Fokker--Planck Dynamics on Star Graphs with Variable Drift: Well-Posedness, Adjoint Analysis, and Numerical Approximation

Stochastic transport processes on networked domains (modelled on metric graphs) arise in a variety of applications where diffusion and drift mechanisms interact with an underlying graph structure. The Fokker--Planck equation provides a natural framework for describing the evolution of probability densities associated with such dynamics. While Fokker--Planck equations on metric graphs have been studied from an analytical viewpoint, their optimal control remains largely unexplored, particularly in settings where the control acts through the drift term. In this paper, we investigate an optimal control problem governed by the Fokker--Planck equation on a star graph, with a bilinear control appearing in the drift. We establish the well-posedness of the state equation and prove the existence of at least one optimal control. The associated adjoint system is derived, and first-order necessary optimality conditions are formulated. A wavelet-based numerical scheme is proposed to approximate the optimal solution, and its performance is illustrated through representative numerical experiments. These results contribute to the analytical and computational understanding of controlled stochastic dynamics on network-like domains.