Dynamical Unbalanced Optimal Transport: Duality, Reachability, and Computation

Abstract

We study dynamical unbalanced optimal transport for nonlinear control-affine systems in which the control input affects both the transportation of mass and its growth or decay. This phenomenon arises naturally in controlled transport-reaction systems such as cancer treatment, and distinguishes the problem from standard unbalanced optimal transport, where transport and mass variation are typically independent. To address this problem, we combine classical relaxed-control theory with a weighted superposition principle to obtain an infinite-dimensional linear programming reformulation over weighted trajectories. Under suitable convexity and compactness assumptions, we show that this relaxation is tight. Based on the relaxation, we derive a Kantorovich dual formulation in terms of Hamilton-Jacobi-Bellman inequalities, and establish strong duality over smooth subsolutions. We then utilize the dual formulation to characterize reachability of the system, namely, when a given terminal measure is reachable from a given initial measure. In addition, tractable sufficient conditions for reachability are provided. Finally, we develop an augmented-Lagrangian algorithm based on the dual formulation for numerical computation of the optimal transport plan. The framework is illustrated on a phenotype-structured cancer treatment problem in which therapy simultaneously drives phenotypic adaptation and population growth or death.