UC Santa Cruz

The need to predict, estimate and control density functions arise across engineeringapplications such as controlling biological and robotic swarms, vehicle guidance-control in uncertain dynamic environments, forecasting and demand response of loads in power systems, and active shaping of chemical concentrations in process control. Notwithstanding this recurring theme in practical applications, there does not exist a systems-control theory of densities. The perspective of this work is to close this gap by developing the theory and algorithms for prediction and control of densities subject to trajectory-level stochastic nonlinear dynamics.

We present theory and algorithms that leverage an emerging geometric interpretationof the equations of density propagation and steering. The governing partial differential equations for density propagation can be viewed as gradient flow of certain Lyapunov functionals with respect to the Wasserstein metric arising from the theory of optimal transport. This metric induces a Riemannian-like geometric structure on the infnite dimensional manifold of joint probability density functions (PDFs) supported on the state space. We leverage this geometric structure to design weighted scattered point cloud-based gradient descent algorithms via recursive evaluation of infinite dimensional proximal operators on the manifold of joint state PDFs. The resulting numerical algorithms avoid function approximation or spatial discretization, and enjoy fast computational speed due to certain conic contraction property that we establish. We provide several numerical examples to elucidate our algorithms.

We show that the Wasserstein proximal recursions can also be leveragedto solve the minimum energy finite horizon density steering, also known as the Schr{\"o}dinger Bridge Problem (SBP), which allows density regulation via feedback synthesis. This is a problem of minimum effort steering of a given joint state PDF to another over a finite time horizon, subject to a controlled stochastic differential evolution of the state vector. The same theory also arises in the study of mean-field dynamics of neural networks. We leverage the same theory to study second-order algorithms to prove their consistency and global convergence.