Sparse optimal control in the Wasserstein space

We analyze sparse optimal control of a non-local continuity equation, where a distribution is guided by finitely many controllable agents. By working in the Wasserstein space, we can treat optimal transport terminal costs that are problematic in finite-dimensional settings. We derive first-order sensitivity results, an adjoint system, and Pontryagin-type necessary conditions, and we illustrate the theory numerically on a distribution-splitting problem.