Inverse stochastic control via generalized Schrödinger problems

We propose a variational formulation of an inverse problem in continuous-time stochastic control, aimed at identifying control costs consistent with a given distribution over trajectories. The formulation is based on minimizing the suboptimality gap of observed behavior. We establish a connection between the inverse problem and a generalized dynamic Schrödinger problem, showing that their optimal values coincide. This result links inverse stochastic control with stochastic optimal transport, offering a new conceptual viewpoint on inverse inference in controlled diffusions.