Hamiltonian Theory and Computation of Optimal Probability Density Control in High Dimensions

We develop a general theoretical framework for optimal probability density control and propose a numerical algorithm that is scalable to solve the control problem in high dimensions. Specifically, we establish the Pontryagin Maximum Principle (PMP) for optimal density control and construct the Hamilton-Jacobi-Bellman (HJB) equation of the value functional through rigorous derivations without any concept from Wasserstein theory. To solve the density control problem numerically, we propose to use reduced-order models, such as deep neural networks (DNNs), to parameterize the control vector-field and the adjoint function, which allows us to tackle problems defined on high-dimensional state spaces. We also prove several convergence properties of the proposed algorithm. Numerical results demonstrate promising performances of our algorithm on a variety of density control problems with obstacles and nonlinear interaction challenges in high dimensions.