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.

💡 Research Summary

This paper presents a comprehensive study on the optimal control of Fokker-Planck dynamics on star-shaped metric graphs, where the control acts bilinearly through the drift term. The Fokker-Planck equation models the evolution of probability densities for stochastic transport processes on networked domains, with applications ranging from neuroscience to fluid dynamics in pipeline networks. While the analysis of such equations on graphs has been explored, their optimal control—particularly via the drift—remains a novel and challenging problem addressed in this work.

The core problem involves minimizing a cost functional that penalizes the deviation of the state (probability density) from a desired trajectory and a target terminal distribution, along with the control effort. The state is governed by a system of coupled Fokker-Planck equations defined on each edge of a star graph, subject to flux conservation and continuity conditions at the central vertex and Dirichlet conditions at the external vertices. The control function is constrained to a bounded set in the space of essentially bounded functions.

The authors first establish the functional-analytic framework, defining appropriate Sobolev-type spaces on the graph. The primary theoretical contribution is the proof of well-posedness for the state equation. They demonstrate the existence and uniqueness of a weak solution in the space W(0,T) for given initial data, source term, and control, relying on Galerkin approximation and energy estimates. This provides a solid foundation for the control analysis.

Subsequently, the existence of at least one optimal control is proven. By showing that the control-to-state operator is weakly-strongly continuous and that the admissible control set is weak-* compact in L∞, the authors establish that minimizing sequences converge to an optimal pair. The next major contribution is the derivation of first-order necessary optimality conditions. Using the Lagrangian method, an adjoint system of backward-in-time Fokker-Planck-type equations is derived. The resulting optimality system consists of the state equation, the adjoint equation, and a variational inequality that characterizes the optimal control in terms of the state and adjoint variables.

To computationally solve this coupled, nonlinear optimality system, the paper proposes a novel numerical scheme based on shifted Legendre wavelets. The unknown state, adjoint, and control functions are approximated using these wavelet bases. A collocation method is then employed to transform the system of PDEs into a system of nonlinear algebraic equations, which can be solved iteratively. The multiscale and localized properties of wavelets make them suitable for discretizing problems on graph edges.

Finally, the theoretical findings are validated through two representative numerical experiments. The first example considers a single edge (interval) to benchmark the method’s accuracy. The second, more complex example involves a star graph with three edges, where independent controls are applied to steer the probability densities towards different desired profiles. The results successfully demonstrate the capability of the proposed control framework and the efficiency of the wavelet-based numerical solver.

In conclusion, this work provides a rigorous analytical and computational framework for the optimal control of stochastic dynamics on network-like structures via the Fokker-Planck equation. It opens avenues for controlling diffusion-drift processes in complex geometries, with potential impacts across multiple scientific and engineering disciplines. Future work may extend these results to more general graph topologies, different boundary conditions, or stochastic parameters.