL;DR
The work tackles stochastic optimal control for Hamiltonian dynamics on graphs by formulating a Hamilton–Jacobi–Bellman equation on the Wasserstein space over graphs. It develops existence and uniqueness results for viscosity solutions using an energy-truncation technique within a doubling of variables framework, thereby reconciling graph-based Wasserstein geometry with Euclidean momentum variables. The approach is demonstrated on stochastic Schrödinger equations on graphs with polynomial and logarithmic nonlinearities, yielding well-posed HJB equations and broad applicability to quantum-graph control problems. The results advance the theory of HJB equations in mixed graph–Wasserstein settings and provide a systematic route to optimal control of SWHSs on graphs with potential impacts in quantum networks and wave propagation on graphs.
Abstract
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 Schrödinger equation with polynomial nonlinearity and stochastic logarithmic Schrödinger 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.