Uniform Sampling from the Reachable Set Using Optimal Transport

Estimating the reachable set of a dynamical system is a fundamental problem in control theory, particularly when control inputs are bounded. Direct simulation using randomly sampled admissible controls often leads to trajectories that cluster near attractors, resulting in poor coverage of the reachable set. To achieve a more uniform distribution of terminal states, we formulate the problem within an Optimal Transport (OT) framework. In this setting, the goal is to steer the system so that the final state distribution, determined by the chosen controls and initial conditions, matches a desired target distribution. Enforcing this condition exactly is not possible since the reachable set is not known. So we introduce an $L_2$-norm based regularization of the terminal distribution that relaxes the constraint while promoting uniform coverage. The resulting formulation can be approximated by a finite-dimensional, particle-based optimal control problem with kernel-coupled terminal cost. We show that this approach converges to the original formulation and demonstrate through numerical examples that it provides significantly more uniform reachable-set sampling than random control strategies.