Constrained minmax density transportation for linear parabolic PDEs: a numerical optimal control perspective

This article introduces a numerical optimal control framework for minmax constrained density control for a class of noisy linear parabolic partial differential equations (PDEs), in particular the noisy heat equation. The goal is to transport an initial density to a target density while minimizing a specified cost with respect to control actions and maximizing it with respect to disturbances, all within a fixed time horizon while satisfying given convex path constraints. To address this, the spatial derivatives in the PDE are discretized using finite-difference approximations, transforming the problem into a system of ordinary differential equations in time. The admissible space of control and disturbance trajectories is then finitely parametrized, and the resulting optimal control problem is formulated as a convex semi-infinite program (SIP) under mild assumptions. By leveraging new numerical tools from convex SIP theory, we establish guarantees for exact solutions that account for constraint satisfaction under an infinite family of disturbance realizations, and we establish an optimization-based computationally efficient algorithm to recover these solutions. Comprehensive numerical examples to demonstrate and validate our findings are included.